# Dynamics of Interaction of Two Soliton Clouds111JETP, 135, 768 (2022)

A. M. Kamchatnov  Institute of Spectroscopy, Russian Academy of Sciences,
Troitsk, Moscow, 108840, Russia  D. V. Shaykin  Moscow Institute of Physics
and Technology, Institutsky lane 9, Dolgoprudny, Moscow region, 141700, Russia

###### Abstract

On the basis of relationship between the kinetic equation for two soliton
clouds in the theory of the Korteweg-de Vries equation and equations of the
Chaplygin gas dynamics it is shown that the existence of waves propagating
without a change in their form is a fundamental property of the nonlinear
dynamics of soliton gases. The solutions of several typical problems in the
soliton gas dynamics are considered and characteristic features of such
dynamics, which make it possible to estimate the effects of interaction of
soliton gases, are indicated.

###### pacs:

05.45.Yv, 47.35.Fg

##  I Introduction

It is well known that the term “solitons” has been introduced zk-65
analogously to the names of elementary particles (electron, proton, etc.) in
view of their elastic interaction with one another for an important class of
nonlinear wave equations. Namely, two solitary waves spaced by a large
distance prior to their “collision”, return to their initial form after their
passage through the interaction stage without forming any additional waves.
Nevertheless, the event of their interaction is not quite traceless: as a
result of this interaction, the trajectories of solitons acquire additional
shifts as compared to their initial trajectories lax-68 . For example, in the
case of the Korteweg-de Vries (KdV) equation, which is a universal wave
equation for describing waves with account for weak nonlinearity and
dispersion effects and which can be written in terms of standard dimensionless
variables in form

$u_{t}+6uu_{x}+u_{xxx}=0,$ (1)

the soliton solution is given by the expression

$u(x,t)=\frac{\kappa^{2}}{2}\cdot\frac{1}{\cosh^{2}[\kappa(x-\kappa^{2}t-x_{0})/2]},$
(2)

i.e., velocity $s=\kappa^{2}$ of a soliton is proportional to its amplitude
$a=\kappa^{2}/2$. If the wave excitation at the initial instant can be
represented to a high degree of accuracy in the form of two soliton pulses(2)
separated by a large distance (the faster soliton with parameter
$\kappa=\kappa_{2}$ has initial coordinate $x_{02}$ on the left of coordinate
$x_{01}$ of the slower soliton with parameter $\kappa_{1}$
($\kappa_{1}<\kappa_{2}$), their initial trajectories
$x=\kappa_{1}^{2}t+x_{01}$, $x=\kappa_{2}^{2}t+x_{02}$ acquire after the
collision the shifts $x=\kappa_{1}^{2}t+x_{01}+\Delta x_{1}$,
$x=\kappa_{2}^{2}t+x_{02}+\Delta x_{2}$, where

$\Delta
x_{1}=-\frac{2}{\kappa_{1}}\ln\frac{\kappa_{2}+\kappa_{1}}{\kappa_{2}-\kappa_{1}},\quad\Delta
x_{2}=\frac{2}{\kappa_{2}}\ln\frac{\kappa_{2}+\kappa_{1}}{\kappa_{2}-\kappa_{1}}.$
(3)

Therefore, the fast soliton with a large amplitude is shifted in the forward
direction, while the slow and lower soliton is shifted in the backward
direction, the shift of the lower soliton being larger (in absolute value)
than the shift of the higher soliton. Remarkably, for an important class of
integrable equations, solutions can be obtained with any number of solitons.
In this case, as a result of simultaneous collision of three or more solitons,
the total shift of each soliton is equal to the sum of shifts of form (3) for
pairwise collisions (see, for example, zmnp-80 ; newell ), i.e., multiple
simultaneous collisions of solitons in some spatial region do not differ in
this sense from sequential spatially separated pair collisions of solitons
with one another.

Let us now suppose that in our wave system, a very large number of solitons,
each is characterized by its parameter $\kappa$, are excited in the system. In
this case, we can speak of a soliton gas and use the concepts of gas kinetics
for its description. If we denote by $f(\kappa,x,t)dxd\kappa$ the number of
solitons with coordinates from interval $(x,x+dx)$, which have parameter
$\kappa$ from interval $(\kappa,\kappa+d\kappa)$ at instant $t$, the evolution
of such a gas is characterized by the time dependence of distribution function
$f(\kappa,x,t)$ in the coordinate and parameter $\kappa$. The problem of
obtaining the kinetic equation for such evolution was formulated by Zakharov
in zakh-71  and was solved by him for a rarefied gas of solitons when their
mutual collisions change velocities $s=\kappa^{2}$ very little. Later, El, who
investigated the special limit of Whitham modulation equations for the
infinite-phase solution of the KdV equation, obtained in el-03  a
generalization of the Zakharov kinetic equation to a dense soliton gas. A
simpler derivation of this equation based on the self-consistent determination
of the mean velocity of a soliton moving through the gas was given in
elkamch-05  where the simplest solution to these equations, which described
the collision of two soliton gas clouds with close parameters $\kappa$ was
also obtained. The formal solutions for any number of such clouds were
analyzed in ekpz-11 ; fp-22 . At present, the theory of kinetic equations for
solitons has become a part of actively developing “generalized hydrodynamics”,
which is applicable to any integrable models of systems of many interacting
particles (see, for example, reviews doyon-20 ; el-21  and references
therein). However, despite such a remarkable formal progress, the physical
explanation of the behavior of soliton gases appears as insufficient since the
number of solved problems that give an idea of characteristic features of the
dynamics of interacting soliton gases is very scarce. At the same time,
experimental investigations of soliton gas dynamics in various physical
systems including waves in water and cold atoms have been launched (see, for
example, redor-19 ; suret-20 ; bd-22 ); this requires the development of well-
elaborated theory of the soliton gas dynamics. In this article, we give
several examples of such typical dynamics, which clarify some essential
features of the soliton gas dynamics, distinguishing it from the dynamics of
conventional gases.

##  II Kinetic equation for a soliton gas

Let us first consider for completeness the brief derivation of the KdV kinetic
equation for a soliton gas following the method proposed in elkamch-05 . It
should be noted above all that in view of the integrability of the KdV
equation discovered in ggkm-67 , the evolution of wave $u(x,t)$ occurs so that
the spectrum of the eigenvalue problem for the time-independent Schrödinger
equation $\psi_{xx}=-(u(x,t)+\lambda)\psi$, which is associated with the KdV
equation, is independent of time $t$ and each soliton corresponds to a certain
value $\lambda<0$ of discrete spectrum. More convenient parameter $\kappa$,
which has been used in the soliton solution in form (2), is connected with
$\lambda$ by relation $\kappa=\sqrt{-\lambda}$. Consequently, during the
evolution of the wave, both the spectrum and the set of $\kappa$-values for
the motion of solitons accompanied with their collisions remain unchanged in
accordance with the KdV equation.

If solitons did not experience shifts (3) during their collisions, their
velocities could be expressed by formula $s=\kappa^{2}$ for all solitons with
parameter $\kappa$. However, collisions modify this velocity. In each
collision of a soliton with slower solitons characterized by parameter
$\eta<\kappa$, the “probe” $\kappa$-soliton advances in the forward direction
through additional distance
$\frac{2}{\kappa}\ln\frac{\kappa+\eta}{\kappa-\eta}$, and the number of such
collisions per unit time is equal to relative velocity $s(\kappa)-s(\eta)$,
multiplied by the density of $\eta$-solitons. Consequently, owing to such
collisions, the $\kappa$-soliton acquires additional velocity

$\int_{0}^{\kappa}\frac{2}{\kappa}\ln\frac{\kappa+\eta}{\kappa-\eta}\cdot[s(\kappa)-s(\eta)]f(\eta)d\eta.$

Analogously, each soliton with parameter $\eta>\kappa$, which overtakes the
probe $\kappa$-soliton, shifts it per unit time backwards by distance

$\begin{split}&-\int_{\kappa}^{\infty}\frac{2}{\kappa}\ln\frac{\eta+\kappa}{\eta-\kappa}\cdot[s(\eta)-s(\kappa))]f(\eta)d\eta=\\\
&=\int_{\kappa}^{\infty}\frac{2}{\kappa}\ln\left|\frac{\kappa+\eta}{\kappa-\eta}\right|\cdot[s(\kappa)-s(\eta)]f(\eta)d\eta,\end{split}$

which can be obtained by multiplying shift
$-\frac{2}{\kappa}\ln\frac{\eta+\kappa}{\eta-\kappa}$ during a single
collision by the number of such collisions $[s(\eta)-s(\kappa))]f(\eta)$ and
integrating with respect to values of $\eta>\kappa$. Adding these corrections
to the unmodified value of velocity $\kappa^{2}$, we obtain the following
self-consistent equation:

$s(\kappa)=\kappa^{2}+\frac{2}{\kappa}\int_{0}^{\infty}\ln\left|\frac{\kappa+\eta}{\kappa-\eta}\right|[s(\kappa)-s(\eta)]f(\eta)d\eta.$
(4)

Therefore, the distribution function for solitons with parameter $\kappa$ is
transferred along the $x$ axis with velocity $s(\kappa)$, defined by integral
equation (4), and the condition of the spectrum conservation during the
evolution of the wave in accordance with the KdV equation can be written in
the form of conservation law

$\frac{\partial f(\kappa,x,t)}{\partial
t}+\frac{\partial[s(\kappa)f(\kappa,x,t)]}{\partial x}=0.$ (5)

This equation, supplemented with integral equation (4), is known as the KdV
kinetic equation for a soliton gas; this equation was obtained by El in el-03
using a different method. It should be noted that the inclusion in Eq. (4) of
pair collisions alone for a dense soliton gas is justified by the
aforementioned addition of shifts for multiple soliton collisions.

If the soliton gas is rarefied, i.e., $\int f(\kappa)d\kappa\ll\kappa_{0}$,
where $\kappa_{0}$ is the characteristic value of parameter $\kappa$ in
distribution $f(\kappa)$, the correction term in formula (4) is small, and we
can substitute unmodified value of $s(\kappa)\approx\kappa^{2}$ into it:

$s(\kappa)=\kappa^{2}+\frac{2}{\kappa}\int_{0}^{\infty}\ln\left|\frac{\kappa+\eta}{\kappa-\eta}\right|(\kappa^{2}-\eta^{2})f(\eta)d\eta.$
(6)

This expression defines the function $s(\kappa)$ by a closed expression in
contrast to integral equation (4) for a dense soliton gas. System (5), (6) was
obtained by Zakharov in zakh-71  together with the approach to the kinetics of
the soliton gas formulated here. Clearly, such kinetic equations can also be
obtained for other completely integrable equations with the isospectral
evolution of nonlinear waves.

##  III Two-stream flows of soliton gases

To get an idea of the dynamics of soliton gases based on kinetic equation (4),
(5), we assume that the distribution function has two very narrow peaks near
$\kappa_{1}$ and $\kappa_{2}$, i.e., this function can be written in form

$f(\kappa,x,t)=f_{1}(x,t)\delta(\kappa-\kappa_{1})+f_{2}(x,t)\delta(\kappa-\kappa_{2}).$
(7)

This means that we consider the dynamics of interacting gases over times, such
that we can disregard the collisions of solitons of the same species with very
close velocities and take into account only collisions between solitons of
different species. Then the substitution of distribution function (7) into
Eqs. (5) and (6) gives conservation laws

$\frac{\partial f_{1}}{\partial t}+\frac{\partial(s_{1}f_{1})}{\partial
x}=0,\quad\frac{\partial f_{2}}{\partial
t}+\frac{\partial(s_{2}f_{2})}{\partial x}=0,$ (8)

where velocities $s_{1},s_{2}$ satisfy equations

$s_{1}=\kappa_{1}^{2}+\alpha_{1}f_{2}(s_{1}-s_{2}),\quad
s_{2}=\kappa_{2}^{2}+\alpha_{2}f_{1}(s_{2}-s_{1}),$ (9)

and we have introduced the following notation for convenience:

$\alpha_{1}=\frac{2}{\kappa_{1}}\ln\left|\frac{\kappa_{1}+\kappa_{2}}{\kappa_{1}-\kappa_{2}}\right|,\quad\alpha_{2}=\frac{2}{\kappa_{2}}\ln\left|\frac{\kappa_{1}+\kappa_{2}}{\kappa_{1}-\kappa_{2}}\right|.$
(10)

Equations (9) make it possible to express the renormalized velocities in terms
of the densities of soliton gases:

$\begin{split}s_{1}=\frac{\kappa_{1}^{2}(1-\alpha_{2}f_{1})-\kappa_{2}^{2}\alpha_{1}f_{2}}{1-\alpha_{1}f_{2}-\alpha_{2}f_{1}},\\\
s_{2}=\frac{\kappa_{2}^{2}(1-\alpha_{1}f_{2})-\kappa_{1}^{2}\alpha_{2}f_{1}}{1-\alpha_{1}f_{2}-\alpha_{2}f_{1}}.\end{split}$
(11)

If, however, we express densities $f_{1,2}$ from Eqs. (9) in terms of
velocities $s_{1,2}$,

$f_{1}=\frac{s_{2}-\kappa_{2}^{2}}{\alpha_{2}(s_{2}-s_{1})},\quad
f_{2}=\frac{\kappa_{1}^{2}-s_{1}}{\alpha_{1}(s_{2}-s_{1})},$ (12)

and substitute these expressions into Eq. (8), this system can be reduced to a
remarkably simple diagonal form:

$\frac{\partial s_{1}}{\partial t}+s_{2}\frac{\partial s_{1}}{\partial
x}=0,\quad\frac{\partial s_{2}}{\partial t}+s_{1}\frac{\partial
s_{2}}{\partial x}=0,$ (13)

where renormalized velocities $s_{1},s_{2}$ are Riemann invariants of this
system of equations of the hydrodynamic type.

Formally, Eqs. (13) resemble the gas dynamic equations in the Riemann form,
but physical properties of the soliton gas differ significantly from the
properties of a conventional gas. It should be noted above all that although
the soliton velocities are renormalized in the overlap region of soliton
clouds, such a change in velocity cannot be interpreted as the acceleration of
solitons under the action of pressure: after the clouds leave the overlap
region, their velocities restore their initial values. Further, the gases flow
freely through each other without experiencing any dissipation processes. In
this case, dynamic equations (8) have the form of conservation laws and can
have discontinuous solutions like in the theory of viscous shock waves.
Finally, we can assume in our case that soliton gases have zero temperature so
that a transition through a discontinuity is not associated with an increase
of entropy; i.e., the Jouguet-Zemplén theorem (see, for example, LL6 ) is
inapplicable, and the discontinuity can have any sign. This also means that
the existence of such formally discontinuous flows is not associated with
soliton pulse breaking followed by formation of a narrow transition layer
connecting both flows with different parameters like in the theory of viscous
shock waves.

This situation is clarified with the help of the interrelation between Eqs.
(13) and the Chaplygin gas theory. Chaplygin chaplygin  noted that the
equation of state of a gas,

$p=A-\frac{B}{\rho}$ (14)

($p$ is the gas pressure, $\rho$ is its density, and $A,B$ are constant
parameters) can serve as a convenient approximation of small segments of the
Poisson adiabat, where the formulas of the theory are simplified
significantly. (Chaplygin noticed also the connection between the gas dynamic
equations for this case and the theory of minimal surfaces.) However, we
approach the Chaplygin gas theory from a different point of view.

Already at the initial stage of development of the theory of shock waves,
Stokes and Kelvin discussed whether gas dynamic equations

$\rho_{t}+(\rho u)_{x}=0,\quad u_{t}+uu_{x}+\frac{c^{2}}{\rho}\rho_{x}=0,$
(15)

($c^{2}=dp/d\rho$, $c$ being the velocity of sound) permit solutions in the
form of a traveling stationary wave. With such a formulation of the problem,
both density $\rho$ and the flow velocity $u$ in the stationary solution are
obviously functions of only the coordinate $\xi=x-Vt$, these quantities are
connected by a one-to-one dependence. Consequently, such a solution must be a
simple wave expressed by the Poisson relation (see LL6 )

$\rho=\rho_{0}[x-(c+u)t],$ (16)

$\rho_{0}(x)$ being the density profile in such a wave propagating with
velocity $c+u$. In this case, the relation between $\rho$ and $u$ is expressed
by the condition of constancy of the Riemann invariant: $u-\int
cd\rho/\rho=r_{-}=\mathrm{const}$ (see LL6 ). A wave with a stationary profile
can exist only if condition $c+u=\mathrm{const}$ is satisfied. Substituting
into this expression relations

$c=\sqrt{\frac{dp}{d\rho}},\quad
u=\int_{0}^{\rho}\sqrt{\frac{dp}{d\rho}}\frac{d\rho}{\rho}+\mathrm{const},$

and differentiating with respect to $\rho$, we obtain the following equation
for gas equation of state $p=p(\rho)$ which permits a stationary wave:

$\frac{d^{2}p}{d\rho^{2}}+\frac{2}{\rho}\frac{dp}{d\rho}=0.$

This equation can be solved easily and gives exactly above dependence (14).
Setting for simplicity $B=1$, we obtain $c^{2}=dp/d\rho=1/\rho^{2}$,
$c=1/\rho$ and the Riemann invariants (see LL6 ) turn out to be

$\begin{split}&s_{1}=u+\int\frac{cd\rho}{\rho}=u-\frac{1}{\rho},\\\
&s_{2}=u-\int\frac{cd\rho}{\rho}=u+\frac{1}{\rho},\end{split}$ (17)

where $u$ and $\rho$ obey the Euler equations for the Chaplygin gas:

$\rho_{t}+(\rho u)_{x}=0,\qquad u_{t}+uu_{x}+\frac{\rho_{x}}{\rho^{3}}=0,$
(18)

and can be expressed in terms of the Riemann invariants:

$\rho=\frac{2}{s_{2}-s_{1}},\qquad u=\frac{1}{2}(s_{2}+s_{1}).$ (19)

If we write Eqs. (18) in terms of Riemann invariants (17), simple calculations
lead to the dynamic equations for a Chaplygin gas in the diagonal Riemann form
coinciding with Eqs. (13) which describe the dynamics of two interacting
soliton gases.

If we now seek the solution to Eq. (18) in the form of a wave traveling with
constant velocity $V$ ($\rho=\rho(\xi),u=u(\xi)$, $\xi=x-Vt$), we can easily
see that these equations are satisfied for any function $\rho(\xi)$, if the
Chaplygin gas flow velocity can be expressed in terms of $\rho(\xi)$ as

$u(\xi)=V+\frac{1}{\rho(\xi)}.$ (20)

In this case, the Riemann invariants are given by

$s_{1}=V,\qquad s_{2}=V+\frac{2}{\rho(\xi)}.$ (21)

These expressions obviously give a solution to Eq. (13) in the form of a
simple wave: if we set, for example, $s_{1}=V=\mathrm{const}$, the second
equation is transformed into $s_{2,t}+Vs_{2,x}=0$ with general solution
$s_{2}=F(x-Vt)$, which coincides with (21), if we write function $F(\xi)$ in
form $F(\xi)=V+2/\rho(\xi)$. In particular, the function $F(\xi)$ can include
a discontinuity propagating without change of form in the overlap region of
the two gases.

The relation between the equations for a Chaplygin gas and the kinetic
equation describing the dynamics of two interacting soliton clouds makes it
possible to construct instructive examples of solutions to the kinetic
equation.

##  IV Collision of two soliton gases

Figure 1:  Coordinate dependence of the densities of soliton gases interacting
after a collision of soliton clouds with constant initial densities and
velocities.

Equations (13) have an obvious degenerate solution, in which $s_{1}$ and
$s_{2}$ are constants depending neither on $x$, nor $t$. Despite the
triviality of this solution, it has a clear physical meaning: it can serve as
a “plateau” connecting two aforementioned solutions in the form of simple
waves. In particular, during the collision of mutually penetrating gases with
constant densities $f_{1},f_{2}$ and, hence, with Riemann invariants
$s_{1},s_{2}$, a region of a two-stream flow is formed, in which the
velocities of solitons of one species are renormalized because of their
interaction with solitons of the other species. Thus, the problem of collision
of two gases is reduced to the determination of densities and velocities of
solitons in the region of their “mixing”, as well as the velocities of the
edges of this region. Although this problem has already been analyzed in
elkamch-05 ; cde-16 , we will briefly consider this problem because the
relevant results will be used in further analysis. We assume that at the
initial instant, there is a discontinuity in the density distributions for
solitons of two species:

$\left\\{\begin{array}[]{lll}f_{2}(x,0)=f_{20},&f_{1}(x,0)=0,&x<0,\\\
f_{2}(x,0)=0,&f_{1}(x,0)=f_{10},&x>0,\end{array}\right.$ (22)

here, we assume that $\kappa_{2}>\kappa_{1}$, so that the faster soliton cloud
with density $f_{20}$ overtakes at instant $t=0$ the slower cloud with density
$f_{10}$ at point $x=0$ and their mutual penetration begins. As usual, in the
case of initial discontinuity the solution must depend only on self-similar
variable $\zeta=x/t$. Therefore, the initial discontinuity leads to the
formation of a plateau between two simple waves, each of which is a
discontinuity with a constant value of one of the Riemann invariants (the
existence of such solutions is determined by the aforementioned properties of
the Chaplygin gas). In the whole, the solution is a sequence of three flows
with constant densities, which are separated by discontinuities:

$f(x,t)=\left\\{\begin{array}[]{ll}f_{20},&x<c^{-}t,\\\
f_{1c}+f_{2c},&c^{-}t<x<c^{+}t,\\\ f_{10},&x>c^{+}t.\end{array}\right.$ (23)

It can clearly be seen from Fig. 1 that the leading edge of the plateau moves
with renormalized velocity $s_{2c}$ of the fast gas, while the rear edge of
the plateau moves with renormalized velocity $s_{1c}$ of the slow gas because
the densities at these edges vanish:

$c^{-}=s_{1c},\qquad c^{+}=s_{2c}.$ (24)

In the two-flow region $c^{-}t<x<c^{+}t$ of the plateau, the soliton
velocities are renormalized by their interaction; in accordance with relations
(11) we have

$\begin{split}&s_{1c}=\frac{\kappa_{1}^{2}(1-\alpha_{2}f_{1c})-\kappa_{2}^{2}\alpha_{1}f_{2c}}{1-\alpha_{1}f_{2c}-\alpha_{2}f_{1c}},\\\
&s_{2c}=\frac{\kappa_{2}^{2}(1-\alpha_{1}f_{2c})-\kappa_{1}^{2}\alpha_{2}f_{1c}}{1-\alpha_{1}f_{2c}-\alpha_{2}f_{1c}}.\end{split}$
(25)

The slow gas flows through the right discontinuity into the plateau region;
equating the expressions for its flux on different sides of the discontinuity
in the reference frame in which it is at rest, we obtain
$f_{1c}(s_{1c}-c^{+})=f_{10}(\kappa_{1}^{2}-c^{+})$. Fast gas flows through
the left discontinuity into the plateau region, and analogous calculation
gives $f_{20}(\kappa_{2}^{2}-c^{-})=f_{2c}(s_{2c}-c^{-})$. With account for
relations (24) and (25) the resulting equalities lead to equations

$\begin{split}&f_{1c}=f_{10}\frac{s_{2c}-\kappa_{1}^{2}}{s_{2c}-s_{1c}}=f_{10}(1-\alpha_{1}f_{2c}),\\\
&f_{2c}=f_{20}\frac{\kappa_{2}^{2}-s_{1c}}{s_{2c}-s_{1c}}=f_{20}(1-\alpha_{2}f_{1c}),\end{split}$

which gives expressions

$\begin{split}&f_{1c}=\frac{f_{10}(1-\alpha_{1}f_{20})}{1-\alpha_{1}\alpha_{2}f_{10}f_{20}},\\\
&f_{2c}=\frac{f_{20}(1-\alpha_{2}f_{10})}{1-\alpha_{1}\alpha_{2}f_{10}f_{20}},\end{split}$
(26)

for the densities of the soliton gases in the two-flow region of the plateau.
Clearly, these expressions have sense only when the following inequality
holds:

$\alpha_{1}f_{1c}+\alpha_{2}f_{2c}<1,$ (27)

when renormalized velocities (25) are positive, i.e., the densities of soliton
gases cannot be too high. Actually, more stringent limitations on the soliton
density are imposed by the condition that variation
$\overline{u^{2}}-\overline{u}^{2}$ of the fluctuating wave field in a soliton
gas must be positive (see el-16 ). The relations derived above are in good
agreement with the results of numerical solution of the KdV equation with the
initial data in the form of a large number of solitons of two different
species, which are located on different sides of the coordinate origin (see
cde-16 ).

##  V Flow of two soliton gases in the form of a simple wave

Figure 2:  (a) Distributions of densities of soliton gases and (b) of their
renormalized velocities as functions of the coordinate for the wave profile
defined by formula (34). Soliton gas parameters: $\kappa_{1}=1$,
$\kappa_{2}=1.8$, $f_{10}=0.05$, $f_{20}=0.1$. The wave velocity coinciding
with the flow velocity of slow solitons is $V=s_{1}=0.174$, while the flow
velocity for the fast soliton gas at infinity is $s_{20}=3.469$, which is
higher than nonrenormalized velocity $\kappa_{2}^{2}=3.24$.

Expressions (21) give a more general form of the solution for the flow of two
soliton gases in the form of a simple wave. Let us suppose that the densities
of the soliton gases at infinity tend to constant values:

$f_{1}\to f_{10},\qquad f_{2}\to f_{20}\qquad\text{as}\qquad x\to\pm\infty.$
(28)

Then the constant value of Riemann invariant $s_{1}$ is equal to the wave
velocity of the soliton gas (see (25))

$V=s_{1}=\frac{\kappa_{1}^{2}(1-\alpha_{2}f_{10})-\kappa_{2}^{2}\alpha_{1}f_{20}}{1-\alpha_{1}f_{20}-\alpha_{2}f_{10}},$
(29)

which coincides with the renormalized velocity of the slow gas, while formulas
(12) give expressions for the densities of soliton gases:

$\begin{split}&f_{1}(\xi)=\frac{1}{\alpha_{2}}\left[1+\frac{1}{2}(V-\kappa_{2}^{2})\rho(\xi)\right],\\\
&f_{2}(\xi)=\frac{1}{2\alpha_{1}}(\kappa_{1}^{2}-V)\rho(\xi).\end{split}$ (30)

It can easily be seen that these densities are connected by relation

$1-\alpha_{1}f_{2}-\alpha_{2}f_{1}=\frac{1}{2}(\kappa_{2}^{2}-\kappa_{1}^{2})\rho,$
(31)

which implies, in particular, that

$\rho_{0}=\frac{2(1-\alpha_{1}f_{20}-\alpha_{2}f_{10})}{\kappa_{2}^{2}-\kappa_{1}^{2}}.$
(32)

Density distributions (30) contain arbitrary function $\rho(\xi)$ and the
distribution of the renormalized velocity of the fast soliton gas can also be
expressed in terms of this function in accordance with the second expression
in (21). At infinity, $s_{2}$ tends to the value (see expressions (25))

$s_{20}=\frac{\kappa_{2}^{2}(1-\alpha_{1}f_{20})-\kappa_{1}^{2}\alpha_{2}f_{10}}{1-\alpha_{1}f_{20}-\alpha_{2}f_{10}}.$
(33)

The resulting solution has physical sense for such values of parameters
$\kappa_{1},\kappa_{2},f_{10},f_{20}$, that the distribution of densities
$f_{1},f_{2}$ of the soliton gases, as well as their renormalized velocities
$s_{1},s_{2}$ are positive everywhere.

Figure 2 illustrates the flow for two soliton gases for function $\rho(\xi)$
given by

$\rho(\xi)=\rho_{0}+\frac{a}{\cosh\xi}.$ (34)

In this case, the profile of density $f_{1}$ of the slow gas moves together
with this component with velocity $V=s_{1}$. The fast gas flows through the
slow gas with local velocity $s_{2}$, which decreases in the region of the
“well” in distribution $f_{1}$, this leads to an increase in density $f_{2}$
in this region.

##  VI General solution for the flow of two soliton gases

The hodograph method, which is well known in gas dynamics (see, for example,
LL6 ) makes it possible to easily obtain the general solution to Eqs. (13), in
the case when both velocities $s_{1}$ and $s_{2}$ vary in space and time.
Performing standard hodograph transformation $t=t(s_{1},s_{2})$,
$x=x(s_{1},s_{2})$, we reduce these equations to the form

$\frac{\partial x}{\partial s_{1}}-s_{1}\frac{\partial t}{\partial
s_{1}}=0,\qquad\frac{\partial x}{\partial s_{2}}-s_{2}\frac{\partial
t}{\partial s_{2}}=0.$ (35)

Eliminating $x$, we obtain

$\frac{\partial^{2}t}{\partial s_{1}\partial
s_{2}}=\frac{1}{s_{1}}\frac{\partial^{2}x}{\partial s_{1}\partial
s_{2}}=\frac{1}{s_{1}}\frac{\partial}{\partial
s_{1}}\left({s_{2}}\frac{\partial t}{\partial
s_{2}}\right)=\frac{s_{2}}{s_{1}}\frac{\partial^{2}t}{\partial s_{1}\partial
s_{2}}.$

Since $s_{1}\neq s_{2}$, we arrive at equation

$\frac{\partial^{2}t}{\partial s_{1}\partial s_{2}}=0,$ (36)

the general solution to which can be written in terms of two arbitrary
functions (it is convenient to denote them as $F^{\prime\prime}(s_{1})$ and
$G^{\prime\prime}(s_{2})$), in form

$t(s_{1},s_{2})=F^{\prime\prime}(s_{1})+G^{\prime\prime}(s_{2}).$ (37)

Equations (35) readily give

$x(s_{1},s_{2})=s_{1}F^{\prime\prime}(s_{1})+s_{2}G^{\prime\prime}(s_{2})-F^{\prime}(s_{1})-G^{\prime}(s_{2}).$
(38)

This solution can be transform to

$\begin{split}&x-s_{1}t=(s_{2}-s_{1})G^{\prime\prime}(s_{2})-F^{\prime}(s_{1})-G^{\prime}(s_{2}),\\\
&x-s_{2}t=-(s_{2}-s_{1})F^{\prime\prime}(s_{1})-F^{\prime}(s_{1})-G^{\prime}(s_{2}).\end{split}$
(39)

In this form, the solution has been obtained in pavlov-87b  by a different
method.

For completeness of analysis, we consider other representations of the general
solution, which may turn out to be more convenient for solving specific
problems. If we turn to the hodograph method in the Tsarev form tsarev-90 ,
the general solution can be written in the form

$x-s_{2}t=\frac{\partial W}{\partial s_{1}},\qquad x-s_{1}t=\frac{\partial
W}{\partial s_{2}},$ (40)

where function $W=W(s_{1},s_{2})$ must satisfy the Euler-Poisson equation:

$\frac{\partial^{2}W}{\partial s_{1}\partial
s_{2}}+\frac{1}{s_{1}-s_{2}}\left(\frac{\partial W}{\partial
s_{1}}-\frac{\partial W}{\partial s_{2}}\right)=0.$ (41)

It can easily be verified that the solutions to this equation can be expressed
in terms of arbitrary functions $F(s_{1})$ and $G(s_{2})$ as follows:

$W(s_{1},s_{2})=(s_{1}-s_{2})[F^{\prime}(s_{1})-G^{\prime}(s_{2})]-2[F(s_{1})+G(s_{2})].$
(42)

Substituting this expression into (40), we obtain the solution in form (39).
Functions $F(s_{1})$ and $G(s_{2})$ should be determined from the initial
conditions to the problem.

As was shown by Riemann (see, for example, sommer-50 ), instead of using this
general solution expressed in terms of arbitrary functions, it is often
convenient to employ the method in which the solution to the problem is
expressed directly in terns of initial conditions with the help of the so-
called Riemann function. In our case, the Riemann function can be obtained
from the general expression for the polytropic gas dynamics with adiabatic
exponent $\gamma=-1$. The hypergeometric function appearing in this expression
reduces to $F(-1,2;1;z)=1-2z$ and the Riemann function for Eqs. (13) takes an
especially simple form:

$R(r_{1},r_{2};s_{1},s_{2})=\frac{(r_{1}+r_{2})(s_{1}+s_{2})-2(r_{1}r_{2}-s_{1}s_{2})}{(r_{1}-r_{2})^{2}},$
(43)

where $(r_{1},r_{2})$ are the current coordinates on the hodograph plane and
$(s_{1},s_{2})$ are the coordinates of point $P$, at which the value of
function $W=W(P)=W(s_{1},s_{2})$ is sought. We will not write here rather
cumbersome general expressions that do not add much for understanding the
dynamics of interacting soliton gases; instead of this, we consider the
solution to one of typical gas dynamic problems, viz., the problem of
expansion of a gas layer. The solution can easily be obtained in elementary
form, and its difference from the solutions to the same problem for other
physical systems demonstrates well the substantial difference between the
conventional gas dynamics and the dynamics of soliton gases.

##  VII Expansion of a layer from a mixture of two soliton gases

One of typical problems in which a domain of the general solution appears is
the problem of expansion of a gas layer. This problem was considered in nk-76
for a classical monatomic gas; in ik-19b  for the Bose-Einstein condensate; in
ik-20  for a two-temperature plasma, and in landau-53 ; khal-54 ; kamch-19
for an ultrarelativistic gas. In all cases, rarefaction waves propagated from
the edges to the bulk of the layer, and after their collision, the general
solution domain bordering the rarefaction waves at the edges was formed.
However, in the case of a soliton gas, there is no solution in the form of
rarefaction waves, and instead of this, discontinuous solutions separated by a
plateau with constant values of the Riemann invariants appears. The solution
constructed in this way makes it possible to draw certain general conclusions
about the evolution of overlapping clouds of soliton gases.

If we assume that a single soliton gas with parameter $\kappa_{1}$ at the
initial moment of time has a density distribution in the form of a plateau
with value $f_{10}$ for $-l/2\leq x\leq l/2$, its motion at subsequent
instants is obvious: this distribution is transferred along the $x$ axis with
velocity $\kappa_{1}^{2}$ without change of its shape. If, however, there is a
mixture of soliton gases with parameters $\kappa_{1}$ and $\kappa_{2}$,
$(\kappa_{1}<\kappa_{2})$ and densities $f_{10}$ and $f_{20}$ at the initial
instant in domain $-l/2\leq x\leq l/2$, then the evolution of these gases is
more complex: in the course of separation of these gases into two clouds
moving with velocities $\kappa_{1}^{2}$ and $\kappa_{2}^{2}$, a two-flow
plateau appears. Our aim is the description of the process of spatial
separation of soliton clouds and the determination of their parameters after
the separation.

Figure 3:  Coordinate dependencies of the densities of two soliton gas clouds
during the evolution of the initial layer of two gases before the instant of
their separation. Bold dashed line shows the density of the slow gas, while
solid line corresponds to the density of the fast gas.

On the plateau formed during the separation, the gases obviously move with
renormalized velocities $s_{1},s_{2}$, preserving their initial densities
$f_{10},f_{20}$. In front of this layer, there appears a fast gas layer with
density $f_{2}$, while behind this layer, a slow gas layer with density
$f_{1}$ is formed. The left edge of the overlap region moves with velocity
$s_{2}$, while its right edge moves with velocity $s_{1}$. Consequently, the
overlap region exists during time

$T=\frac{l}{s_{2}-s_{1}}$ (44)

and the gases are separated at $t>T$. It can easily be seen from Fig. 3, that
the lengths of the clouds of slow and fast gases at the instant of separation
are given by

$l_{1}=l\cdot\frac{s_{2}-\kappa_{1}^{2}}{s_{2}-s_{1}},\qquad
l_{2}=l\cdot\frac{\kappa_{2}^{2}-s_{1}}{s_{2}-s_{1}},$ (45)

respectively, both of them being smaller than the initial length $l$, because
$s_{1}<\kappa_{1}^{2}$ and $s_{2}>\kappa_{2}^{2}$. From the conservation of
the number of solitons in each component, we find amplitudes

$f_{1}=f_{10}\cdot\frac{s_{2}-s_{1}}{s_{2}-\kappa_{1}^{2}},\qquad
f_{2}=f_{20}\cdot\frac{s_{2}-s_{1}}{\kappa_{2}^{2}-s_{2}}.$ (46)

At the instant of separation of the clouds, the centers of mass of the slow
soliton cloud and the fast soliton cloud are, respectively, at points

$x_{1}(T)=\frac{l}{2}\cdot\frac{s_{1}+\kappa_{1}^{2}}{s_{2}-s_{1}},\qquad
x_{2}(T)=\frac{l}{2}\cdot\frac{s_{2}+\kappa_{2}^{2}}{s_{2}-s_{1}}.$ (47)

If solitons did not interact with one another, these clouds would move with
nonrenormalized velocities and would be at points
$x_{10}(T)=\kappa_{1}^{2}l/(s_{2}-s_{1})$,
$x_{20}(T)=\kappa_{2}^{2}l/(s_{2}-s_{1})$ at this instant; i.e., because of
the interaction, the clouds are shifted through distances

$\begin{split}&\Delta
x_{1}=x_{1}(T)-x_{10}(T)=\frac{l}{2}\cdot\frac{s_{1}-\kappa_{1}^{2}}{s_{2}-s_{1}}<0,\\\
&\Delta
x_{2}=x_{2}(T)-x_{20}(T)=\frac{l}{2}\cdot\frac{s_{2}-\kappa_{2}^{2}}{s_{2}-s_{1}}>0.\end{split}$
(48)

Therefore, because of their interaction, the clouds are narrowed, the soliton
densities in both clouds increase, and the slow cloud is shifted backwards,
while the fast cloud is shifted in the forward direction. A transition to
these values of the shifts are illustrated in Fig. 4, which is obtained from
the numerical solution for the Chaplygin gas. At the initial instant, the
centers of mass of both gases were at the origin of coordinates, and in the
course of separation of the clouds, the center of mass of the fast gas moved
more rapidly than it would do with its nonrenormalized velocity, while the
center of mass of the slow gas would lag behind the analogous motion with its
nonrenormalized velocity. After the instant of separation, the centers of mass
of both gases continue their motion with nonrenormalized velocities, acquiring
additional shifts due to the interaction with solitons of other species. Good
agreement between the values of shifts (48) predicted by the theory and the
result of numerical solution means that our idea of the formation of simple
waves with discontinuities from the initial discontinuities corresponds to the
actual dynamics of interacting soliton clouds in accordance with the kinetic
equation.

Figure 4:  Shifts of the centers of mass of clouds from their positions in the
absence of interaction, which are obtained by numerical solution of equations
for the Chaplygin gas for the initial values of parameters $f_{10}=0.05$,
$f_{20}=0.1$, $\kappa_{1}=1.3$, $\kappa_{2}=2.5$, $l=10$. At instant $T=1.6$
these parameters assume the theoretical values given by formulas (48).

##  VIII Conclusion

In this study, it is shown that a fundamental property of the dynamics of two
interacting soliton gases is the existence of simple waves propagating without
a change in form. In particular, such simple waves can have discontinuities;
therefore, in problems of the “dam breaking”, instead of rarefaction waves in
the dynamic of conventional gases, simple waves appear with discontinuities,
which are connected (instead of the general solution in which both Riemann
invariants change) by a plateau region with constant Riemann invariants. This
property of the dynamics of soliton gases differ drastically from the
properties of gas dynamics of conventional gases. At the same time, based on
the aforementioned pattern, simple analytic solutions to typical problems,
which are confirmed by exact numerical solutions of the kinetic equation
reduced to the equivalent form of equations for the dynamics of the Chaplygin
gas, can be constructed. The equations constructed in this way lead to an
intuitively clear pattern of the behavior of soliton gas clouds: as a result
of the interaction, the cloud of fast solitons is shifted in the forward
direction, while the cloud of slow solitons is shifted backwards, both clouds
becoming narrower and soliton densities increase in both of them. The
resulting expressions make it possible to estimate these effects in current
experimental investigations of soliton gases in waves on the water surface, in
nonlinear optics, and in the physics of cold atoms (see, for example, redor-19
; suret-20 ; bd-22 ).

The authors are grateful to G. A. El for fruitful discussions. This study was
supported by the Russian Foundation for Basic Research (project no.
20-01-00063).

## References

  * (1) N. J. Zabusky, M. D. Kruskal, Phys. Rev. Lett. 15, 240 (1965).
  * (2) P. D. Lax, Comm. Pure Appl. Math., 21, 467 (1968).
  * (3) V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: The Inverse Scattering Method, (Nauka, Moscow, 1980; Springer, Berlin, 1984).
  * (4) A. C. Newell, Solitons in Mathematics and Physics, (SIAM, Philadelphia, 1985).
  * (5) V. E. Zakharov, Sov. Phys. JETP 33, 538 (1971).
  * (6) G. A. El, Phys. Lett. A, 311, 374 (2003).
  * (7) G. A. El, A. M. Kamchatnov, Phys. Rev. Lett., 95, 204101 (2005).
  * (8) G. A. El, A. M. Kamchatnov, M. V. Pavlov, S. A. Zykov, J. Nonlinear Sci., 21, 151 (2011).
  * (9) E. V. Ferapontov, M. V. Pavlov, J. Nonlinear Sci., 32, 26 (2022).
  * (10) B. Doyon, SciPost Phys. LectNotes, 18, 1 (2020).
  * (11) G. A. El, J. Stat. Mech. (2021) 114001.
  * (12) I. Redor, E. Barthélemy, H. Michallet, M. Onorato, N. Mordant, Phys. Rev. Lett. 122, 214502 (2019).
  * (13) P. Suret, A. Tikan, F. Bonnefoy, F. Copie, G. Ducrozet, A. Gelash, G. Prabhudesai, G. Michel, A. Cazaubiel, E. Falcon, G. El, S. Randoux, Phys. Rev. Lett. 125, 264101 (2020).
  * (14) I. Bouchoule, J. Dubail, J. Stat. Mech. 2022, 014003 (2022).
  * (15) S. C. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura, Phys. Rev. Lett., 19, 1095 (1967).
  * (16) L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 6: Fluid Mechanics, (Pergamon, New York, 1987; Fizmatlit, Moscow, 2001).
  * (17) S. A. Chaplygin, About Gas Jets, (Univ. Tip., Moscow, 1902) [in Russian]; S. A. Chaplygin, Collection of Scientific Works, (OGIZ GITTL, Moscow, 1948), Vol. 2, p. 19 [in Russian].
  * (18) F. Carbone, D. Dutykh, G. A. El, EPL, 113, 30003 (2016).
  * (19) G. A. El, Chaos, 26, 023105 (2016).
  * (20) M. V. Pavlov, Sov. J. Theor. Math. Phys. 73, 1242 (1987).
  * (21) S. P. Tsarev, Izv. Akad. Nauk SSSR, Ser. Mat. 54, 1048 (1990).
  * (22) A. Sommerfeld, Partial Differential Equations in Physics, Vol. 6 of Lectures on Theoretical Physics (Academic, New York, 1964).
  * (23) V. G. Nosov and A. M. Kamchatnov, Sov. Phys. JETP 43, 397 (1976).
  * (24) S. K. Ivanov, A. M. Kamchatnov, Phys. Rev. A 99, 013609 (2019).
  * (25) S. K. Ivanov, A. M. Kamchatnov, Phys. Fluids, 32, 126115 (2020).
  * (26) L. D. Landau, Izv. Akad. Nauk SSSR, Ser. Fiz. 17, 51 (1953).
  * (27) I. M. Khalatnikov, Zh. Eksp. Teor. Fiz. 27, 529 (1954).
  * (28) A. M. Kamchatnov, J. Exp. Theor. Phys. 129, 607 (2019).


# Increasing the second uniform indiscernible by strongly ssp forcing

Ben De Bondt  Institut de Mathématiques de Jussieu (IMJ-PRG)
Université Paris Cité
Bâtiment Sophie Germain
8 Place Aurélie Nemours
75013 Paris, France<EMAIL_ADDRESS>https://perso.imj-prg.fr/ben-
debondt/ and  Boban Veličković  Institut de Mathématiques de Jussieu (IMJ-PRG)
Université Paris Cité
Bâtiment Sophie Germain
8 Place Aurélie Nemours
75013 Paris, France<EMAIL_ADDRESS>https://webusers.imj-prg.fr/
boban.velickovic/

###### Abstract.

We introduce a new and natural stationary set preserving forcing
$\mathbb{P}^{c\text{-}c}(\lambda,\mu)$ that (under $\mathsf{NS}$ precipitous +
existence of $H_{\theta}^{\\#}$ for a sufficiently large regular $\theta$)
increases the second uniform indiscernible $\mathbf{u}_{2}$ beyond some given
ordinal $\lambda.$ The forcing $\mathbb{P}^{c\text{-}c}$ shares this property
with forcings defined in [9] and [2]. As a main tool we use certain natural
open two player games which are of independent interest, viz. the capturing
games $\mathbf{G}^{cap}_{M}(X)$ and the catching-capturing games
$\mathbf{G}^{c\text{-}c}_{M}(X)$. In particular, these games are used to
isolate a special family of countable elementary submodels $M\prec H_{\theta}$
that occur as side conditions in $\mathbb{P}^{c\text{-}c}$ and thus allow to
control the forcing in a strong way.

###### Key words and phrases:

Stationary Set Preserving Forcing, Side Conditions, Second Uniform
Indiscernible, Games.

###### 2020 Mathematics Subject Classification:

03E57, 03E50, 03E05

##  1\. Introduction

To date, the study of forcing axioms is one of the foremost lines in set
theoretic research. Reflecting upon the nature of this phenomenon, one could
argue that the true reason for the tremendous success of a forcing axiom as
$\mathsf{PFA}$ is largely twofold. On the one hand, proper forcings are
sufficiently well-behaved to allow a very fruitful iteration theory to be
developed. Secondly, and just as crucially, the class of proper forcings is
sufficiently rich to allow for a variety of very different proper forcings,
giving in turn rise to many applications. A versatile technique for
constructing such proper forcings, allowing them to be tailored to perform
various different tasks, was introduced by Todorcevic ([17] and [18], see also
[19], [20], …) in the 1980’s and proceeds by including in a forcing condition
finitely many working parts, together with a finite chain of countable
elementary substructures of $H_{\theta}.$ The behaviour of such forcing is
determined by carefully fixing the allowed configurations of working parts and
models occuring in the conditions. One can contrast this with the situation
for stationary set preserving (from now on: ssp) forcings. The satisfactory
iteration theory of semiproper forcings makes up for the lack of a general
iteration theory for ssp forcings and allows to prove the consistency relative
to a supercompact cardinal of the ssp-forcing axiom $\mathsf{MM}.$
Furthermore, the power of Woodin’s $\mathbb{P}_{\max}$-forcing technique, in
conjunction with the recent results by Asperó and Schindler ([1]) rephrasing
(under suitable conditions) Woodin’s axiom $(\ast)$ as a bounded forcing
axiom, suggest many possible applications of ssp forcing strictly beyond
proper forcing. However, a technique for constructing ssp forcings just as
versatile as Todorcevic’s side condition technique has largely remained
absent, even though the $\mathcal{L}$-forcing technique developed by Jensen
([8]) and further developed in [2],[5] and ultimately in [1], may be
considered a candidate. In this article we exhibit one rudimentary instance of
an alternative approach towards constructing specific ssp forcings that stays
much closer to the side condition method for proper forcings. More precisely,
we consider a reasonably simple forcing $\mathbb{P}^{c\text{-}c}$ that is
almost, but not entirely, a pure side condition forcing. The twists are the
following: the side conditions are (morally) allowed to grow through the
forcing procedure and working parts are included in the forcing conditions
with the sole purpose of guiding this growth of the models. To guarantee that
these models still fulfill their task as side conditions (and thus make the
resulting forcing ssp) they can however not be allowed to absorb any new
countable ordinals. Consequently, the crucial idea in this prima facie quite
flexible adaptation of the proper-style side condition method is that only
models $M\prec H_{\theta}$ exhibiting the right kind of growth behaviour are
admissable as side conditions. The gain for the particular forcing
$\mathbb{P}^{c\text{-}c}(\lambda,\mu)$ is that instead of just adding any
sequence $(M^{G}_{\alpha}:\alpha<\omega_{1})$ of countable models with union
equal to $H_{\theta}$ (as a trivial pure side condition forcing would do), it
will add a special such sequence that Mostowski collapses to a generic
iteration with $\omega_{1}$-th iterate of the form $H_{\lambda}.$ Moreover
under the existence of $H_{\lambda}^{\\#}$, it can be arranged that the first
model in this generic iteration is iterable. Recall that under the presence of
sharps for reals, all $\omega_{1}$-th iterates $\mathfrak{m}_{\omega_{1}}$
occuring in some generic iteration
${\mathbf{It}}=(\mathfrak{m}_{\beta},j_{\alpha\beta},G_{\alpha}:\alpha<\beta\leq\omega_{1})$
of a countable transitive iterable model $\mathfrak{m}_{0}$ are, by standard
boundedness arguments, bounded in rank by the second uniform indiscernible
ordinal ${\bf u}_{2}$. Thus, it makes sense to consider the forcing
$\mathbb{P}^{c\text{-}c}(\lambda,\mu)$ within the light of the still poorly
understood problem of controlling the size of ${\bf u}_{2}$ by forcing. This
problem has a significant history, starting with [16] where Steele and Van
Wesep force over suitable models of $\mathsf{ZF}+\mathsf{AD}$ to obtain the
consistency of $\mathsf{ZFC}$ \+ $\mathsf{NS}$ saturated + ${\bf
u}_{2}=\omega_{2}$. Later, Woodin showed [21, Theorem 3.17] that in fact ${\bf
u}_{2}=\omega_{2}$ follows from $\mathsf{NS}$ saturated +
$\mathscr{P}(\omega_{1})^{\\#}$ exists. Thus, ${\bf u}_{2}=\omega_{2}$ can be
forced over a $\mathsf{ZFC}$-model from a Woodin cardinal with a measurable
above it using the forcing introduced by Shelah to make $\mathsf{NS}$
saturated [21, Theorem 2.64]. In [15], Steel and Welch showed that

$(\forall x\in\mathbb{R})\ x^{\\#}\text{ exists }+{\bf u}_{2}=\omega_{2}$

implies the existence of an inner model with a strong cardinal. The above
premise also follows from the theory $\mathsf{BMM}$ \+ $\mathsf{NS}$
precipitous, whose precise consistency strength is itself unclear. That
$\mathsf{BMM}$ \+ $\mathsf{NS}$ precipitous implies ${\bf u}_{2}=\omega_{2}$
is in turn a consequence of the existence of ssp forcings such as
$\mathbb{P}^{c\text{-}c}$, which, working under the theory
$\mathsf{ZFC}+\mathsf{NS}$ precipitous + existence of sufficient sharps,
increase the value of $\mathbf{u}_{2}$ beyond some arbitrarily high ordinal.
So far, two different such ssp forcings had been described in the literature,
namely a Namba-like forcing in [9] and an $\mathcal{L}$-forcing in [2]. It is
worth to note that both these two forcings, as well as the forcing
$\mathbb{P}^{c\text{-}c}$ described here, use the same argument to affect the
value of $\mathbf{u}_{2}$ which essentially goes back to Woodin’s argument
[21, Theorem 3.17], namely they all proceed through adding a countable
transitive iterable model $\mathfrak{m}$ together with a suitable
length-$\omega_{1}$ generic iteration that blows it up to arbitrarily large
rank. Also in some other aspects all three forcings behave very much alike. In
particular, all three share with Namba forcing the property that they give
some large regular cardinal cofinality $\omega$. We believe that a main
advantage of the construction as presented here is that the inclusion of
suitable elementary substructures into the forcing conditions elucidates the
proof of preservation of stationarity of ground model stationary subsets of
$\omega_{1}.$ Indeed, the argument for proving that $\mathbb{P}^{c\text{-}c}$
is ssp becomes entirely standard once the right class of models to be used as
side conditions has been identified and the existence of projective stationary
many such models has been proved. This additionally allows us to abstractly
define a property of forcings that we call _strongly ssp_ and is reminiscent
of a similar notion of strong properness that is commonly satisfied by proper
side condition forcings. Strongly ssp forcings enjoy additional strong
properties beyond being merely ssp such as not adding any fresh functions
$f:\omega_{1}\to V.$ The forcing $\mathbb{P}^{c\text{-}c}$ is strongly ssp and
therefore must add the many new reals that thin out the uniform
indiscernibles, without adding any fresh subsets of $\omega_{1}.$

##  2\. Preliminaries

We gather here several notions and results that will be needed below.

###  2.1. Iterable structures

$\mathsf{ZFC}^{-}$ is the theory obtained from $\mathsf{ZF}$ by removing the
powerset axiom and adding the collection axiom and the axiom that every set
can be wellordered. Let $\mathsf{ZFC}^{\bullet}$ be the theory

$\mathsf{ZFC}^{-}+\ \mathscr{P}(\mathscr{P}(\omega_{1}))\text{ exists
}+\mathsf{NS}\text{ is precipitous.}$

Whenever $\mathfrak{H}$ is a transitive model of $\mathsf{ZFC}^{-}$ and
$M\prec\mathfrak{H}$ is a countable elementary substructure, then $\delta(M)$
denotes the countable ordinal $M\cap\omega_{1},$ and for every $x\in\bigcup
M,$ we define

$\operatorname{Hull}(M,x)=\\{f(x):f\in M,x\in\operatorname{dom}(f)\\}.$

Note that under these conditions, $\operatorname{Hull}(M,x)$ is the minimal
countable elementary submodel of $\mathfrak{H}$ that $\subseteq$-extends $M$
and contains $x$ as an element. The meaning of the statement

$x\parallel_{\omega_{1}}M,$

is by definition that

$\operatorname{Hull}(M,x)\cap\omega_{1}=M\cap\omega_{1}.$

$\mathsf{NS}$ denotes the non-stationary ideal on $\omega_{1},$
$\mathsf{NS}^{+}$ is the set of stationary subsets of $\omega_{1}.$ We also
use $\mathsf{NS}^{+}$ as a shorthand for the partial order
$\displaystyle(\mathsf{NS}^{+},\subseteq)$, that is, whenever
$\mathsf{NS}^{+}$ is without further indication treated as partial order, we
have silently endowed it with the partial order $S\leq T\Leftrightarrow
S\subseteq T.$

######  Definition 2.1.

A countable model $M\prec\mathfrak{H}\vDash\mathsf{ZFC}^{\bullet}$ is called
_selfgeneric_ if for every $\mathcal{A}\in M$ such that

$\mathfrak{H}\vDash\mathcal{A}\subseteq\mathsf{NS}^{+}\text{ is a maximal
antichain},$

there exists $S\in\mathcal{A}\cap M$ such that $\delta(M)\in S$.

######  Definition 2.2.

Let $\mathfrak{m}$ be a countable transitive model of the theory
$\mathsf{ZFC}^{\bullet}.$

A _generic iteration_ of $\mathfrak{m}=\mathfrak{m}_{0}$ of length $\gamma$
(with $\gamma\leq\omega_{1}+1$) is a commuting system
${\mathbf{It}}=(\mathfrak{m}_{\beta},j_{\alpha\beta},G_{\alpha}:\alpha<\beta<\gamma)$
satisfying

  1. (a)

for every $\alpha+1<\gamma$,
$G_{\alpha}\subset(\mathsf{NS}^{+})^{\mathfrak{m}_{\alpha}}$ is a filter that
is $(\mathsf{NS}^{+})^{\mathfrak{m}_{\alpha}}$-generic over
$\mathfrak{m}_{\alpha},$

  2. (b)

for every $\alpha+1<\gamma$, $\mathfrak{m}_{\alpha+1}$ is the (transitivised)
generic ultrapower of $\mathfrak{m}_{\alpha}$ induced by $G_{\alpha}$,
$j_{\alpha,\alpha+1}$ is the corresponding ultrapower map,

  3. (c)

for every limit ordinal $\beta<\gamma$,
$(\mathfrak{m}_{\beta},(j_{\alpha\beta})_{\alpha<\beta})$ is the
(transitivised) direct limit of the system
$(\mathfrak{m}_{\beta^{\prime}},j_{\alpha\beta^{\prime}}:\alpha<\beta^{\prime}<\beta)$.

The $\mathsf{ZFC}^{\bullet}$-model $\mathfrak{m}$ is called _iterable_ if for
every limit ordinal $\gamma\leq\omega_{1}$ and every generic iteration
${\mathbf{It}}=(\mathfrak{m}_{\beta},j_{\alpha\beta},G_{\alpha}:\alpha<\beta<\gamma)$
of length $\gamma$ of $\mathfrak{m}$, the direct limit of the system
$(\mathfrak{m}_{\beta},j_{\alpha\beta}:\alpha<\beta<\gamma)$ is well-founded.

We will make use of the following standard way of constructing iterable models
from sharps.

######  Lemma 2.3.

Suppose $\mathfrak{H}$ is a transitive $\mathsf{ZFC}^{\bullet}$-model with
$\omega_{1}\subseteq\mathfrak{H}$ and $\theta\in\mathfrak{H}$ is an ordinal
such that

  1. (a)

$\mathfrak{H}\vDash\theta$ is a regular cardinal and
$\theta>|\mathscr{P}(\mathscr{P}(\omega_{1}))|,$

  2. (b)

$\mathfrak{H}\vDash$ $H_{\theta}$ and $H_{\theta}^{\\#}$ both exist.

If $M\prec\mathfrak{H}$ is countable and contains $\theta,$ then the
transitive collapse of $M\cap H^{\mathfrak{H}}_{\theta}$ is an iterable
$\mathsf{ZFC}^{\bullet}$-model. ∎

A proof of Lemma 2.3 can be found by applying standard modifications to the
argument given in [21] to prove [21, Theorem 3.14].

######  Definition 2.4.

Let $\mathfrak{H}$ be a transitive $\mathsf{ZFC}^{\bullet}$-model and let
$(M_{\alpha}:\alpha<\omega_{1})$ be a continuous $\subseteq$-increasing
sequence of countable elementary submodels of $\mathfrak{H}.$ We call
$(M_{\alpha}:\alpha<\omega_{1})$ an _iteration-inducing factorisation_ of
$\mathfrak{H}$ if

$\bigcup_{\alpha<\omega_{1}}M_{\alpha}=\mathfrak{H},$

and additionally the following two conditions hold for every
$\alpha<\omega_{1}:$

  * •

$M_{\alpha}$ is selfgeneric,

  * •

$M_{\alpha+1}=\operatorname{Hull}(M_{\alpha},\delta(M_{\alpha})).$

There is a straightforward way to translate between generic iterations of
length $\omega_{1}$ and iteration-inducing factorisations. Indeed, the
following is easily checked.

######  Lemma 2.5.

Let $\mathfrak{H}$ be a transitive $\mathsf{ZFC}^{\bullet}$-model.

  1. (1)

If
${\mathbf{It}}=(\mathfrak{m}_{\beta},j_{\alpha\beta},G_{\alpha}:\alpha<\beta<\omega_{1})$
is a generic iteration of length $\omega_{1}$ with (transitivised) direct
limit equal to $\mathfrak{H},$ then
$(j_{\alpha\omega_{1}}[\mathfrak{m}_{\alpha}]:\alpha<\omega_{1})$ is an
iteration-inducing factorisation of $\mathfrak{H},$

  2. (2)

if $(M_{\alpha}:\alpha<\omega_{1})$ is an iteration-inducing factorisation of
$\mathfrak{H},$ then there exists a unique generic iteration
${\mathbf{It}}=(\mathfrak{m}_{\beta},j_{\alpha\beta},G_{\alpha}:\alpha<\beta<\omega_{1})$
of length $\omega_{1}$ such that the following diagrams
$(\alpha<\beta<\omega_{1})$ commute:

$\begin{array}[]{lcl}M_{\alpha}&\quad\subseteq&M_{\beta}\\\
\bigg{\downarrow}\pi_{\alpha}&&\bigg{\downarrow}\pi_{\beta}\\\
\mathsf{m}_{\alpha}&\quad\xrightarrow{j_{\alpha\beta}}&\mathsf{m}_{\beta}\end{array}$

with $\pi_{\alpha}:M_{\alpha}\to\mathsf{m}_{\alpha}$ for each
$\alpha<\omega_{1}$ being the Mostowski isomorphism. Moreover, the direct
limit of the system ${\mathbf{It}}$ is equal to $\mathfrak{H}.$

∎

###  2.2. The second uniform indiscernible

If $x^{\\#}$ exists for every real, then there exists a proper class of
ordinals which are $x$-indiscernible for every real $x,$ these are called the
uniform indiscernibles. Clearly, every uncountable cardinal is a uniform
indiscernible and $\omega_{1}$ is the minimal uniform indiscernible ordinal.
The second uniform indiscernible ${\bf u}_{2}\leq\omega_{2}$ deserves special
interest because it is simultaneously the second projective ordinal
$\boldsymbol{\delta}_{2}^{1}$. The projective ordinals

$\boldsymbol{\delta}_{n}^{1}=\sup\\{\alpha:\alpha\text{ is the length of a
}\boldsymbol{\Delta}^{1}_{n}\text{ prewellordering of }\mathbb{R}\\}$

were introduced by Moschovakis ([12]) to gauge the extent to which
$\mathsf{CH}$ can fail in a projectively definable way. One can check that if
the sharp of every real exists, additionally

$\boldsymbol{\delta}_{2}^{1}=\sup\\{\omega_{1}^{+L[x]}:x\in\mathbb{R}\\}.$

Results of respectively Martin ([10]) and Woodin ([21, Theorem 3.17]), give
that under both $(\mathsf{ZF}+)\mathsf{AD}$ and under $\mathsf{NS}$ saturated
+ $\mathscr{P}(\omega_{1})^{\\#},$ the second projective ordinal
$\boldsymbol{\delta}_{2}^{1}$ has the maximal value it could have, namely
$\boldsymbol{\delta}_{2}^{1}=\omega_{2}.$ We need the following lemma which is
also essentially used in Woodin’s argument ([21, Theorem 3.17]) for

$\mathsf{NS}\text{ saturated }+\mathscr{P}(\omega_{1})^{\\#}\text{ exists
}\Rightarrow\boldsymbol{\delta}_{2}^{1}=\omega_{2}.$

######  Lemma 2.6 ([21, Lemma 3.15]).

Let $\mathfrak{m}$ be a countable transitive model of $\mathsf{ZFC}^{\bullet}$
that is iterable. Let $x$ be a real that codes both $\mathfrak{m}$ and some
countable ordinal $\alpha.$ Then for every generic iteration
${\mathbf{It}}=(\mathfrak{m}_{\beta},j_{\alpha\beta},G_{\alpha}:\alpha<\beta<\alpha+1)$
of $\mathfrak{m}$ of length $\alpha+1,$

$\operatorname{rank}(\mathfrak{m}_{\alpha})<\omega_{1}^{+L[x]}.$

∎

Finally, we will also make use of the well known fact that existence of sharps
of all (transitive) sets in a fragment of the universe is not destroyed by
small forcing. See e.g. [14, Lemma 3.4] for a much finer result of this type,
for the present purposes the following lemma suffices.

######  Lemma 2.7.

Let $\kappa$ be a cardinal and suppose that $H_{\kappa^{+}}^{\\#}$ exists.
Then for every forcing $\mathbb{P}$ of size at most $\kappa$ and every filter
$G$ that is $\mathbb{P}$-generic over $V,$

$V[G]\vDash X^{\\#}\text{ exists for every transitive set }X\in
H_{\kappa^{+}}.$

##  3\. The strongly ssp property

######  Notation 3.1.

For a set $X$ and a cardinal $\theta,$ we write $X\ll\theta$ to mean that
$\mathscr{P}(\mathscr{P}(X))\in H_{\theta}.$

######  Definition 3.2.

Given a $\mathbb{P}$-condition $p^{*},$ the condition $p^{*}$ is called
$(M,\mathbb{P})$-_semigeneric_ if for every $q\leq p^{*}$ and every maximal
antichain $\mathcal{A}\subseteq\mathbb{P}$ with $\mathcal{A}\in M,$ there
exists some $r\in\mathcal{A}$ such that

$r\parallel q\text{ and }r\parallel_{\omega_{1}}M.$

Equivalently, $p^{*}$ is $(M,\mathbb{P})$-semigeneric iff for every
$\mathbb{P}$-name $\tau\in M$ for a countable ordinal,

$p^{\ast}\Vdash\tau<\delta(M).$

The condition $p^{*}$ is called _strongly $(M,\mathbb{P})$-semigeneric_ if for
every $q\leq p^{*}$ there exists $\operatorname{tr}(q|M)\in\mathbb{P}$ such
that $\operatorname{tr}(q|M)\parallel_{\omega_{1}}M$ and for every
$r\in\operatorname{Hull}(M,\operatorname{tr}(q|M))\cap\mathbb{P},$

$r\leq\operatorname{tr}(q|M)\Rightarrow r\parallel q.$

$\mathbb{P}$ is said to be _(strongly) semiproper_ with respect to $M\prec
H_{\theta}$ if for every $p\in\mathbb{P}\cap M,$ there exists $p^{*}\leq p$
such that $p^{*}$ is (strongly) $(M,\mathbb{P})$-semigeneric.

Recall that a set $X\subseteq[H_{\theta}]^{\omega}$ is _projective stationary_
if and only if for every stationary subset $S\subseteq\omega_{1}$ the set
$\\{M\in X:M\cap\omega_{1}\in S\\}$ is stationary in $[H_{\theta}]^{\omega}.$

######  Lemma 3.3.

If $\mathbb{P}$ is strongly semiproper with respect to $M\prec H_{\theta},$
then $\mathbb{P}$ is semiproper with respect to $M\prec H_{\theta}.$ If
$\mathbb{P}$ is semiproper with respect to projective stationary many $M\prec
H_{\theta},$ then $\mathbb{P}$ is ssp.

###### Proof.

The first statement is clear because every condition that is strongly
$(M,\mathbb{P})$-semigeneric is also $(M,\mathbb{P})$-semigeneric. To prove
the second statement, let be given a stationary set $S\subseteq\omega_{1},$ a
name for a club $\dot{C}\in H_{\theta}$ and an arbitrary $p\in\mathbb{P}.$ By
assumption there exists $M\prec H_{\theta}$ such that $\mathbb{P}$ is
semiproper relative to $M$ and such that $p,\dot{C}\in M$ and
$M\cap\omega_{1}\in S.$ By semiproperness there exists $p^{*}\leq p$ that is
$(M,\mathbb{P})$-semigeneric. Then $p^{*}$ forces that
$M[G]\cap\omega_{1}=M\cap\omega_{1}$ and hence also that
$M\cap\omega_{1}\in\dot{C}.$ ∎

In fact (see [6, Lemma 4.8]), a forcing is ssp if and only if for every
sufficiently large regular cardinal $\theta,$ there exist projective
stationary many countable $M\prec H_{\theta}$ such that $\mathbb{P}$ is
semiproper with respect to $M.$ Replacing here semiproper by _strongly_
semiproper results in the following stronger notion.

######  Definition 3.4.

A forcing $\mathbb{P}$ is _strongly ssp_ if for every regular cardinal
$\theta$ which is sufficiently large ($\theta\gg\mathbb{P}$), there exist
projective stationary many countable $M\prec H_{\theta}$ such that
$\mathbb{P}$ is strongly semiproper with respect to $M.$

The following lemma makes apparent that even ccc forcings and countably closed
forcings can fail to be strongly ssp.

######  Lemma 3.5.

If $\mathbb{P}$ is strongly ssp, then $\mathbb{P}$ does not add any fresh
functions $\omega_{1}\to V.$

###### Proof.

Let $\dot{f}$ be a $\mathbb{P}$-name and suppose $p\in\mathbb{P}$ such that

$p\Vdash\dot{f}:\omega_{1}\to
V\quad\text{and}\quad(\forall\alpha<\omega_{1})\,\dot{f}\upharpoonright\alpha\in
V.$

Choose a sufficiently large regular $\theta$ and let $M\prec H_{\theta}$
countable such that $\mathbb{P},\dot{f},p\in M$ and such that $\mathbb{P}$ is
strongly semiproper with respect to $\mathbb{P}.$ Then pick $p^{*}\leq p$
which is strongly $(M,\mathbb{P})$-generic and also pick $q\leq p^{*}$ and
$g:M\cap\omega_{1}\to V$ such that

$q\Vdash\dot{f}\upharpoonright M\cap\omega_{1}=g.$

We claim that $\operatorname{tr}(q|M)$ decides $\dot{f}.$
Suppose not, then there exist
$r_{1},r_{2}\in\mathbb{P}\,\cap\,\operatorname{Hull}(M,\operatorname{tr}(q|M)),$
$\alpha\in M\cap\omega_{1}$ and $x\neq y$ such that

$r_{1},r_{2}\leq\operatorname{tr}(q|M),\qquad
r_{1}\Vdash\dot{f}(\alpha)=x,\qquad r_{2}\Vdash\dot{f}(\alpha)=y.$

This contradicts $r_{1}\parallel q$, $r_{2}\parallel q$ and
$q\Vdash\dot{f}(\alpha)=g(\alpha).$ ∎

As a consequence, a strongly ssp forcing adds no new uncountable branches to a
ground-model tree of height $\omega_{1}.$

######  Remark 3.6.

Of course, the strongly ssp property is inspired by Mitchell’s ([11]) notion
of strong properness. A forcing $\mathbb{P}\in H_{\theta}$ is strongly proper
with respect to $M\prec H_{\theta}$ if for every $p\in M\cap\mathbb{P},$ there
exists $p^{*}\leq p$ that is strongly $(M,\mathbb{P})$-generic. $p^{*}$ being
strongly $(M,\mathbb{P})$-generic means that for every $q\leq p^{*},$ there
exists $\operatorname{tr}(q|M)\in\mathbb{P}\cap M$ such that

$(\forall r\in\mathbb{P}\cap M)\ r\leq\operatorname{tr}(q|M)\Rightarrow
r\parallel q.$

$\mathbb{P}$ is strongly proper if for every regular $\theta\gg\mathbb{P}$
there exist club many countable $M\prec H_{\theta}$ such that $\mathbb{P}$ is
strongly proper with respect to $M.$
Unsurprisingly, it is easy to check that every strongly proper forcing is
strongly ssp.

##  4\. Games: catching and capturing

In this section three types of games will be introduced,

  * •

the antichain catching games $\mathbf{G}^{cat}_{M},$

  * •

the capturing games $\mathbf{G}^{cap}_{M}(X),$

  * •

and the catching-capturing games $\mathbf{G}^{c\text{-}c}_{M}(X).$

All three types of games will be played by two players $P1$ (_the Challenger_)
and $P2$ (_the Constructor_). $P1$ and $P2$ thank their nicknames to the fact
that the moves of $P2$ are used to construct a growing sequence
$(M_{n}:n<\omega)$ of countable elementary submodels of some $H_{\theta},$
while the moves of $P1$ can be thought of as challenges for $P2$ to make the
models $M_{n}$ satisfy certain conditions. All games considered have length
$\omega.$ If $\mathbf{G}$ is one of these games, we write
$P1\nearrow\mathbf{G}$ for the statement that $P1$ has a winning strategy for
the game $\mathbf{G}$ (and likewise for $P2\nearrow\mathbf{G}$).

We proceed straight to the definitions.

######  Definition 4.1 (The antichain catching game).

Let $\theta$ be a regular cardinal and suppose $\mathscr{P}(\omega_{1})\in
H_{\theta}.$ Let also $M\prec H_{\theta}$ be countable. The _antichain
catching game_ $\mathbf{G}^{cat}_{M}$ for $M$ is defined as follows. Define
$M_{0}=M.$ In round $n$ of the game,

  * •

$P1$ plays some maximal antichain
$\mathcal{A}_{n}\subseteq\mathsf{NS}_{\omega_{1}}^{+}$ with
$\mathcal{A}_{n}\in M_{n}$,

  * •

$P2$ plays some $S_{n}\in\mathcal{A}_{n}$ such that $M_{n}\cap\omega_{1}\in
S_{n}$,

  * •

define $M_{n+1}=\operatorname{Hull}(M_{n},S_{n})$.

Thus, a play of the game $\mathbf{G}^{cat}_{M}$ can be pictured as follows.

$\begin{array}[]{lcccccc}\phantom{P1}&M_{0}&\stackrel{{\scriptstyle+S_{0}}}{{\rightsquigarrow}}&M_{1}&\stackrel{{\scriptstyle+S_{1}}}{{\rightsquigarrow}}&M_{2}&{\ldots}\\\
\phantom{P2}&\rotatebox[origin={c}]{90.0}{$\in$}&\phantom{S_{0}}&\rotatebox[origin={c}]{90.0}{$\in$}&\phantom{f_{1}}&\rotatebox[origin={c}]{90.0}{$\in$}&\phantom{\ldots}\end{array}$

$\begin{array}[]{l|cccccc}P1&\mathcal{A}_{0}&\phantom{\stackrel{{\scriptstyle+S_{0}}}{{\rightsquigarrow}}}&\phantom{i}\mathcal{A}_{1}&\phantom{\stackrel{{\scriptstyle+f_{1}}}{{\rightsquigarrow}}}&\phantom{i}\mathcal{A}_{2}&\ldots\\\
\hline\cr\\\\[-6.99997pt] P2&&S_{0}&\phantom{nn}&S_{1}&&\ldots\end{array}$

A winner is assigned as decided by the following rules.

  * If there is a player who first disobeys the rules, this player loses,

  * else, $P2$ wins iff $\forall n$ $M_{n}\cap\omega_{1}=M\cap\omega_{1}$.

The catching game is clearly just one of the many variations on the classic
theme of antichain catching and it is easy to observe that $P1$ has a winning
strategy for the catching game $\mathbf{G}^{cat}_{M}$ if and only if there
exists some countable selfgeneric $N\prec H_{\theta}$ such that $M\subseteq N$
and $M\cap\omega_{1}=N\cap\omega_{1}.$

######  Definition 4.2 (The capturing game).

Let $\theta$ be a regular cardinal and $X\in H_{\theta}$. Suppose $M\prec
H_{\theta}$ is countable and $X\in M$. The $X$-_capturing game_
$\mathbf{G}^{cap}_{M}(X)$ for $M$ is defined as follows. Define $M_{0}=M$. In
round $n$ of the game,

  * •

$P1$ plays some $x_{n}\in X$ ($x_{n}$ not necessarily belonging to $M_{n}$),

  * •

$P2$ then has to play some function $f_{n}:\omega_{1}\to X$, with
$f_{n}(M_{n}\cap\omega_{1})=x_{n}$,

  * •

define $M_{n+1}=\operatorname{Hull}(M_{n},f_{n})$.

Thus, a play of the game $\mathbf{G}^{cap}_{M}(X)$ can be pictured as follows.

$\begin{array}[]{lcccccc}\phantom{P1}&M_{0}&\stackrel{{\scriptstyle+f_{0}}}{{\rightsquigarrow}}&M_{1}&\stackrel{{\scriptstyle+f_{1}}}{{\rightsquigarrow}}&M_{2}&{\ldots}\\\
\end{array}$

$\begin{array}[]{l|cccccc}P1&x_{0}&\phantom{\stackrel{{\scriptstyle+f_{0}}}{{\rightsquigarrow}}}&\phantom{i}x_{1}&\phantom{\stackrel{{\scriptstyle+f_{1}}}{{\rightsquigarrow}}}&\phantom{i}x_{2}&\ldots\\\
\hline\cr\\\\[-6.99997pt] P2&&f_{0}&\phantom{nn}&f_{1}&&\ldots\end{array}$

The winner is again assigned in accordance with the following same rules.

  * If there is a player who first disobeys the rules, this player loses,

  * else, $P2$ wins iff $\forall n$ $M_{n}\cap\omega_{1}=M\cap\omega_{1}$.

Finally, the catching-capturing game $\mathbf{G}^{c\text{-}c}_{M}(X)$ is
simply a mix of the two above games $\mathbf{G}^{cat}_{M}$ and
$\mathbf{G}^{cap}_{M}(X)$.

######  Definition 4.3 (The catching - capturing game).

Let $\theta$ be a regular cardinal and $X\in H_{\theta}$. Suppose $M\prec
H_{\theta}$ is countable and $X\in M$. The _catching-capturing game_
$\mathbf{G}^{c\text{-}c}_{M}(X)$ for $M$ is defined as follows. In round $n,$
player $P1$ now has two options:

  * •

either $P1$ makes a move $\mathcal{A}_{n}\in M_{n}$ as in the game
$\mathbf{G}^{cat}_{M}$,

    * –

the player $P2$ then has to answer with some $S_{n},$ respecting the same
restrictions as in the game $\mathbf{G}^{cat}_{M}$,

    * –

define $M_{n+1}=\operatorname{Hull}(M_{n},S_{n})$.

  * •

or $P1$ decides to play some $x_{n}\in X$, as in the game
$\mathbf{G}^{cap}_{M}(X)$.

    * –

$P2$ then has to answer with some function $f_{n}:\omega_{1}\to X$, respecting
the same restrictions as in the game $\mathbf{G}^{cap}_{M}(X)$,

    * –

define $M_{n+1}=\operatorname{Hull}(M_{n},f_{n})$.

A play of the game $\mathbf{G}^{c\text{-}c}_{M}(X)$ can be pictured as
follows.

$\begin{array}[]{lcccccccc}\phantom{P1}&M_{0}&\stackrel{{\scriptstyle+S_{0}}}{{\rightsquigarrow}}&M_{1}&\stackrel{{\scriptstyle+f_{1}}}{{\rightsquigarrow}}&M_{2}&\stackrel{{\scriptstyle+S_{2}}}{{\rightsquigarrow}}&M_{3}&\phantom{\ldots}\\\
\phantom{P2}&\rotatebox[origin={c}]{90.0}{$\in$}&\phantom{S_{0}}&&\phantom{f_{1}}&\rotatebox[origin={c}]{90.0}{$\in$}&\phantom{S_{2}}&&\phantom{\ldots}\end{array}$

$\begin{array}[]{l|cccccccc}P1&\mathcal{A}_{0}&\phantom{\stackrel{{\scriptstyle+S_{0}}}{{\rightsquigarrow}}}&\phantom{n}x_{1}&\phantom{\stackrel{{\scriptstyle+f_{1}}}{{\rightsquigarrow}}}&\mathcal{A}_{2}&\phantom{\stackrel{{\scriptstyle+S_{0}}}{{\rightsquigarrow}}}&\phantom{n}x_{3}&\ldots\\\
\hline\cr\\\\[-6.99997pt]
P2&&S_{0}&&f_{1}&\phantom{nn}&S_{2}&&\ldots\end{array}$

The winning conditions remain unaltered:

  * If there is a player who first disobeys the rules, this player loses,

  * else, $P2$ wins iff $\forall n$ $M_{n}\cap\omega_{1}=M\cap\omega_{1}$.

Note that the games
$\mathbf{G}^{cat}_{M},\mathbf{G}^{cap}_{M}(X),\mathbf{G}^{c\text{-}c}_{M}(X)$
are all determined, the winning conditions for $P2$ being closed. We will
allow countable models $M\prec H_{\theta}$ to occur as side conditions in the
forcing $\mathbb{P}^{c\text{-}c}$ only if
$P2\nearrow\mathbf{G}^{c\text{-}c}_{M}(X)$ for a sufficiently large set $X\in
H_{\theta}.$ Thus it will be crucial for this approach that sufficiently many
models exist for which $P2$ has a winning strategy in
$\mathbf{G}^{c\text{-}c}_{M}(X)$. If the non-stationary ideal on $\omega_{1}$
is precipitous, this is warranted by the following proposition.

######  Proposition 4.4.

Suppose $\theta\gg\omega_{1}$ is a regular cardinal and $X\in H_{\theta}$. The
following are all equivalent.

  1. (a)

there exist projective stationary many countable $M\prec H_{\theta}$ for which
$P2\nearrow\mathbf{G}^{c\text{-}c}_{M}(X),$

  2. (b)

there exist projective stationary many countable $M\prec H_{\theta}$ for which
$P2\nearrow\mathbf{G}^{cat}_{M}$,

  3. (c)

there exist projective stationary many countable selfgeneric $M\prec
H_{\theta},$

  4. (d)

$\mathsf{NS}$ is precipitous.

###### Proof.

$\boxed{(d)\Rightarrow(a)}$ We first prove that (d) implies (a). The argument
is a relative of those in [4, Theorem 3.8] and [9, Proposition 2.4].
Let $A\subseteq\omega_{1}$ be stationary and let $F:H_{\theta}\to H_{\theta}$
be arbitrary. We use an elementarity-argument to prove that there exists a
full countable model $M\prec H_{\theta}$ which is closed under $F$ and
satisfies $M\cap\omega_{1}\in A$. Let $G$ be a filter for forcing with
stationary subsets of $\omega_{1}$ that is generic over $V$ and contains $A$.
Using precipitousness of $\mathsf{NS},$ let $W$ be the corresponding
transitivised ultrapower of $V$ and let $j:V\to W$ be the corresponding
elementary embedding. Select (in $V$) an arbitrary countable elementary
submodel $N\prec H_{\theta}$ that contains $X$ and has the property that for
every function $f:\omega_{1}\to H_{\theta}$ which is in $N,$ also the function
$F\circ f:\omega_{1}\to H_{\theta}$ is in $N.$
Next, we continue working in $W.$ Note that, in $W$, $j(N)$ is a countable
elementary submodel of $H_{j(\theta)}$. Hence, the model

$\widetilde{N}:=\\{f(\xi):\xi\in\omega_{1}^{V},f\in j(N)\text{ with
}\xi\in\operatorname{dom}(f)\\},$

is, in $W$, a countable elementary submodel of $H_{j(\theta)}.$
By construction we have that
$\widetilde{N}\cap\omega_{1}^{W}=\omega_{1}^{V}\in j(A)$ and it clearly
follows from the choice of $N$ that $\widetilde{N}$ is closed under the
function $j(F):H_{j(\theta)}\to H_{j(\theta)}.$ Using elementarity of the
embedding $j,$ the proof will be completed if we can establish that

$(\text{ the model }\widetilde{N}\prec H_{j(\theta)}\text{ satisfies
}P2\nearrow\mathbf{G}^{c\text{-}c}_{\widetilde{N}}(j(X))\,)^{W}.$

Arguing by contradiction, suppose that this is not the case. Then there exists
in $W$ some winning strategy $\Sigma$ for $P1$ (the Challenger) in the game
$\mathbf{G}^{c\text{-}c}_{\widetilde{N}}(j(X)).$ Let $T$ be the tree (as
defined in $W$) of all partial plays of the game
$\mathbf{G}^{c\text{-}c}_{\widetilde{N}}(j(X))$ in which $P1$ plays by the
strategy $\Sigma$ and in which $P2$ has not lost yet, then

$(\text{ the tree }T\text{ is well-founded })^{W}.$

We now leave $W$ and describe in $V[G]$ an infinite branch $B$ in the tree
$T,$ thus arriving at the desired contradiction (since $W$ is transitive,
well-foundedness is absolute between $W$ and $V[G]$). The branch $B$ can be
defined as follows:

  * •

$P1$ makes a $\mathbf{G}^{cap}_{\widetilde{N}}(j(X))$-move in round $n$:
Suppose $P1$ plays the set $x\in j(X).$ There exists $f:\omega_{1}^{V}\to X$
such that $x=j(f)(\omega_{1}^{V}).$
$P2$ replies by playing the function $j(f).$

  * •

$P1$ makes a $\mathbf{G}^{cat}_{\widetilde{N}}$-move in round $n$:
Suppose $P1$ plays the maximal antichain $\mathcal{A}_{n},$ then it is of the
form $j(f(\bar{y})),$ where $\bar{y}$ is the sequence of moves that have been
played by $P2$ in the first $n$ moves and where

$f:X^{n}\to\\{\mathcal{A}\subseteq I^{+}:\mathcal{A}\text{ is a maximal
antichain in }\mathsf{NS}\\}^{V}$

is a function which belongs to $N.$ Then

$\mathcal{B}:=f(\bar{y})\in\\{\mathcal{A}\subseteq I^{+}:\mathcal{A}\text{ is
a maximal antichain in }\mathsf{NS}\\}^{V},$

so there exists some $S\in\mathcal{B}\cap G$.
$P2$ replies by playing the set $j(S).$

To check that $P2$ does not lose by playing this way, we note that

$M_{n}\subseteq\\{f(\xi,j(w)):\xi\in\omega_{1},w\in H_{\theta}^{<\omega},f\in
j(\widetilde{N})=j[\widetilde{N}]\\}\subseteq j[H_{\theta}]$

and

$j[H_{\theta}]\cap\omega_{1}^{W}=\omega_{1}^{V}=\widetilde{N}\cap\omega_{1}^{W}.$

$\boxed{(a)\Rightarrow(b)}$ The implication (a)$\Rightarrow$(b) is trivial.

$\boxed{(b)\Rightarrow(c)}$ It was already noted above that

$P2\nearrow\mathbf{G}^{cat}_{M}$

is equivalent to the statement that there exists an extension of $M$ to a
countable selfgeneric $N\prec H_{\theta}$ containing the same countable
ordinals. It is easily seen to follow from this that (b) and (c) are
equivalent.

$\boxed{(c)\Rightarrow(d)}$ Finally, the direction (c)$\Rightarrow$(d) is well
known and can be found in [4, Lemma 3.6]. We repeat this short argument for
the sake of completeness. Suppose $\mathsf{NS}$ were not precipitous. There
would exist $S\in\mathsf{NS}^{+}$ such that

$\begin{split}S\Vdash\text{there exists an infinite
}<_{G}&\text{-decreasing}\text{ sequence}\\\ &\text{ of ground-model functions
}f_{n}:\omega_{1}\to Ord.\end{split}$

Let $M\prec H_{\theta}$ be countable and selfgeneric with $S\in M$ and
$\delta:=M\cap\omega_{1}\in S.$ Let $\pi:M\to\mathfrak{m}$ be the Mostowski
collapse of $M.$ Then inside the transitive $\mathsf{ZFC}^{\bullet}$-model
$\mathfrak{m}$, $\pi(S)$ is a stationary subset of $\omega_{1}$ and

$\begin{split}\pi(S)\Vdash\text{there exists an infinite
}<_{G}&\text{-decreasing}\text{ sequence}\\\ &\text{ of ground-model functions
}f_{n}:\omega_{1}\to Ord.\end{split}$

Moreover, by selfgenericity,

$G_{\delta}=\\{\pi(T):T\in\mathsf{NS}^{+}\text{ with }\delta\in T\\}$

is a $\pi(\mathsf{NS}^{+})$-filter that is generic over $\mathfrak{m}$. This
contradicts the prior statement about $\pi(S)$ forcing ill-foundedness of
$\operatorname{Ult}(V,G)$ because $\pi(S)\in G_{\delta}$ and
$\operatorname{Ult}(\mathfrak{m},G_{\delta})$ is well-founded as it is
$\in$-isomorphic to $\operatorname{Hull}(M,\delta).$

∎

######  Question 4.5.

Let $\lambda<\theta$ both be sufficiently large, what is the strength of the
statement that there exist projective stationary many countable $M\prec
H_{\theta}$ for which $P2\nearrow\mathbf{G}^{cap}_{M}(H_{\lambda})$, in
particular is it also equivalent to precipitousness of $\mathsf{NS}$?

Although not needed for the discussion in the next section of the forcing
$\mathbb{P}^{c\text{-}c},$ we end this section with some results that show
that the equivalences in Proposition 4.4 fall apart when one replaces
_projective stationary many_ by _club many_.

It is well known that the existence of club many selfgeneric models is
equivalent to saturation of the non-stationary ideal.

######  Proposition 4.6.

Let $\theta\gg\omega_{1}$ be a regular cardinal. The following statements are
equivalent.

  1. (a)

there exist club many countable $M\prec H_{\theta}$ for which
$P2\nearrow\mathbf{G}^{cat}_{M},$

  2. (b)

there exist club many countable selfgeneric $M\prec H_{\theta},$

  3. (c)

$\mathsf{NS}$ is $\omega_{2}$-saturated.

###### Proof.

See e.g. [7, Vol. 2, Lemma 13.3.46]. ∎

In contrast, the existence of club many countable models $M\prec H_{\theta}$
for which $P2\nearrow\mathbf{G}^{cap}_{M}(X)$ can be translated into strong
versions of Chang’s conjecture.

######  Definition 4.7.

(cof-Strong Chang’s Conjecture, [3, Definition 4.3])
$\mathsf{SCC}^{\textrm{cof}}$ is the statement that for all regular cardinals
$\theta\gg\omega_{2}$ and every countable $M\prec H_{\theta}$ there exist
cofinally many $\xi\in\omega_{2}$ such that $\xi\parallel_{\omega_{1}}M.$

######  Definition 4.8.

(Global Strong Chang’s Conjecture, [3, Definition 4.9])
$\textrm{Glob-}\mathsf{SCC}^{\textrm{cof}}$ is the statement that for all
regular cardinals $\theta\gg\lambda\geq\omega_{2},$ for every $\lambda\in
M\prec H_{\theta}$ and every $X\in[H_{\lambda}]^{\omega_{1}},$ there exists
$Y\in[H_{\lambda}]^{\omega_{1}}$ such that

$X\subseteq Y\text{ and }Y\parallel_{\omega_{1}}M.$

######  Lemma 4.9.

The following three statements are equivalent.

  1. (a)

$\mathsf{SCC}^{\textrm{cof}},$

  2. (b)

for every regular $\theta\gg\omega_{2}$ and every $M\prec H_{\theta}$
countable, $P2$ has as a winning strategy in the $\omega_{2}$-capturing game
$\mathbf{G}^{cap}_{M}(\omega_{2})$ for $M,$

  3. (c)

letting $X=\omega_{2}\setminus\omega_{1},$ for every regular
$\theta\gg\omega_{2}$ and every $M\prec H_{\theta}$ countable, $P2$ has as a
winning strategy in the $X$-capturing game $\mathbf{G}^{cap}_{M}(X)$ for $M.$

###### Proof.

$\boxed{(a)\Rightarrow(b)}$ Suppose first $P1$ plays $\rho\in\omega_{1}.$ Use
$\mathsf{SCC}^{\textrm{cof}}$ to find $M\subseteq M^{*}\prec H_{\theta}$ such
that $M\cap\omega_{1}=M^{*}\cap\omega_{1}$ and such that there exists
$\xi^{*}\in M^{*}\cap\omega_{2}$ satisfying
$\operatorname{otp}(M^{*}\cap\xi^{*})=\rho.$ Choose inside $M^{*}$ a bijection
$g$ from $\omega_{1}$ onto $\xi^{*}.$ Let $f:\omega_{1}\to\omega_{1}$ be
defined by $f(\alpha)=\operatorname{otp}(g[\alpha]).$ Then
$f\parallel_{\omega_{1}}M$ and $f(\delta(M))=\rho.$
Suppose then that $P1$ plays $\rho\in\omega_{2}\setminus\omega_{1}.$ By
$\mathsf{SCC}^{\textrm{cof}}$, we can assume that there exists $\xi\in M$ such
that $\rho<\xi.$ Choose in $M$ a bijection $g$ from $\omega_{1}$ onto $\xi.$
Using the argument in the foregoing case, one finds
$f:\omega_{1}\to\omega_{1}$ such that $f\parallel_{\omega_{1}}M$ and
$f(\delta(M))=g^{-1}(\rho).$
$\boxed{(b)\Rightarrow(c)}$ Trivial.
$\boxed{(c)\Rightarrow(a)}$ Given $\eta\in X=\omega_{2}\setminus\omega_{1},$
it follows from $P2\nearrow\mathbf{G}^{cap}_{M}(X)$ that there exists
$f:\omega_{1}\to\omega_{2}$ such that $f\parallel_{\omega_{1}}M$ and
$f(\delta(M))=\eta.$ Then $\xi=\sup(f)$ satisfies $\xi>\eta$ and
$\xi\parallel_{\omega_{1}}M.$ ∎

######  Proposition 4.10.

The following two statements are equivalent.

  1. (a)

$\textrm{Glob-}\mathsf{SCC}^{\textrm{cof}},$

  2. (b)

for every regular $\theta\gg\omega_{2},$ every set $X\in H_{\theta}$ and every
$M\prec H_{\theta}$ with $X\in M,$ $P2$ has a winning strategy in the
$X$-capturing game $\mathbf{G}^{cap}_{M}(X)$ for $M.$

###### Proof.

$\boxed{(a)\Rightarrow(b)}$ $\textrm{Glob-}\mathsf{SCC}^{\textrm{cof}}$
implies that for every $x\in X$ there exists some $\rho<\omega_{1}$ and
$f:\omega_{1}\to\omega_{1}$ such that $f(\rho)=x$ and
$f\parallel_{\omega_{1}}M.$ $\textrm{Glob-}\mathsf{SCC}^{\textrm{cof}}$ also
implies $\mathsf{SCC}^{\textrm{cof}}$, which by Lemma 4.9 implies that there
exists a function $g:\omega_{1}\to\omega_{1}$ such that $g(\delta(M))=\rho$
and $g\parallel_{\omega_{1}}M.$
$\boxed{(b)\Rightarrow(a)}$ Given $\lambda\in M\prec H_{\theta}$ and
$X\in[H_{\lambda}]^{\omega_{1}},$ find a function
$g:\omega_{1}\to[H_{\lambda}]^{\omega_{1}}$ such that $g(\delta(M))=X$ and
$g\parallel_{\omega_{1}}M.$ Then $Y=\bigcup_{\alpha\in\omega_{1}}g(\alpha)$
satisfies

$|Y|=\aleph_{1},\quad X\subseteq Y,\quad Y\parallel_{\omega_{1}}M.$

∎

##  5\. The forcing $\mathbb{P}^{c\text{-}c}(\lambda,\mu)$

Throughout this section, fix two regular cardinals
$\omega_{1}\ll\lambda\ll\mu.$ We further write $\mathbf{G}^{c\text{-}c}_{M}$
for $\mathbf{G}^{c\text{-}c}_{M}(H_{\lambda}).$ If $z$ is a finite run of the
game $\mathbf{G}^{c\text{-}c}_{M},$ we write $P2(z)$ for the finite set of all
moves that have been made by $P2$ in $z.$ The conditions in the forcing
$\mathbb{P}^{c\text{-}c}(\lambda,\mu)$ will be finite sets of special triples,
we call those triples _building blocks_.

######  Definition 5.1.

A set $B\in H_{\mu}$ is called a _building block_ if it is a three-tuple
$B=(M,z,\Sigma)$, with:

  * •

$M\prec H_{\mu}$ countable with $\lambda\in M$ and such that
$P2\nearrow\mathbf{G}^{c\text{-}c}_{M},$

  * •

$\Sigma$ a winning strategy for $P2$ in $\mathbf{G}^{c\text{-}c}_{M}$,

  * •

$z$ a finite play of the game $\mathbf{G}^{c\text{-}c}_{M}$ in which $P2$
follows $\Sigma$.

######  Notation 5.2.

Let $B=(M,z,\Sigma)$ and $B^{\prime}=(M^{\prime},z^{\prime},\Sigma^{\prime})$
be building blocks. We define

  * •

$\text{Ext}(B):=\operatorname{Hull}(M,P2(z))\prec H_{\mu},$

  * •

$B\sqsubseteq_{\text{game}}B^{\prime}$ holds if and only if
$M=M^{\prime},\Sigma=\Sigma^{\prime}$ and $z\subseteq z^{\prime}.$

We are now ready to define the main forcing
$\mathbb{P}^{c\text{-}c}(\lambda,\mu)$.

######  Definition 5.3.

$p$ is a $\mathbb{P}^{c\text{-}c}(\lambda,\mu)$-_condition_ if

  * •

$p$ is a finite set of building blocks,

  * •

for every two building blocks $B_{1},B_{2}\in p$,

${\text{Ext}(B_{1})\ni B_{2}\text{ or }B_{1}=B_{2}\text{ or
}B_{1}\in\text{Ext}(B_{2})}.$

The order on $\mathbb{P}^{c\text{-}c}(\lambda,\mu)$ is defined as follows. If
$p,q\in\mathbb{P}^{c\text{-}c}(\lambda,\mu)$, then

${q\leq p\iff(\forall B\in p)(\exists B^{\prime}\in
q)B\sqsubseteq_{\text{game}}B^{\prime}}.$

In analysing the forcing $\mathbb{P}^{c\text{-}c}(\lambda,\mu)$ it will be
convenient to use the following additional notation.

######  Notation 5.4.

  * •

Call a building block $B$ an $M$-block if and only if $B=(M,z,\Sigma)$ for
certain $z,\Sigma,$

  * •

let $\lessdot$ be the strict partial order defined on building blocks by

$B_{1}\lessdot B_{2}\iff B_{1}\in\operatorname{Ext}(B_{2}).$

######  Lemma 5.5.

Suppose $B=(M,z,\Sigma)$ is a building block and $p^{*}$ is a
$\mathbb{P}^{c\text{-}c}(\lambda,\mu)$-condition that contains $B$ as an
element. If $\mu\ll\theta$ regular and $N\prec H_{\theta}$ is countable with
$\lambda,\mu\in N$ and satisfies

$N\cap H_{\mu}=M,$

then $p^{*}$ is strongly
$(N,\mathbb{P}^{c\text{-}c}(\lambda,\mu))$-semigeneric.

###### Proof.

To check that $p^{*}$ is strongly
$(N,\mathbb{P}^{c\text{-}c}(\lambda,\mu))$-semigeneric, consider a
$\mathbb{P}^{c\text{-}c}(\lambda,\mu)$-condition $q\leq p^{*}$ and define

$\operatorname{tr}(q|N)=q\cap\operatorname{Ext}(B_{M}^{q})\in\mathbb{P}^{c\text{-}c}(\lambda,\mu),$

where $B_{M}^{q}$ is the unique $M$-building block that belongs to $q.$ From
$\operatorname{tr}(q|N)\in\operatorname{Ext}(B^{q}_{M})$ follows

$\operatorname{tr}(q|N)\parallel_{\omega_{1}}N.$

Suppose then that
$r\in\mathbb{P}^{c\text{-}c}(\lambda,\mu)\cap\operatorname{Hull}(N,\operatorname{tr}(q|N))$
and $r\leq\operatorname{tr}(q|N).$ To prove that $r\parallel q,$ one checks
that $r\cup(q\setminus\operatorname{tr}(q|N))$ is a
$\mathbb{P}^{c\text{-}c}(\lambda,\mu)$-condition. This follows because for
every two building blocks $B_{1}\in r$ and $B_{2}\in
q\setminus\operatorname{tr}(q|N),$ either $B_{1}\lessdot B_{M}^{q}=B_{2}$ or
$B_{1}\lessdot B_{M}^{q}\lessdot B_{2}.$ ∎

The foregoing lemma allows to follow the well known side condition pattern for
proving (semi)properness: _to find $p^{*}\leq p$ that is a (semi)generic
condition for $M,$ simply add $M$ on top of $p$._

######  Proposition 5.6.

If $\mathsf{NS}$ is precipitous, then $\mathbb{P}^{c\text{-}c}(\lambda,\mu)$
is strongly ssp.

###### Proof.

Let $\theta\gg\mu$ be regular. Using Proposition 4.4, it suffices to check
that $\mathbb{P}^{c\text{-}c}(\lambda,\mu)$ is strongly ssp with respect to
every countable model $M\prec H_{\theta}$ that contains $\lambda,\theta$ and
satisfies $P2\nearrow\mathbf{G}^{c\text{-}c}_{M}.$ Note that in this case also
$P2\nearrow\mathbf{G}^{c\text{-}c}_{M\cap H_{\mu}}.$ Thus, if $p\in
M\cap\mathbb{P}^{c\text{-}c}(\lambda,\mu),$ choose $\Sigma_{M}$ such that
$B_{M}=(M\cap H_{\mu},\emptyset,\Sigma_{M})$ is a building block, define
$p^{*}=p\cup\\{B_{M}\\}$ and note that
$p^{*}\in\mathbb{P}^{c\text{-}c}(\lambda,\mu).$ By Lemma 5.5, $p^{*}$ is
strongly $(M,\mathbb{P}^{c\text{-}c}(\lambda,\mu))$-semigeneric. ∎

Throughout the rest of this section, we assume that $\mathsf{NS}$ is
precipitous and thus that $\mathbb{P}^{c\text{-}c}(\lambda,\mu)$ is strongly
ssp. Under this assumption, we analyse the generic object.

######  Definition 5.7.

Let $G$ be a $\mathbb{P}^{c\text{-}c}(\lambda,\mu)$-filter generic over $V.$

  * •

Define

$C^{G}=\\{\delta(M):(\exists p\in G)(\exists B\in p)\ B=(M,z,\Sigma)\text{ for
certain }z,\Sigma\\}.$

Clearly, $C^{G}$ is unbounded in $\omega_{1},$ so

  * •

let $(\delta_{\alpha}^{G}:{\alpha<\omega_{1}})$ be the increasing enumeration
of $C^{G}$.

  * •

Define for every $\alpha<\omega_{1}$,

$M_{\alpha}^{G}=\bigcup_{p\in G}\\{\operatorname{Ext}(B)\cap
H_{\lambda}^{V}:B\in p\text{ is an }M\text{-block and
}M\cap\omega_{1}=\delta_{\alpha}^{G}\\}.$

Note that inside $V[G],$ $M_{\alpha}^{G}$ is a countable elementary submodel
of $H^{V}_{\lambda}$. Countability of $M_{\alpha}^{G}$ follows because for
every two $p,q\in G,$ and every $(M_{1},z_{1},\Sigma_{1})=B_{1}\in p$,
$(M_{2},z_{2},\Sigma_{2})=B_{2}\in q$:

$M_{1}\cap\omega_{1}=M_{2}\cap\omega_{1}\quad\Rightarrow\quad
B_{1}\sqsubseteq_{\text{game}}B_{2}\text{ or
}B_{2}\sqsubseteq_{\text{game}}B_{1}.$

######  Proposition 5.8.

The sequence $(M_{\alpha}^{G}:\alpha<\omega_{1})$ is an iteration-inducing
factorisation of $H_{\lambda}^{V}.$

###### Proof.

This is checked through the following series of density arguments.

######  Claim 5.8.1.

$(M_{\alpha}^{G}:\alpha<\omega_{1})$ is a continuous sequence of countable
elementary substructures of $H_{\lambda}^{V}$.

###### Proof.

Suppose $\beta<\omega_{1}$ is limit, then

$\bigcup_{\alpha<\beta}M^{G}_{\alpha}=M^{G}_{\beta}$

because
$\boxed{\subseteq}$ if $\alpha<\beta$ and $x\in M^{G}_{\alpha}$, then there
exists $p\in G$ with $(M_{1},z_{1},\Sigma_{1}),(M_{2},z_{2},\Sigma_{2})\in p$
such that $M_{1}\cap\omega_{1}=\delta_{\alpha}^{G}$,
$M_{2}\cap\omega_{1}=\delta_{\beta}^{G}$ and
$x\in\operatorname{Hull}(M_{1},P2(z_{1}))\cap H_{\lambda}^{V}.$ Then
$(M_{1},z_{1},\Sigma_{1})\lessdot(M_{2},z_{2},\Sigma_{2}),$ so
$x\in\operatorname{Hull}(M_{2},P2(z_{2}))\cap H_{\lambda}^{V}$.
$\boxed{\supseteq}$ if $x\in M^{G}_{\beta}$, then there exists $p\in G$ and
$B_{1},B_{2}\in p$ such that

$\displaystyle B_{1}$ $\displaystyle=\max\\{B\in p:B\lessdot B_{2}\\},$
$\displaystyle B_{2}$
$\displaystyle=(M_{2},z_{2},\Sigma_{2}),M_{2}\cap\omega_{1}=\delta_{\beta}^{G},\
x\in\operatorname{Ext}(B_{2}),$ $\displaystyle B_{1}$
$\displaystyle=(M_{1},z_{1},\Sigma_{1}),M_{1}\cap\omega_{1}=\delta_{\alpha}^{G}\
(\text{for some }\alpha<\beta).$

Consider the play $z_{1}^{\prime}$ which is an extension of $z_{1}$ with one
round in which $P1$ plays $x$ as a $\mathbf{G}^{cap}$-move and $P2$ answers
following $\Sigma_{1}$. Then
$p^{\prime}=p\setminus\\{B_{1}\\}\cup\\{(M_{1},z_{1}^{\prime},\Sigma_{1})\\}\in\mathbb{P}^{c\text{-}c}(\lambda,\mu)$
forces that $x\in M^{G}_{\alpha+1}$. ∎

######  Claim 5.8.2.

Every $M^{G}_{\alpha}$ is selfgeneric.

###### Proof.

Suppose that $p\in\mathbb{P}^{c\text{-}c}(\lambda,\mu)$ with $(M,z,\Sigma)\in
p$, $M\cap\omega_{1}=\delta_{\alpha}^{G}$ and $\mathcal{A}$ is a maximal
antichain for $\mathsf{NS}^{+}$ such that
$\mathcal{A}\in\operatorname{Hull}(M,P2(z))$. Then consider the play
$z^{\prime}$ which is an extension of $z$ with one round in which $P1$ plays
$\mathcal{A}$ as a $\mathbf{G}^{cat}$-move and $P2$ answers following
$\Sigma$, say the answer is $S\in\mathcal{A}$. Then
$p^{\prime}=p\setminus\\{(M,z,\Sigma)\\}\cup\\{(M,z^{\prime},\Sigma)\\}\in\mathbb{P}^{c\text{-}c}(\lambda,\mu)$
forces that $S\in M^{G}_{\alpha}\cap\mathcal{A}$ and
$M^{G}_{\alpha}\cap\omega_{1}=\delta_{\alpha}^{G}\in S$. ∎

######  Claim 5.8.3.

$M^{G}_{\alpha+1}=\operatorname{Hull}(M^{G}_{\alpha},\delta_{\alpha}^{G})$.

###### Proof.

$\boxed{\supseteq}$ is already clear.
$\boxed{\subseteq}$ Suppose that $p\in\mathbb{P}^{c\text{-}c}(\lambda,\mu)$
with $(M,z,\Sigma),(M^{\prime},z^{\prime},\Sigma^{\prime})\in p$,
$M\cap\omega_{1}=\delta_{\alpha}^{G}$,
$M^{\prime}\cap\omega_{1}=\delta_{\alpha+1}^{G}$ and
$x\in\operatorname{Hull}(M^{\prime},P2(z^{\prime}))\cap H_{\lambda}^{V}$ then
consider the play $\tilde{z}$ which is an extension of $z$ with one round in
which $P1$ plays $x$ as a $\mathbf{G}^{cap}$-move and $P2$ answers following
$\Sigma$. Say this answer of $P2$ is the function $f$. Then
$\tilde{p}=p\setminus\\{(M,z,\Sigma)\\}\cup\\{(M,\tilde{z},\Sigma)\\}\in\mathbb{P}^{c\text{-}c}(\lambda,\mu)$
forces that
$x=f(\delta^{G}_{\alpha})\in\operatorname{Hull}(M^{G}_{\alpha},\delta_{\alpha}^{G}).$
∎

∎

######  Definition 5.9.

For every $\alpha<\omega_{1},$ define $\mathbf{m}_{\alpha}^{G}$ to be the
transitive collapse of $M_{\alpha}^{G}.$

We now put the foregoing together.

######  Theorem 5.10.

Suppose $\mathsf{NS}$ is precipitous. If $G$ is a
$\mathbb{P}^{c\text{-}c}(\lambda,\mu)$-filter generic over $V,$ then in $V[G]$
there exists a generic iteration
${\mathbf{It}}=(\mathfrak{m}_{\beta},j_{\alpha\beta},G_{\alpha}:\alpha<\beta<\omega_{1})$
of length $\omega_{1}$ of the model $\mathbf{m}_{0}^{G}$ with direct limit
equal to $H_{\lambda}^{V}.$ Moreover, if $H_{\lambda}^{\\#}$ exists in $V,$
then the model $\mathbf{m}_{0}^{G}$ is iterable.

###### Proof.

This follows from Proposition 5.8 together with Lemma 2.5. Iterability of
$\mathbf{m}_{0}^{G}$ follows from Lemma 2.3. ∎

By combining Theorem 5.10 and Woodin’s Lemma 2.6, one derives the following
conclusions. Note in the first point of Corollary 5.11 that the
$\mathbb{P}^{c\text{-}c}(\lambda,\mu)$-extension is closed under sharps for
reals because of Lemma 2.7. In the third point of Corollary 5.11, we reobtain
a result of Claverie and Schindler ([2, Corollary 21]), it is implied by the
second point of Corollary 5.11 together with Schindler’s theorem that
$\mathsf{BMM}$ implies the existence of sharps for all sets (see [13, Theorem
1.3]).

######  Corollary 5.11.

Suppose $\mathsf{NS}$ is precipitous.

  1. (1)

If $H_{\theta}^{\\#}$ exists for $\theta=(2^{<\mu})^{+}$, then

$\mathbb{P}^{c\text{-}c}(\lambda,\mu)\Vdash\mathbf{u}_{2}>\lambda.$

  2. (2)

If $H_{\lambda}^{\\#}$ exists and the bounded forcing axiom for the forcing
$\mathbb{P}^{c\text{-}c}(\lambda,\mu)$ holds,

$\text{ then }\mathbf{u}_{2}=\omega_{2}.$

  3. (3)

[2, Corollary 21] If Bounded Martin’s Maximum $\mathsf{BMM}$ holds, then
$\mathbf{u}_{2}=\omega_{2}.$

######  Remark 5.12.

  1. (1)

By [5, Theorem 5.7], the forcing $\mathbb{P}^{c\text{-}c}(\lambda,\mu)$ is
semiproper if and only if the principle $(\dagger)$ holds, that is
$\mathbb{P}^{c\text{-}c}(\lambda,\mu)$ is semiproper if and only if every ssp
forcing is semiproper. Also by [5, Theorem 5.7], $(\dagger)$ is equivalent to
$\textrm{Glob-}\mathsf{SCC}^{\textrm{cof}}$ which by Proposition 4.10 is
equivalent to the existence of club many countable $M\prec H_{\theta}$ for all
$\theta\gg\lambda$ such that
$P2\nearrow\mathbf{G}^{c\text{-}c}_{M}(H_{\lambda}).$

  2. (2)

It is interesting to compare the forcing
$\mathbb{P}^{c\text{-}c}(\lambda,\mu)$ with the forcings from [9] and [2]; and
although all three are very similar, the exact relation between these forcings
remains largely open. We note that the model $M_{0}^{G},$ added by
$\mathbb{P}^{c\text{-}c}(\lambda,\mu),$ has the same strong properties as the
countable family of functions that occurs on the generic branch of the Namba
forcing in [9]. Namely:

     * •

every countable elementary submodel of $H_{\lambda}^{V}$ that contains
$M_{0}^{G}$ as a subset is selfgeneric,

     * •

$M_{0}^{G}$ contains as a subset a countable set of functions
$\\{g_{n}:n<\omega_{1}\\}$ from $\omega_{1}^{<\omega}$ to $H_{\lambda}$ such
that $H_{\lambda}=\bigcup_{n<\omega}\operatorname{ran}(g_{n}).$

The second of these two points is already clear from the foregoing, the first
one is slightly more surprising and can be checked through a more elaborate
density argument.

  3. (3)

The proof of [5, Lemma 2.1] can directly be adapted to show that if
$\mathsf{NS}$ is precipitous, $H_{\theta}^{\\#}$ exists for
$\theta=(2^{<\mu})^{+}$, and the bounded forcing axiom for the forcing
$\mathbb{P}^{c\text{-}c}(\lambda,\mu)$ holds, then also the admissible club
guessing principle $\mathsf{ACG}$ holds. $\mathsf{ACG}$ states that the club-
filter on $\omega_{1}$ is generated by sets of the form
$A_{x}=\\{\alpha<\omega_{1}:L_{\alpha}[x]\vDash\mathsf{KP}\\}$ with
$x\in\mathbb{R}.$

######  Acknowledgements.

BV is supported by an Emergence en recherche grant from the Université Paris
Cité (IdeX). BDB is supported by a Cofund MathInParis PhD fellowship. This
project has received funding from the European Union’s Horizon 2020 research
and innovation programme under the Marie Skłodowska-Curie grant agreement No.
754362.

## References

  * [1] David Asperó and Ralf Schindler, _Martin’s Maximum ++ implies Woodin’s axiom $(\ast)$_, Ann. Math. 193 (2021), 793–835.
  * [2] Benjamin Claverie and Ralf Schindler, _Increasing $u_{2}$ by a stationary set preserving forcing_, The Journal of Symbolic Logic 74 (2009), no. 1, 187–200.
  * [3] Sean Cox, _Adjoining only the things you want: a survey of Strong Chang’s Conjecture and related topics_ , To appear in Research Trends in Contemporary Logic (2020).
  * [4] Sean Cox and Martin Zeman, _Ideal projections and forcing projections_ , The Journal of Symbolic Logic 79 (2014), no. 4, 1247–1285.
  * [5] Philipp Doebler and Ralf Schindler, _$\Pi_{2}$ consequences of $\mathsf{BMM}+\mathsf{NS}_{\omega_{1}}$ is precipitous and the semiproperness of all stationary set preserving forcings_, Math. Res. Letters 16 (2009), 797–815.
  * [6] Qi Feng, Thomas Jech, and Jindřich Zapletal, _On the structure of stationary sets_ , Sci China Ser A 50 (2007), no. 4, 615–627.
  * [7] Matthew Foreman and Akihiro Kanamori, _Handbook of set theory_ , Springer, Dordrecht, 2010.
  * [8] Ronald Jensen, _Subcomplete forcing and $\mathcal{L}$-forcing_, E-recursion, Forcing And C∗-algebras, Lecture notes series, Institute for Mathematical Sciences, National University of Singapore, vol. 27, 2012, pp. 83–181.
  * [9] Richard Ketchersid, Paul Larson, and Jindřich Zapletal, _Increasing $\delta_{2}^{1}$ and Namba-Style Forcing_, Journal of Symbolic Logic 72 (2007), no. 4, 1372–1378.
  * [10] Donald A. Martin, _Projective sets and cardinal numbers: some questions related to the continuum problem_ , Lecture Notes in Logic, p. 484–508, Cambridge University Press, 2011.
  * [11] William J. Mitchell, _Adding closed unbounded subsets of $\omega_{2}$ with finite forcing_, Notre Dame Journal of Formal Logic 46 (2005), no. 3, 357–371.
  * [12] Yiannis N. Moschovakis, _Determinacy and prewellorderings of the continuum_ , Mathematical Logic and Foundations of Set Theory (Yehoshua Bar-Hillel, ed.), Studies in Logic and the Foundations of Mathematics, vol. 59, Elsevier, 1970, pp. 24–62.
  * [13] Ralf Schindler, _Semi-proper forcing, remarkable cardinals, and Bounded Martin’s Maximum_ , Mathematical Logic Quarterly 50 (2004), no. 6, 527–532.
  * [14] Philipp Schlicht, _Thin equivalence relations and inner models_ , Annals of Pure and Applied Logic 165 (2014), no. 10, 1577–1625.
  * [15] John R. Steel and Philip D. Welch, _$\Sigma^{1}_{3}$ absoluteness and the second uniform indiscernible_, Israel Journal of Mathematics 104 (1998), 157–190.
  * [16] John R. Steel and Robert Van Wesep, _Two consequences of determinacy consistent with choice_ , Transactions of the American Mathematical Society 272 (1985), 67–85.
  * [17] Stevo Todorcevic, _A note on the proper forcing axiom_ , Axiomatic set theory (Providence, Rhode Island), Contemp. Math., vol. 31, AMS, 1984, pp. 209–218.
  * [18] by same author, _Directed sets and cofinal types_ , Trans. Amer. Math. Soc. 290 (1985), no. 2, 711–723.
  * [19] by same author, _Partition problems in topology_ , Contemporary Mathematics, vol. 84, AMS, 1989.
  * [20] by same author, _Irredundant sets in Boolean algebras_ , Trans. Amer. Math. Soc. 339 (1993), no. 1, 35–44.
  * [21] W. Hugh Woodin, _The axiom of determinacy, forcing axioms, and the nonstationary ideal_ , De Gruyter, Berlin, New York, 2010.