# Dispersive shocks in diffusive-dispersive approximations of elasticity and
quantum-hydrodynamics

Daria Bolbot
Computer, Electrical and Mathematical Science and Engineering Division
King Abdullah University of Science and Technology (KAUST)
Thuwal 23955-6900, Saudi Arabia<EMAIL_ADDRESS>,  Dimitrios
Mitsotakis
Victoria University of Wellington
School of Mathematics and Statistics
PO Box 600
Wellington 6140
New Zealand<EMAIL_ADDRESS>and  Athanasios E. Tzavaras
Computer, Electrical and Mathematical Science and Engineering Division
King Abdullah University of Science and Technology (KAUST)
Thuwal 23955-6900, Saudi Arabia<EMAIL_ADDRESS>To
Constantine Dafermos, who keeps inspiring us, with friendship and admiration

###### Abstract.

The aim is to assess the combined effect of diffusion and dispersion on shocks
in the moderate dispersion regime. For a diffusive dispersive approximation of
the equations of one-dimensional elasticity (or p-system), we study
convergence of traveling waves to shocks. The problem is recast as a
Hamiltonian system with small friction, and an analysis of the length of
oscillations yields convergence in the moderate dispersion regime
$\varepsilon,\delta\to 0$ with $\delta=o(\varepsilon)$, under hypotheses that
the limiting shock is admissible according to the Liu E-condition and is not a
contact discontinuity at either end state. A similar convergence result is
proved for traveling waves of the quantum hydrodynamic system with artificial
viscosity as well as for a viscous Peregrine-Boussinesq system where traveling
waves model undular bores, in all cases in the moderate dispersion regime.

##  1\. Introduction

Systems exhibiting interplay of diffusion, dispersion and nonlinear response
have been extensively studied in the field of conservation laws, starting with
works on the subject of phase transitions and undercompressive shocks, e.g.
[23, 21, 14, 1, 2]. An alternative perspective arose from the field of
nonlinear dispersive equations with the objective to study dispersive or
dissipative-dispersive shocks [4, 9]. Similar problems appear in a variety of
fluid mechanics settings, like shallow water flows [6, 8], undular waves in
atmospheric flows or water waves [19, 5]. Peregrine [18] introduced weakly
nonlinear and dispersive wave equations in order to study undular bores, a
wave appearing in rivers, atmospheric flows and also in blood vessels composed
of a solitary wave followed by undulations.

The Burgers-Korteweg de Vries (KdV) equation

$u_{t}+f(u)_{x}=\varepsilon u_{xx}-\delta u_{xxx}\ ,$ (1)

has been a testing ground for assessing the interplay of diffusion, dispersion
and nonlinearity. Various perspectives of study exist: (a) to assess the
effect of dispersion in the KdV or modified KdV equation ($\varepsilon=0$) on
expanding wavetrain solutions connecting two constant states arising via
Whitham modulation theory and termed in the field of dispersive equations as
dispersive shock waves; (b) to assess the limiting behavior of traveling wave
solutions when both diffusion and dispersion are present. We refer to [9] for
an in depth presentation of these viewpoints and their relation. Here, we
focus on a specific aspect of (b), relevant from the viewpoint of systems of
conservation laws, namely how oscillatory traveling waves for diffusive-
dispersive systems of two conservation laws approach shocks in the limit
$\varepsilon,\delta\to 0$.

Traveling wave solutions have been a focal point for assessing the interplay
of diffusive-dispersive systems with studies for the Burgers-KdV equation with
$f(u)=u^{2}$ [4], the modified Burgers-KdV equation with $f(u)=u^{3}$ [14],
diffusive-dispersive approximations of elasticity [12, 3, 1] or hyperbolic-
elliptic models for phase transitions [23, 2]. Comprehensive presentations can
be found in [17, 10]. The related convergence results from shock profiles to
shock waves generally hold in the weak dispersion regime
$\delta=O(\varepsilon^{2})$. Based on such studies and related convergence
results from (1) to the inviscid Burgers equation [22, 13] it was believed for
a while that $\delta=O(\varepsilon^{2})$ might be the threshold for
convergence to Kruzhkov entropy solutions. This was refuted in [20] where
traveling wave solutions for genuinely nonlinear Burgers-KdV equations were
shown to converge to shocks in the range $\delta=o(\varepsilon)$. In that
range traveling waves present relatively strong oscillatory behavior and
dispersive effects are significant, hence it was termed moderate dispersion
regime.

The aim here is to examine the convergence of diffusive-dispersive traveling
waves for systems of conservation laws in the moderate dispersion regime. We
note that the use of genuine nonlinearity is avoided and replaced by the Liu
shock admissibility condition and a requirement that the shock is not a (right
or left) contact discontinuity. The analysis is developed for the system of
elasticity and extended to other situations where diffusive-dispersive effects
play a role: the quantum hydrodynamic system with diffusion and to diffusive-
dispersive models modeling undular bores.

The specific cases analyzed are the following: We first consider a diffusive-
dispersive approximation of the elasticity system

$\displaystyle u_{t}$ $\displaystyle=v_{x}\ ,$ (2) $\displaystyle v_{t}$
$\displaystyle=\sigma(u)_{x}+\varepsilon v_{xx}-\delta u_{xxx}\ ,$

where $(t,x)\in\mathbb{R_{+}}\times\mathbb{R}$ and $\varepsilon,\delta>0$. The
hyperbolic part of (2) is known as the p-system and is expressed in Lagrangian
coordinates. The hyperbolicity condition $\sigma^{\prime}(u)>0$ is employed
throughout but use of genuine nonlinearity is avoided. Existence for traveling
waves and convergence to shocks in the range $\delta=O(\varepsilon^{2})$
appear in [12, 3, 1] using a dynamical systems approach. The convergence from
traveling waves to shock waves is extended here to the moderate dispersion
regime $\delta=o(\varepsilon)$ – as contrasted to the weak dispersion regime
$\delta=O(\varepsilon^{2})$ – for a shock satisfying the Liu E-condition and
avoiding contact discontinuities at the end states. This convergence covers
the regime of moderate dispersion where oscillations have a significant
presence. Our analysis does not cover undercompressive shocks or non-monotone
stress-strain relations appearing in phase transitions or Van der Waals
fluids; we refer to [10, 17] and references therein for reviews of those
subjects.

Next, consider the Quantum hydrodynamics system with artificial viscosity

$\displaystyle\rho_{t}+j_{x}$ $\displaystyle=0\ ,$ (3) $\displaystyle
j_{t}+\left(\frac{j^{2}}{\rho}+\rho^{\gamma}\right)_{x}$
$\displaystyle=\varepsilon
j_{xx}+\delta\rho\left(\frac{\sqrt{\rho}_{xx}}{\sqrt{\rho}}\right)_{x}\ .$

Here, $\rho=\rho(t,x)>0$ denotes the density, $j=j(t,x)$ momentum,
$(t,x)\in\mathbb{R_{+}}\times\mathbb{R}$ while $\rho^{\gamma}$ stands for the
pressure with $\gamma\geq 1$. This system with $\varepsilon=0$ is used to
model semiconductors or superfluidity, the dispersive term
$\rho\left(\frac{\sqrt{\rho}_{xx}}{\sqrt{\rho}}\right)_{x}$ is called quantum
Bohm potential [26], while the term $\varepsilon j_{xx}$ with $\varepsilon>0$
models artificial viscosity. Traveling wave analysis and convergence to shock
waves in the regime $\delta=O(\varepsilon^{2})$ is performed in [15, 16]. It
is here improved by showing that diffusive-dispersive shock profiles converge
to shocks in the regime $\delta=o(\varepsilon)$.

The third example is the dissipative Peregrine-Boussinesq system

$\displaystyle\eta_{t}+u_{x}+(\eta u)_{x}=0\ ,$ (4) $\displaystyle
u_{t}+\eta_{x}+uu_{x}-\delta u_{xxt}-\varepsilon u_{xx}=0\ .$

We consider traveling wave solutions $(\eta(\tau),u(\tau))$,
$\tau=-\frac{x-st}{\sqrt{s\delta}}$ for $s>1$, under the limiting conditions

$\lim_{\tau\to-\infty}(\eta,u)=(\eta^{-},u^{-})=(0,0),\quad\lim_{\tau\to+\infty}(\eta,u)=(\eta^{+},u^{+})\
,$ (5)

Here $\eta$ is the elevation of the free surface, and $u$ the horizontal
velocity of the fluid measured at some height above a flat bottom. This model
has been used for the prediction of undular bores, [5], as a balance of
nonlinear shock formation of the shallow water equations and dispersive
effects of water waves. For some values of Froude number though, the
oscillations can disappear and classical shock waves can be formed. Existence
of traveling waves can be found in [5]; this result is complemented here by
convergence to shock waves in the regime $\delta=o(\varepsilon)$.

A key ingredient is the analysis of the length of the oscillatory tail
inspired by the approach of [20]. The traveling wave problem is recast as
Hamiltonian system with friction of size $c=\varepsilon/\sqrt{\delta}s$, with
$s$ the shock speed, like in [14, 20]. The regime of moderate dispersion
corresponds to small friction $c<c^{*}$, where $c^{*}$ is some critical
threshold. In contrast to [20] the genuine nonlinearity hypothesis is replaced
by the use of Liu E-condition for shock admissibility. The main result
concerning the size of oscillations is stated in Proposition 6. It is used to
show convergence of traveling wave solutions of (2) to the equations of
elasticity in the limit $\varepsilon,\delta\to 0$ with
$\delta=o(\varepsilon)$, see Theorem 2. The same methodology is applied to
show convergence from traveling waves to shocks for the quantum hydrodynamics
system with artificial viscosity (3) and a similar result for the Peregrine-
Boussinesq system with viscosity (4); in both cases in the regime
$\delta=o(\varepsilon)$.

The manuscript is organized as follows: In Section 2 the traveling wave
problem for the diffusive-dispersive regularization of the elasticity system
(2) is reduced to a Hamiltonian system with friction (26)–(27). Moderate
dispersion leads to the study of a regime of weak friction, carried out in
Section 3. The length of the oscillatory tail for traveling wave solutions is
estimated in Proposition 6. As a corollary, convergence from traveling waves
to shocks for (2) holds for $\delta=o(\varepsilon)$, stated in Theorem 2. In
Section 4, the quantum hydrodynamic system with viscosity is considered for
genuinely nonlinear pressures. The problem of traveling waves is recast in the
form of the problem (26)–(27), and convergence to Lax shocks is shown in the
moderate dispersion regime $\delta=o(\varepsilon)$, see Theorem 7. The
dissipative Peregrine-Boussinesq system (4) is studied in section 5 and
convergence in the moderate dispersion regime is again based in Proposition 6.

##  2\. Diffusion-dispersion approximation of the elasticity system

We consider a diffusive-dispersive approximation for the one-dimensional
elasticity system

$\displaystyle u_{t}$ $\displaystyle=v_{x}\ ,$ (6) $\displaystyle v_{t}$
$\displaystyle=(\sigma(u))_{x}+\varepsilon v_{xx}-\delta u_{xxx}\ ,$

where $(t,x)\in\mathbb{R_{+}}\times\mathbb{R}$ are the time and space
variables, $u$ is the strain, $v$ the velocity and $\sigma(u)$ the stress
depending on $u$. The parameters $\varepsilon>0$, $\delta>0$ in (6) measure
respectively the sizes of diffusion and dispersion. Throughout this work we
assume that $\sigma^{\prime}(u)>0$.

###  2.1. Preliminaries: Shocks for the p-system

The system (6) is a regularization of the equations of one-dimensional
nonlinear elasticity, also called $p$-system:

$\displaystyle\partial_{t}u$ $\displaystyle=\partial_{x}v\,,$ (7)
$\displaystyle\partial_{t}v$ $\displaystyle=\partial_{x}\sigma(u)\;.\ $

There are interpretations of (7) representing both longitudinal and shear
motions; for longitudinal motions $u>0$ is interpreted as the longitudinal
strain, for shear motions $u\in\mathbb{R}$ is a shear strain. When $u$
represents longitudinal motions $\varepsilon v_{xx}$ needs to be replaced by
$(\frac{\varepsilon}{u}v_{x})_{x}$ but we will not consider such issues here
focusing on the essential behaviors.

When $\sigma^{\prime}(u)>0$, the system (7) is strictly hyperbolic with wave
speeds $\lambda_{1}=-\sqrt{\sigma^{\prime}(u)}$,
$\lambda_{2}=\sqrt{\sigma^{\prime}(u)}$. Shocks are generated by solving the
Rankine-Hugoniot conditions

$\displaystyle-s(u_{+}-u_{-})$ $\displaystyle=(v_{+}-v_{-})\ ,$ (8)
$\displaystyle-s(v_{+}-v_{-})$ $\displaystyle=(\sigma(u_{+})-\sigma(u_{-}))\
,$

where $s$ is the shock speed and $(u_{-},v_{-})$, $(u_{+},v_{+})$ the left and
right states, respectively. The shock speed is computed by

$s^{2}=(\sigma(u_{+})-\sigma(u_{-}))/(u_{+}-u_{-})\;,$ (9)

and (7) admits two types of shocks: 1-shocks with $s<0$ moving backwards and
2-shocks with $s>0$ moving forward. When $\sigma^{\prime\prime}(u)\neq 0$ the
system is called genuinely nonlinear; this assumption is not, in general,
adopted here and $\sigma(u)$ will be allowed to have inflection points.

Admissibility conditions are imposed on shocks, motivated by either stability
considerations or from requesting that admissible shocks emerge as limits of
traveling waves for viscosity regularizations. We refer to [7, Ch VIII] for an
in depth discussion of shock-admissibility criteria and [24, Ch 18] for the
construction of shock curves and the solution of the Riemann problem for (7).

For (7), under the hyperbolicity assumption $\sigma^{\prime}(u)>0$, the usual
admissibility criteria are:

  * (i)

For genuinely nonlinear systems $\sigma^{\prime\prime}(u)\neq 0$ admissible
shocks are selected by the Lax-shock admissibility criterion, stating that
admissible 1-shocks ($s<0$) satisfy

$-\sqrt{\sigma^{\prime}(u_{+})}<s<-\sqrt{\sigma^{\prime}(u_{-})}\;,$ (10)

while admissible 2-shocks ($s>0$) satisfy

$\sqrt{\sigma^{\prime}(u_{+})}<s<\sqrt{\sigma^{\prime}(u_{-})}\;.$ (11)

  * (ii)

When $\sigma^{\prime\prime}(u)$ changes sign admissible shocks of (7) are
selected by the Wendroff E-condition, which for 2-shocks dictates that
$\sigma$ satisfies

$\frac{\sigma(u)-\sigma(u_{-})}{u-u_{-}}\geq
s^{2}\geq\frac{\sigma(u)-\sigma(u_{+})}{u-u_{+}}\ ,$ (HE)

for any $u$ between $u_{-}$ and $u_{+}$. The Wendroff E-condition for 1-shocks
reads like (HE) with the inequalities reversed.

A discriminating criterion capturing the internal stability of shocks and used
for the solution of the Riemann problem for general fluxes is the Liu shock
admissibility criterion, [7, Sec 8.4]. At the level of the particular system
(7) the Liu shock admissibility criterion is equivalent to the Wendroff
E-condition. The reader can easily check that the latter ((HE) for $s>0$)
implies at the endpoints a weak version of the Lax inequality (11), where
strict inequality might be replaced by equality in which case the shock
becomes a (right or left) contact discontinuity.

###  2.2. Traveling waves

We look for a traveling wave solution
$(u^{\varepsilon,\delta},v^{\varepsilon,\delta})$ of (6) in the form

$\displaystyle u^{\varepsilon,\delta}(x,t)$
$\displaystyle=u\left(\frac{x-st}{\sqrt{\delta}}\right)=u(\tau)\;,$ (12)
$\displaystyle v^{\varepsilon,\delta}(x,t)$
$\displaystyle=v\left(\frac{x-st}{\sqrt{\delta}}\right)=v(\tau)\,,$

connecting states $(u_{-},v_{-})$ and $(u_{+},v_{+})$ that satisfy the
Rankine-Hugoniot conditions (8). Setting $\tau=\frac{x-st}{\sqrt{\delta}}$ and
retaining the notation $(u(\tau),v(\tau))$ for the traveling wave, we need to
solve a system of ordinary differential equations

$\displaystyle-su^{\prime}-v^{\prime}$ $\displaystyle=0\ ,$ (13)
$\displaystyle-sv^{\prime}-\sigma(u)^{\prime}$
$\displaystyle=\frac{\varepsilon}{\sqrt{\delta}}v^{\prime\prime}-u^{\prime\prime\prime}\,.$

Existence of traveling waves is a well studied problem; the objective is to
provide conditions that guarantee convergence of the traveling wave as
$\varepsilon,\delta\to 0$ to the associated shock of (7) in the moderate
dispersion regime $\delta=o(\varepsilon)$.

Consider the problem of constructing traveling wave solutions of (13)
satisfying

$\lim_{\tau\to\pm\infty}u=u_{\pm},\quad\lim_{\tau\to\pm\infty}v=v_{\pm},\quad\lim_{\tau\to\pm\infty}v^{\prime}=\lim_{\tau\to\pm\infty}u^{\prime\prime}=0\,.$

Integrating (13) in $(-\infty,\tau)$ leads to solve the boundary value problem

$\displaystyle
u^{\prime\prime}+\frac{s\varepsilon}{\sqrt{\delta}}u^{\prime}-\big{(}\sigma(u)-\sigma(u_{-})-s^{2}(u-u_{-})\big{)}$
$\displaystyle=0\ ,$ (14) $\displaystyle u(\pm\infty)=u_{\pm}$ $\displaystyle\
,$ (15)

and define $v$ via the equation

$v-v_{-}=-s(u-u_{-})\,.$ (16)

Necessary for the existence of traveling waves is that the end states satisfy
(8).

In summary, denoting $c=\frac{s\varepsilon}{\sqrt{\delta}}$, traveling waves
are constructed by solving the ordinary differential equation

$\displaystyle u^{\prime\prime}+cu^{\prime}+\phi(u)$ $\displaystyle=0\ ,$ (17)
$\displaystyle u(\pm\infty)$ $\displaystyle=u_{\pm}\ ,$ $\displaystyle\phi(u)$
$\displaystyle:=-\big{(}\sigma(u)-\sigma(u_{-})-s^{2}(u-u_{-})\big{)}\ .$

Following the approach for traveling waves of the KPP equation in [11] and for
the viscous Burgers-KdV equation in [20], Problem (17) is viewed as a
Hamiltonian system with friction, by setting $w=u^{\prime}$ and writing

$\displaystyle\frac{du}{d\tau}$ $\displaystyle=w\;,$ (18)
$\displaystyle\frac{dw}{d\tau}$ $\displaystyle=-\phi(u)-cw\;.$

Define the potential $\Phi(u)$ by $\frac{d\Phi}{du}=\phi(u)$ and
$\Phi(u_{+})=0$, namely

$\Phi(u)=\int^{u}_{u_{+}}\phi(z)\leavevmode\nobreak\
dz=-\int^{u}_{u_{+}}\big{(}\sigma(z)-\sigma(u_{-})-s^{2}(z-u_{-})\big{)}\leavevmode\nobreak\
dz\ .$ (19)

The energy of (18), $E(u,w)=\frac{w^{2}}{2}+\Phi(u)$, satisfies

$\frac{d}{d\tau}E(u(\tau),w(\tau))=-cw(\tau)^{2}\,.$ (20)

The associated Hamiltonian system of (18) (when $c=0$) evolves on orbits of
constant energy $\tfrac{1}{2}w^{2}+\Phi(u)=E$, and its trajectories are
identified by integrating the differential equations

$\frac{du}{d\tau}=\pm\sqrt{2(E-\Phi(u))}\ .$ (21)

###  2.3. Existence of traveling waves for $\varepsilon$, $\delta>0$ fixed

Existence and uniqueness (up to translation) results for traveling waves of
(6) have been presented by several authors: Hagan-Slemrod [12], Boldrini [3]
and Bedjaoui-Lefloch [1, 2]. Traveling waves for the viscous Burgers-KdV (1)
lead to the same problem and were constructed in [4]. These references show
existence for $\varepsilon$, $\delta>0$ fixed and convergence to shock waves
in the regime $\delta=O(\varepsilon^{2})$. An outline of existence is provided
in Theorem 1 following Hagan-Slemrod [12].

The main theme is the study of convergence of traveling waves to shocks in the
regime $o(\varepsilon)\leq\delta<O(\varepsilon^{2})$ where traveling waves
have oscillatory tails. We first review the framework of pertinent hypotheses
and their relation with shock-admissibility criteria, and then we prove
convergence to shock waves. We follow an approach devised in Perthame-Ryzhik
[20] for traveling waves of scalar viscous Burgers-KdV equations (1) extending
their analysis to systems (6) with no genuine nonlinearity assumptions.

####  2.3.1. Hypotheses on $\sigma(u)$.

We assume hyperbolicity $\sigma^{\prime}(u)>0$ and consider the general case
that $\sigma^{\prime\prime}(u)$ changes sign. For definiteness we restrict to
(forward moving) 2-shocks $s>0$ and states $u_{-}<u_{+}$. (A similar analysis
can be performed for (backward moving) 1-shocks $s<0$ with $u_{-}>u_{+}$.) We
require the shock satisfies the Lax shock condition

$\sqrt{\sigma^{\prime}(u_{+})}<s<\sqrt{\sigma^{\prime}(u_{-})}\,,\qquad\qquad\qquad$
(HL)

as well as the condition

$\sigma(u)-\sigma(u_{-})-s^{2}(u-u_{-})>0\quad\mbox{ for $u\in(u_{-},u_{+})$}\
.$ (HsE)

Assume also that $u_{s}>u_{+}$ is such that $\Phi(u_{s})=\Phi(u_{-})$ and that
no root of the function $\phi(u)$ exists in the interval $(u_{+},u_{s})$, that
is

$\sigma(u)-\sigma(u_{+})-s^{2}(u-u_{+})<0\quad\mbox{ for $u\in(u_{+},u_{s})$}\
.$ (HoE)

The analysis we present will also apply to 1-shocks $s<0$ with $u_{+}<u_{-}$
by imposing the Lax shock condition (10) and reversing the inequalities in
(HsE), (HoE).

Figure 1. Typical graph of $\sigma(u)$ Figure 2. Typical graph of $\Phi(u)$

The following remarks on the hypotheses are in order: Condition (HsE) is a
strengthened version of the Wendroff E-condition (HE). Indeed, (HsE) implies
the left inequality in (HE) as a strict inequality and, using (9), we obtain
the right inequality in (HE) again as a strict inequality. Hypothesis (HL)
excludes the possibility of having a contact discontinuity at the end-points
$u_{\pm}$ while (HsE) excludes a composite shock with an internal contact
discontinuity. (We remark that excluding contact discontinuities at the end-
points can conceivably be avoided by imposing assumptions on the order of
tangency between the shock and the graph $\sigma(u)$ at the states $u_{\pm}$;
we do not pursue that point here).

Hypothesis (HoE) ensures the potential function $\Phi(u)$ has no extrema other
than the critical points in the region $[u_{-},u_{s}]$. We refer to Figure 1
for a geometric interpretation of the position of the graph of $\sigma(u)$ and
to Figure 2 for the form of the potential $\Phi(u)$.

If $\sigma$ is genuinely nonlinear and for definiteness we focus on a concave
$\sigma$ and a forward moving shock $s>0$, then one easily checks that the
conditions

$\displaystyle\sigma^{\prime\prime}(u)$ $\displaystyle<0\ ,$ (Hgn)
$\displaystyle\sqrt{\sigma^{\prime}(u_{+})}<s$
$\displaystyle<\sqrt{\sigma^{\prime}(u_{-})}\ ,$ $\displaystyle u_{-}<u_{+}$
$\displaystyle\mbox{and}\,\,\,v_{-}>v_{+}\ ,$

imply the framework of hypotheses (HL), (HsE) and (HoE). (An analogous remark
holds for backward moving shocks $s<0$ with $u_{-}>u_{+}$.)

####  2.3.2. Existence of traveling waves.

Consider now the problem (18), where $s>0$ satisfies (9),
$c=\frac{s\varepsilon}{\sqrt{\delta}}>0$,

$\displaystyle\phi(u)$
$\displaystyle=-\big{(}\sigma(u)-\sigma(u_{-})-s^{2}(u-u_{-})\big{)}\ ,$ (22)
$\displaystyle\Phi(u)$
$\displaystyle=\int_{u}^{u_{+}}\big{(}\sigma(z)-\sigma(u_{-})-s^{2}(z-u_{-})\big{)}\leavevmode\nobreak\
dz\,.$

We adopt the hypotheses (HL), (HsE), (HoE) and prove the following theorem:

######  Theorem 1.

For $\varepsilon,\delta>0$ fixed there exists a unique up to horizontal
translations traveling wave solution $(u(\tau),v(\tau))$ of the form (12) to
the system (6), connecting the state $(u_{-},v_{-})$ on the left to the state
$(u_{+},v_{+})$ on the right.

###### Proof.

There are only two equilibrium points in the range $[u_{-},u_{s}]$ of the
system (18), namely $(u_{-},0)$, $(u_{+},0)$, and the linearized equations
around these equilibria become

$\frac{d}{d\tau}\begin{pmatrix}U\\\ W\end{pmatrix}=\begin{pmatrix}0&1\\\
-\phi^{\prime}(u_{\pm})&-c\end{pmatrix}\begin{pmatrix}U\\\ W\end{pmatrix}\ .$

Consider the equilibrium $(u_{-},0)$. The eigenvalues are computed by

$\lambda^{2}+c\lambda-\alpha_{-}=0\qquad\mbox{where
\quad$\alpha_{-}=-\phi^{\prime}(u_{-})=\sigma^{\prime}(u_{-})-s^{2}>0$}\ ,$

they are

$\lambda_{\pm}=-\tfrac{c}{2}\pm\tfrac{1}{2}\sqrt{c^{2}+4\alpha_{-}}\ ,$

both real and satisfy $\lambda_{-}<0<\lambda_{+}$ and thus $(u_{-},0)$ is a
saddle. The directions of the stable and unstable manifolds are given by the
two corresponding eigenvectors $r_{-}=\left(\begin{matrix}1\\\
\lambda_{-}\end{matrix}\right)$ for the unstable and
$r_{+}=\begin{pmatrix}1\\\ \lambda_{+}\end{pmatrix}$ for the stable manifold.
Later we will need the property that

$0<\lambda_{+}=:g(c)=-\tfrac{c}{2}+\tfrac{1}{2}\sqrt{c^{2}+4\alpha_{-}}<\sqrt{\alpha_{-}}\
,$ (23)

which is true because $g(c)$ satisfies $g^{\prime}(c)<0$,
$g^{\prime\prime}(c)>0$ and thus $g(c)<g(0)$.

Next for the equilibrium $(u_{+},0)$, the eigenvalues are computed by

$\Lambda^{2}+c\Lambda-\alpha_{+}=0\qquad\mbox{where
\quad$\alpha_{+}=-\phi^{\prime}(u_{+})=\sigma^{\prime}(u_{+})-s^{2}<0$}\ ,$

and they are

$\Lambda_{\pm}=-\tfrac{c}{2}\pm\tfrac{1}{2}\sqrt{c^{2}+4\alpha_{+}}\ .$

We distinguish two cases: (i) When $c^{2}>4|\alpha_{+}|$ there are two real
roots with $\Lambda_{-}<\Lambda_{+}<0$ and the equilibrium is a stable node.
(ii) By contrast, in the range of weak friction $c^{2}<4|\alpha_{+}|$ the
eigenvalues are complex

$\Lambda_{\pm}=-\tfrac{c}{2}\pm\tfrac{1}{2}i\sqrt{|c^{2}+4\alpha_{+}|}\ ,$

with negative real part and $(u_{+},0)$ is an attracting spiral.

Consider the region $\mathcal{D}$ bounded by the curves

$w^{2}=2\big{(}\Phi(u_{-})-\Phi(u)\big{)}=2\int_{u_{-}}^{u}\sigma(z)-\sigma(u_{-})-s^{2}(z-u_{-})\;dz\
.$ (24)

The curves are symmetric with respect to the $u$-axis. For $u\sim u_{-}$ we
compute that

$w^{2}\sim
2\int_{u_{-}}^{u}(\sigma^{\prime}(u_{-})-s^{2})(z-u_{-})\,dz=(\sigma^{\prime}(u_{-})-s^{2})(u-u_{-})^{2}\
,$

therefore

$\frac{dw}{du}(0)\sim\pm\left(\sqrt{\sigma^{\prime}(u_{-})-s^{2}}\right)(u-u_{-})\quad\mbox{
for $u>u_{-}$, $u\sim u_{-}$}\ .$

Next, in the range $u<u_{s}$ and $u\sim u_{s}$ we have

$\displaystyle w^{2}$
$\displaystyle=2\big{(}\Phi(u_{-})-\Phi(u)\big{)}=2\big{(}\Phi(u_{s})-\Phi(u)\big{)}$
$\displaystyle=-2\int_{u}^{u_{s}}\sigma(z)-\sigma(u_{-})-s^{2}(z-u_{-})\;dz\
.$

Observe that by (HoE) there exist $\alpha>0$, $A>0$ such that

$s^{2}-A<\frac{\sigma(u)-\sigma(u_{-})}{u-u_{+}}<s^{2}-\alpha\ ,$

and thus we can show that for $u\sim u_{s}$, $u<u_{s}$ we have

$\alpha(u_{s}-u)(u+u_{s}-2u_{+})<w^{2}<A(u_{s}-u)(u+u_{s}-2u_{+})\ ,$

which shows that the derivative $\frac{dw}{du}\sim\sqrt{u_{s}-u}$ in the
vicinity of $u\sim u_{s}$.

The domain $\mathcal{D}$ is enclosed by the curves
$\tfrac{1}{2}w^{2}+\Phi(u)=\Phi(u_{-})$ which is the homoclinic orbit of the
Hamiltonian system (18) with $c=0$. The normal to this curve is
$N=(\phi(u),w)$. Then we compute that along the flow of (18) it is

$\left(\frac{du}{d\tau},\frac{dw}{d\tau}\right)\cdot N=-cw^{2}\leq 0\ .$

The domain $\mathcal{D}$ is positively invariant along the flow of (18) and,
by (23), the unstable manifold of the linearized system at $(u_{-},0)$ points
inside $\mathcal{D}$ for $c>0$.

The heteroclinic orbit is constructed as follows. Pick a point on the unstable
manifold of (18) at $(u_{-},0)$ which for $u\sim u_{-}$ is inside
$\mathcal{D}$. The flow from this point backwards in time will converge to
$(u_{-},0)$. Going forward in time the flow cannot escape $\mathcal{D}$ and by
the Poincarè-Bendixon theorem it will converge to $(u_{+},0)$, giving the
desired heteroclinic orbit. This is a one-dimensional object and unique up to
time-shifts. ∎

Two regimes distinguish the behavior of the heteroclinic orbit. For
$c^{2}>4|\alpha_{+}|$ the orbit is monotone. For $0<c^{2}<4|\alpha_{+}|$ using
the stable manifold theorem the orbit has an oscillatory behavior (see [4]).
In the following section we study the size of oscillations of the orbit.

##  3\. The effect of weak friction on Hamiltonian systems

The aim of this section is to study the limit of traveling wave solutions of
(6) as $\varepsilon,\delta\to 0$. The technical vehicle is to study the
oscillatory behavior for solutions $(w_{c}(\tau),u_{c}(\tau))$ to (18) in the
regime of weak friction

$0<c<c^{\star}\quad\mbox{ where
$c^{\star}=2\sqrt{|\alpha_{+}|}=2\sqrt{s^{2}-\sigma^{\prime}(u_{+})}$}\ .$
(25)

The $c$-dependence of solutions will be suppressed except when necessary. We
prove the following theorem:

######  Theorem 2.

Under hypotheses (HL), (HsE), (HoE) and for $0<c<c^{\star}$ as in (25), there
exists a unique, up to translations, traveling wave solution
$(u(\tau),v(\tau))$ to (6) connecting $(u_{-},v_{-})$ on the left to
$(u_{+},v_{+})$ on the right. The solution converges strongly, as
$\varepsilon,\delta\to 0$ with $\delta=o(\varepsilon)$ to a shock wave for (7)
that satisfies the Wendroff E-condition (or the Liu shock admissibility
criterion).

The method of proof extends to the elasticity system an approach developed for
scalar genuinely nonlinear equations in [20]. It proceeds as follows:

  * (a)

We analyze the oscillatory behavior of the heteroclinic orbit $(u_{c},w_{c})$
of (18) in the range (25), by estimating the energy drop in each cycle and the
distance between the minima and maxima for small values of $c>0$. This is
presented in Sections 3.1 and 3.2 leading to an estimate of the size of the
oscillatory structure in Figure 3 as a function of $c$.

  * (b)

The information obtained in (a) is then translated at the level of traveling
wave solutions $(u^{\varepsilon,\delta}(\tau),v^{\varepsilon,\delta}(\tau))$
of (6) by means of rescaling.

The oscillatory behavior in this regime is illustrated by a numerical
computation for genuinely nonlinear stress $\sigma(u)=\sqrt{u}$ and critical
points $u_{-}=4$, $u_{+}=5$. Figure 3 presents the phase portrait of $(u,w)$
on the right and the form of $u(\tau)$ on the left for $c=0.004$, $u_{-}=4$,
$u_{+}=5$.

Figure 3. (a) Solution $u$; (b) Phase portrait $(u,w)$. ($\sigma(u)=\sqrt{u}$,
$c=0.004$, $u_{-}=4$, $u_{+}=5$)

Recall that for definiteness we consider the case $u_{-}<u_{+}$,
$c=s\varepsilon/\sqrt{\delta}>0$, and let $u_{c}$ solve

$\displaystyle u^{\prime\prime}+cu^{\prime}+\phi(u)=0\ ,$ (26)

with $\phi(u)=-\big{(}\sigma(u)-\sigma(u_{-})-s^{2}(u-u_{-})\big{)}$,
$\phi(u_{-})=\phi(u_{+})=0$ and

$\Phi(u)=\int_{u_{-}}^{u}\phi(z)dz\ .$ (27)

Hypotheses (HsE), (HoE) imply the only extrema of $\Phi(u)$ in (22) in the
range $[u_{-},u_{s}]$ are $u_{-}$ a strict local maximum and $u_{+}$ a strict
local minimum. $\Phi$ is strictly decreasing on $(u_{-},u_{+})$, strictly
increasing on $(u_{+},u_{s})$ and, by (HL),

$\displaystyle\frac{d^{2}\Phi}{du^{2}}$
$\displaystyle=-(\sigma^{\prime}(u)-s^{2})\,,\qquad\begin{cases}\Phi^{\prime\prime}(u_{-})<0&\\\
\Phi^{\prime\prime}(u_{+})>0&\end{cases}\,.$

Select $\alpha,\beta>0$ such that
$\Phi(u_{+}-\alpha)=\Phi(u_{+}+\beta)=E_{m}>0$ and

$\frac{d^{2}\Phi}{du^{2}}(u)=s^{2}-\sigma^{\prime}(u)>0\quad\mbox{ for \;
$u\in(u_{+}-\alpha,u_{+}+\beta)$}\ .$ (28)

The range of energies is thus split to

$\displaystyle\mbox{the big energies}:\qquad E_{m}<E<E_{max}=\Phi(u_{-})\ ,$
(29) $\displaystyle\mbox{the small energies}:\qquad\Phi(u_{+})=0<E<E_{m}\ .$

Using (HsE), (HoE), there exist $0<\eta<H$ such that

$\eta(u-u_{-})<\sigma(u)-\sigma(u_{-})-s^{2}(u-u_{-})<H(u-u_{-}),\quad
u\in(u_{-},u_{+}-\alpha]\,,$ (30)

and there exist $0<m<M$ such that

$-M(u-u_{+})<\sigma(u)-\sigma(u_{+})-s^{2}(u-u_{+})<-m(u-u_{+}),\quad
u\in(u_{+},u_{s}]\,.$ (31)

######  Remark 3.

In terms of the potential function $\Phi(u)$ the properties that used in the
sequel can be summarized as:

$\Phi^{\prime\prime}(u_{-})=\phi^{\prime}(u_{-})<0\,,\quad\Phi^{\prime\prime}(u_{+})=\phi^{\prime}(u_{+})>0\,.$
(32)

For $\alpha$, $\beta$ fixed as above, there exist $\eta,H>0$ and $m,M>0$ such
that

$\displaystyle\eta(u-u_{-})$
$\displaystyle<\phi(u_{-})-\phi(u)<H(u-u_{-}),\,\quad$ $\displaystyle
u\in(u_{-},u_{+}-\alpha]\ ,$ (33) $\displaystyle m(u-u_{+})$
$\displaystyle<\phi(u)-\phi(u_{+})<M(u-u_{+}),\,\quad$ $\displaystyle
u\in(u_{+},u_{s}]\ .$

###  3.1. Periods of orbits of the Hamiltonian system

When $c=0$, the system becomes Hamiltonian

$\begin{cases}&\frac{du}{d\tau}=w\;,\\\
&\frac{dw}{d\tau}=\phi(u)\;.\end{cases}$ (34)

The orbit emanating from the saddle $(u_{-},0)$ with energy
$E=E_{max}=\Phi(u_{-})$ is a homoclinic. The remaining orbits for
$0<E<E_{max}$ are periodic. Using (21) the period is computed by

$T(E)=2\int_{u_{1}(E)}^{u_{2}(E)}\frac{du}{\sqrt{2(E-\Phi(u))}}\;,$ (35)

where $u_{1}(E)$ and $u_{2}(E)$ satisfy $\Phi(u_{1}(E))=\Phi(u_{2}(E))=E$, see
Fig. 2. First we prove the following:

######  Lemma 4.

Under hypotheses (HL), (HsE), (HoE), there exists $T_{0}>0$ such that

$T(E)\geq T_{0}>0\,,\quad\forall E>0\,.$

and $T(E)\to\infty$ as $E\to E_{max}$.

###### Proof.

We estimate the period first for large energies. The integral in (35) is split
in three parts

$\frac{1}{2}T(E)=\left(\int_{u_{1}(E)}^{u_{+}-\alpha}+\int_{u_{+}-\alpha}^{u_{+}+\beta}+\int_{u_{+}+\beta}^{u_{2}(E)}\right)\frac{du}{\sqrt{2(E-\Phi(u))}}\
.$ (36)

The main contribution comes from the interval $(u_{1}(E),u_{+}-\alpha)$ and is
estimated as follows: From $\Phi(u_{1}(E))=E$, (22) and (30), we have

$\displaystyle E-\Phi(u)$
$\displaystyle=\int_{u_{1}(E)}^{u}\big{(}\sigma(z)-\sigma(u_{-})-s^{2}(z-u_{-})\big{)}\leavevmode\nobreak\
dz$ $\displaystyle>\int_{u_{1}(E)}^{u}\eta(z-u_{-})\leavevmode\nobreak\ dz$
$\displaystyle=\eta(u-u_{1}(E))\left(\frac{u+u_{1}(E)}{2}-u_{-}\right),\qquad
u_{-}<u_{1}(E)<u<u_{+}-\alpha\ ,$

and similarly

$\displaystyle E-\Phi(u)$
$\displaystyle<\int_{u_{1}(E)}^{u}H(z-u_{-})\leavevmode\nobreak\ dz$
$\displaystyle=H(u-u_{1}(E))\left(\frac{u+u_{1}(E)}{2}-u_{-}\right),\qquad
u_{-}<u_{1}(E)<u<u_{+}-\alpha\,.$

Hence,

$\displaystyle\frac{1}{\sqrt{H}}F(\rho)\leq\int_{u_{1}(E)}^{u_{+}-\alpha}\frac{du}{\sqrt{2(E-\Phi(u))}}\leq\frac{1}{\sqrt{\eta}}F(\rho)\
,$

where $\rho=2(u_{1}(E)-u_{-})$,

$F(\rho):=\int_{0}^{(u_{+}-\alpha-
u_{1}(E))}\frac{ds}{\sqrt{s(s+\rho)}}<\infty\quad\mbox{for}\quad\rho>0\ ,$

and $u_{+}-\alpha-u_{1}(E)$ is bounded away from zero in the range of large
energies. Using the monotone convergence theorem,

$F(\rho)\to\infty\quad\mbox{as \quad$\rho=2(u_{1}(E)-u_{-})\to 0$}\ ,$

that is the contribution of that integral to the period becomes infinite as
$E\to E_{max}$ and the periodic orbit approaches a homoclinic orbit.

Again we consider large energies and focus on the complementary region
$u\in(u_{+},u_{2}(E))$. Using (22), (9) and (31), we deduce

$\displaystyle E-\Phi(u)=\Phi(u_{2}(E))-\Phi(u)$
$\displaystyle=-\int_{u}^{u_{2}(E)}\big{(}\sigma(z)-\sigma(u_{+})-s^{2}(z-u_{+})\big{)}\leavevmode\nobreak\
dz$ $\displaystyle>m\int_{u}^{u_{2}(E)}(z-u_{+})\leavevmode\nobreak\ dz$
$\displaystyle=\tfrac{m}{2}(u_{2}(E)-u)\left(u_{2}(E)+u-2u_{+}\right)\ ,$
$\displaystyle\int_{u_{+}}^{u_{2}(E)}\frac{du}{\sqrt{2(E-\Phi(u)})}$
$\displaystyle\leq\frac{1}{\sqrt{m}}\int_{0}^{u_{2}(E)-u_{+}}\frac{ds}{\sqrt{s\big{(}2(u_{2}(E)-u_{+})-s\big{)}}}<\infty\,,$

that is the contribution of this integral to the period is finite. Finally,
for $E>E_{m}$, in the region $u\in[u_{+}-\alpha,u_{+}+\beta]$, the potential
energy satisfies $0<\Phi(u)<E_{m}$ and the contribution of the middle integral
to the period is easily seen to be bounded,

$\frac{\beta-\alpha}{\sqrt{2E}}<\int_{u_{+}-\alpha}^{u_{+}+\beta}\frac{du}{\sqrt{2(E-\Phi(u))}}\leq\frac{\beta-\alpha}{\sqrt{2(E-E_{m})}}\,.$

Next, consider the regime of small energies $0<E<E_{m}$ and analyze first the
range $u_{+}-\alpha<u_{1}(E)\leq u\leq u_{+}$. Setting
$J_{\alpha}=[u_{+}-\alpha,u_{+}]$ and using (28) we have

$\displaystyle\min_{J_{\alpha}}\sigma^{\prime}(u)$
$\displaystyle<\frac{\sigma(u_{+})-\sigma(u)}{u_{+}-u}<\max_{J_{\alpha}}\sigma^{\prime}(u)\
,$
$\displaystyle-\left(\max_{J_{\alpha}}\Phi^{\prime\prime}(u)\right)(u_{+}-u)$
$\displaystyle<\sigma(u_{+})-\sigma(u)-s^{2}(u_{+}-u)<-\left(\min_{J_{\alpha}}\Phi^{\prime\prime}(u)\right)(u_{+}-u)\
.$

Set $m=\min_{J_{\alpha}}\Phi^{\prime\prime}(u)>0$,
$M=\max_{J_{\alpha}}\Phi^{\prime\prime}(u)>0$ and use

$E-\Phi(u)=\int_{u_{1}(E)}^{u}\sigma(z)-\sigma(u_{+})-s^{2}(z-u_{+})dz\ ,$

to obtain

$m\int_{u_{1}(E)}^{u}(u_{+}-z)\leavevmode\nobreak\
dz<E-\Phi(u)<M\int_{u_{1}(E)}^{u}(u_{+}-z)\leavevmode\nobreak\ dz\ ,$

and hence

$\displaystyle\frac{1}{\sqrt{M}}I(E)$
$\displaystyle\leq\int_{u_{1}(E)}^{u_{+}}\frac{du}{\sqrt{2(E-\Phi(u))}}<\frac{1}{\sqrt{m}}I(E)\
,$ (37) $\displaystyle\mbox{where}\qquad I(E)$
$\displaystyle=\int_{0}^{u_{+}-u_{1}(E)}\frac{ds}{\sqrt{s\big{(}2(u_{+}-u_{1}(E))-s\big{)}}}\
.$

Again for small energies $0<E<E_{m}$ we consider next the range $u_{+}\leq
u<u_{2}(E)<u_{+}+\beta$ and similarly obtain the bound

$\displaystyle\frac{1}{\sqrt{M}}J(E)\leq\int_{u_{+}}^{u_{2}(E)}\frac{du}{\sqrt{2(E-\Phi(u))}}\leq\frac{1}{\sqrt{m}}J(E)\
,$ (38) $\displaystyle\mbox{where}\qquad
J(E)=\int_{0}^{u_{2}(E)-u_{+}}\frac{ds}{\sqrt{s\big{(}2(u_{2}(E)-u_{+})-s\big{)}}}\
,$

$m=\min_{J_{\beta}}\Phi^{\prime\prime}(u)$,
$M=\max_{J_{\beta}}\Phi^{\prime\prime}(u)$, $J_{\beta}=[u_{+},u_{+}+\beta]$
and $m,M>0$.

We conclude from (37), (38) that periodic orbits with small energies have
periods of the order of $I(E)+J(E)$. The limiting behavior as $E\to 0$ is
computed via the integral

$K(a):=\int_{0}^{a}\frac{ds}{\sqrt{s(2a-s)}}=\int_{0}^{a}\frac{ds}{\sqrt{a^{2}-(a-s)^{2}}}=\arcsin\frac{\tau}{a}\;\Bigg{|}_{0}^{a}=\frac{\pi}{2}\
.$ (39)

Therefore, $J(E)=I(E)=\frac{\pi}{2}$ and $T(E)$ remains bounded from below as
$E\to 0$. The limit of $T(E)$ can be calculated but we do not pursue this
point further. ∎

###  3.2. Effect of friction on orbits

We study the oscillatory behavior of solutions $(u_{c}(\tau),w_{c}(\tau))$ to
(18) in the range $0<c<c^{\star}$. Let $x_{n}$ be the consecutive maxima of
$u_{c}(\tau)$ and $y_{n}$ the minima. By a shift of the independent variable
we set the first maximum at $x_{0}=0$; this gives the ordering
$y_{0}=-\infty<x_{0}=0<y_{1}<x_{1}<\cdots<y_{n}<x_{n}<\cdots$.

Large energies. The following lemma, for large energies, estimates the
distance between two consecutive extrema, and the energy drop between these
points.

######  Lemma 5.

For $c>0$ sufficiently small and energies $E_{m}\leq{E(y_{n})}<E_{max}$, there
is a constant $K>0$ such that

$E(y_{n+1})-E(y_{n})\leq-Kc\,,\qquad y_{n+1}-y_{n}\leq\frac{K}{(cn)^{1/2}}\ .$

###### Proof.

Solutions of (18) satisfy the energy dissipation equation (20). Using the
normalization $u^{\prime}(0)=0$, we find that the energy drop between
$-\infty$ and $\tau_{0}=0$ is

$\ E_{max}-E(0)=c\int_{-\infty}^{0}w^{2}(t)\leavevmode\nobreak\ dt\;.$ (40)

Let $(u_{c}(\tau),w_{c}(\tau))$ be a solution of (18) and
$(u_{0}(\tau),w_{0}(\tau))$ be a solution of the Hamiltonian system (34).
Suppose that both solutions emanate from the same initial data, that is

$(u_{c}(\tau_{0}),w_{c}(\tau_{0}))=(u_{0}(\tau_{0}),w_{0}(\tau_{0}))=(u_{0},w_{0})\in\mathcal{D}\
,$

where $\mathcal{D}$ is the domain in (24). Since $\mathcal{D}$ is an invariant
domain for both (18) and (34), using Gronwall’s lemma, we obtain

$\displaystyle|w_{c}(\tau)-w_{0}(\tau)|+|u_{c}(\tau)-u_{0}(\tau)|$
$\displaystyle\leq cK_{T}\sup_{|\zeta-\tau_{0}|\leq{T}}|w_{c}(\zeta)|$ (41)
$\displaystyle\leq c\,\bar{K}_{T}\qquad\quad\text{
for}\;|\tau-\tau_{0}|\leq{T},$

where $K_{T}$ depends on the Lipschitz constant (on the domain $\mathcal{D}$)
and on $T$.

We will show that

$\int_{-\infty}^{0}w_{c}^{2}(\tau)\leavevmode\nobreak\ d\tau\geq\kappa>0\;.$

Indeed, without loss of generality, taking $\tau_{0}=0$ in (41), we have
$|w_{c}(\tau)-w_{0}(\tau)|\leq c\bar{K}$ for $\tau\in[-1,0]$. Hence,

$\int_{-\infty}^{0}w_{c}^{2}(\tau)\leavevmode\nobreak\
d\tau\geq\int_{-1}^{0}w_{c}^{2}(\tau)\leavevmode\nobreak\
d\tau\geq\int_{-1}^{0}w_{0}^{2}(\tau)\leavevmode\nobreak\ d\tau-\tilde{K}c\geq
K\ ,$ (42)

and

$E(0)\leq E_{max}-Kc\;.$

Let now $x_{n}$ be a maximum and $y_{n}$, $y_{n+1}$ be consecutive minima of
$u_{c}(\tau)$. Consider two orbits $(u_{c},w_{c})$ and $(u_{0},w_{0})$ that
meet at the same point in phase space, with $u_{c}(y_{n})=u_{0}(y_{n})$,
$w_{c}(y_{n})=w_{0}(y_{n})=0$, and let $T$ be the period of the periodic
orbit. We claim that

$|(x_{n}-y_{n})-\tfrac{T}{2}|\leq
o(1)\quad\text{and}\quad|(y_{n+1}-y_{n})-T|\leq o(1)\quad\mbox{as $c\to
0$}\,.$ (43)

This comparison between the period $T$ and the time elapsed between two
consecutive extrema $x_{n}-y_{n}$ is a consequence of the fact that the orbits
$(u_{c},w_{c})$ converge to the orbit $(u_{0},w_{0})$ as $c\to 0$. To see that
observe that by (41),

$|u_{c}(x_{n})-u_{0}(x_{n})|\leq
K_{(x_{n}-y_{n})}c\,,\quad|u_{c}(y_{n}+\tfrac{T}{2})-u_{0}(y_{n}+\tfrac{T}{2})|\leq
K_{T}c\,,$

and then use (35) and Lemma 4 to obtain

$\displaystyle(x_{n}-y_{n})-\tfrac{T}{2}$
$\displaystyle=\int_{u_{0}(y_{n})}^{u_{0}(x_{n})}\frac{du}{\sqrt{2(E-\Phi(u))}}-\int_{u_{0}(y_{n})}^{u_{0}\big{(}y_{n}+\tfrac{T}{2}\big{)}}\frac{du}{\sqrt{2(E-\Phi(u))}}$
$\displaystyle=\int_{u_{0}\big{(}y_{n}+\tfrac{T}{2}\big{)}}^{u_{c}(x_{n})+O(c)}\frac{du}{\sqrt{2(E-\Phi(u))}}\to
0\qquad\mbox{as $c\to 0$}\ .$

Similarly is proved the second identity in (43).

This shows that for $E_{m}\leq{E(y_{n})}<E_{max}$ we have $y_{n+1}-y_{n}\geq
T_{0}$ where $T_{0}$ does not depend on $n$, and thus

$E(y_{n+1})-E(y_{n})\leq-c\int_{y_{n}}^{y_{n+1}}w_{c}^{2}(\tau)\leavevmode\nobreak\
d\tau\leq-Kc\ ,$ (44)

and for the energy at $y_{n}$,

$E(y_{n})\leq E_{max}-Kcn\;.$ (45)

For $(u(\tau),w(\tau))$ solution of (18), we proceed to estimate the distance
between two consecutive minimum at $y_{n}$ and maximum at $x_{n}$. The domain
$[y_{n},x_{n}]$ is split to three regions:

  * (I)

$u_{-}<u(y_{n})<u(a_{n})=u_{+}-\alpha$ for $y_{n}\leq\tau\leq a_{n}$

  * (II)

$u_{+}-\alpha=u(a_{n})<u_{+}<u(b_{n})=u_{+}+\beta$ for $a_{n}\leq\tau\leq
b_{n}$

  * (III)

$u_{+}+\beta=u(b_{n})<u(x_{n})<u_{s}$ for $b_{n}\leq\tau\leq x_{n}$

In Region (II) we have

$0<u(b_{n})-u(a_{n})=w(\tau_{\star})(b_{n}-a_{n})\qquad\mbox{for some
$\tau_{\star}\in[a_{n},b_{n}]$}\ .$

Since the orbit of $(u(\tau),w(\tau)$ is near the orbit
$(u_{0}(\tau),w_{0}(\tau))$ we have $w(\tau)\geq\kappa>0$ and we conclude
$b_{n}-a_{n}\leq K$.

Consider next the Region (I) and observe that using (45), (22), (30),

$\displaystyle Knc$ $\displaystyle\leq\Phi(u_{-})-\Phi(u(y_{n}))$
$\displaystyle=(-\phi(v))(u(y_{n})-u_{-})\qquad\mbox{for some
$u_{-}<v<u_{+}-\alpha$}$
$\displaystyle\leq\left(\max_{u_{-}<v<u_{+}-\alpha}(-\phi(v))\right)(u(y_{n})-u_{-})$
$\displaystyle\leq\max_{u_{-}<v<u_{+}-\alpha}\big{(}H(v-u_{-})\big{)}(u(y_{n})-u_{-})$
$\displaystyle\leq K^{\prime}(u(y_{n})-u_{-})$

Using (17) and (30),

$u^{\prime\prime}+cu^{\prime}=-\phi(u)>\eta(u-u_{-})>\eta(u(y_{n})-u_{-})>Knc$

whence

$\big{(}u(\tau)-u_{-}\big{)}^{\prime}\geq Kn\big{(}1-e^{-c(\tau-
y_{n})}\big{)}$

and integrating once again and using $e^{-cx}\geq
1-cx+\tfrac{1}{2}c^{2}x^{2}-\tfrac{1}{6}c^{3}x^{3}$ we derive

$\displaystyle u(\tau)-u(y_{n})$ $\displaystyle\geq\frac{Kn}{c}\Big{(}c(\tau-
y_{n})+e^{-c(\tau-y_{n})}-1\Big{)}$ $\displaystyle\geq Knc\tfrac{1}{2}(\tau-
y_{n})^{2}\big{(}1-\frac{1}{3}c(\tau-y_{n})\big{)}$ $\displaystyle\geq
Knc\tfrac{1}{4}(\tau-y_{n})^{2}\qquad\mbox{provided
$c(x_{n}-y_{n})<\tfrac{3}{2}$}$

We conclude

$\tau-y_{n}\leq\frac{K}{\sqrt{nc}}\qquad\mbox{for $y_{n}\leq\tau\leq a_{n}$}\
.$

On the Region (III) $b_{n}\leq\tau\leq x_{n}$ the estimate proceeds along
similar lines: First using (31),

$Knc\leq\Phi(u_{s})-\Phi(u(x_{n}))\leq M(u_{s}-u(x_{n}))\ .$

Using (17) and (31)

$(u_{s}-u)^{\prime\prime}+c(u_{s}-u)^{\prime}=\phi(u)\geq\min_{u\in[u_{+}+\beta,u_{s}]}\phi(u)=:A>0\
,$

and after an integration

$u^{\prime}(\tau)\geq\frac{K}{c}\big{(}e^{c(x_{n}-\tau)}-1\big{)}\qquad
b_{n}\leq\tau\leq x_{n}\ ,$

and another one

$\displaystyle u(x_{n})-u(\tau)$
$\displaystyle\geq\frac{K}{c}\int_{\tau}^{x_{n}}\big{(}e^{c(x_{n}-z)}-1\big{)}dz$
$\displaystyle\geq\frac{K}{2}(x_{n}-\tau)^{2}\qquad\mbox{for
$b_{n}\leq\tau\leq x_{n}$}\ .$

We conclude that

$x_{n}-\tau\leq\Big{(}\frac{2(u(x_{n})-u(\tau))}{K}\Big{)}^{\tfrac{1}{2}}\leq
K^{\prime}\ .$

Putting all together we obtain

$x_{n}-y_{n}\leq\frac{K}{(nc)^{1/2}}\ ,$ (46)

and similarly by a symmetric argument $y_{n+1}-x_{n}\leq\frac{K}{(nc)^{1/2}}$
which completes the proof. ∎

At this point we estimate the ”length” of the highly oscillatory part of the
solution. Since the energy drop per period is of size $Kc$ the total number of
oscillations is $N=K/c$. The length of this regiion is

$L=\sum_{n=1}^{N=K/c}y_{n+1}-y_{n}\leq\sum_{n=1}^{N=K/c}\frac{K}{(cn)^{1/2}}=\frac{K}{c^{1/2}}\int_{1}^{\tfrac{K}{c}}\frac{dx}{x^{1/2}}\leq\frac{K^{\prime}}{c}\
.$ (47)

Small Energies. For energies $E(\tau)\in(0,E_{m})$, we show the exponential
damping behavior of the solution. Since $\Phi^{\prime\prime}(u)>0$ in this
region, the situation is analogous to the analysis in [20]. Using the energy
dissipation (20), we obtain

$E^{\prime}(\tau)\geq-cE(\tau)\ ,$

and thus

$E(\tau)\geq E(\tau_{0})e^{-c(\tau-\tau_{0})}\ .$

To show the opposite inequality note that in this region
$\sup|w|\leq\sqrt{2E(\tau_{0})}$. Then Gronwall’s inequality (41) implies

$|u_{c}(\tau)-u_{0}(\tau)|+|w_{c}(\tau)-w_{0}(\tau)|\leq
c\sqrt{E(\tau_{0})}e^{L(\tau-\tau_{0})}\,.$ (48)

From the analysis of (34) in section 3.1 for small energies, the distance
between consecutive maxima satisfies $x_{n}-x_{n-1}\geq\alpha>0$ for some
$\alpha>0$ independent of $n$, and same for the minima $y_{n}$. Following
analogous to the large energies case steps in (44), we obtain an upper bound
for the energy

$E(x_{n})\leq E(x_{n-1})\big{(}1-Kc(x_{n}-x_{n-1})\big{)}\;.$

We conclude that for $E<E_{m}$ the energy decays exponentially at a rate $cK$,

$E(x_{n})\leq E(x_{0})e^{-Kc(x_{n}-x_{0})}\,,$

where $x_{0}$ is the first point of maximum such that $E(x_{0})\leq E_{m}$. It
follows that $u_{c}(\tau)\to u_{+}$ on a length scale of order $1/c$.

The result that has been proved can be expressed entirely based on the second
order equation (26) and properties of the function $\phi(u)$. We summarize the
result in a proposition.

######  Proposition 6.

Let $u_{c}(\tau)$ be heteroclinic connections of (26) with
$u_{c}(\pm\infty)=u_{\pm}$ with $0<c<c^{\star}$, where $\Phi(u)$ is defined in
(27),

(i) Let $u_{-}<u_{+}$ and assume $\phi(u)$ satisfies

$u_{-}$, $u_{+}$ are the only solutions of $\phi(u)=0$ in $[u_{-},u_{s}]$
(Hϕ0) $\displaystyle
0=\Phi(u_{+})<\Phi(u)<\Phi(u_{-})=\Phi(u_{s})=E_{max}\quad\text{for}\quad
u\in(u_{-},u_{s})\ ,$

as well as

$\displaystyle\phi(u)<0\,,\qquad u_{-}<u<u_{+}\ ,$ (Hϕ1)
$\displaystyle\phi(u)>0\,,\qquad u_{+}<u\leq u_{s}\ ,$ (Hϕ2)
$\displaystyle\phi^{\prime}(u_{-})<0\,,\quad\phi^{\prime}(u_{+})>0\ .$ (Hϕ3)

Let $u_{c}(\tau)$ be normalized by setting $u^{\prime}_{c}(0)=0$. Then the
domain $\tau>0$ is split into two regions:

  * •

the region where the solution $u_{c}(\tau)$ has large amplitude oscillations
with energy $E_{m}<E<E_{max}$ and which has length of size $O(\tfrac{1}{c})$.

  * •

the region where the solution $u_{c}(\tau)$ has small amplitude oscilations of
energy $0<E<E_{m}$ which has again length of size $O(\tfrac{1}{c})$.

One can easily check that (Hϕ1), (Hϕ2), (Hϕ3) imply that and $\phi(u)$
satisfies (32) and (33) with $\alpha,\beta$ as defined in (28), which are the
key ingredients on which the analysis of section 3 is based.

An analogous result can be proved for the case $u_{-}>u_{+}$ and $c>0$ under
the hypotheses that $\Phi$ has a maximum at $u_{-}$, a (nondegenerate) minimum
at $u_{+}$ and $\phi(u)$ satisfies conditions analogous to (32), (33) in the
rest of the domain $[u_{s},u_{-}]$ where $\Phi(u_{s})=\Phi(u_{-})$. These two
results provide, by performing a change of direction $\tau\to-\tau$, two
analogous results valid for the case that the shock speed $s<0$.

###  3.3. Returning to the original variables

Consider now the convergence of traveling waves for (2) as
$\varepsilon,\delta\to 0$. Recall the traveling wave is expressed via

$\displaystyle u^{\varepsilon,\delta}(x,t)$
$\displaystyle=U^{\varepsilon,\delta}\left(x-st\right)=u_{c}\left(\frac{x-st}{\sqrt{\delta}}\right)\;,$
(49) $\displaystyle v^{\varepsilon,\delta}(x,t)$
$\displaystyle=V^{\varepsilon,\delta}\left(x-st\right)=v_{c}\left(\frac{x-st}{\sqrt{\delta}}\right)\,,$

where $u_{c}$ solves (17), $v_{c}$ is determined by (16) and
$c=s\tfrac{\varepsilon}{\sqrt{\delta}}$ (and here we consider the case that
$s>0$).

We fix the shift of the traveling wave so that $\theta=0$ at the first maximum
of the function $u_{c}(\tau)$. Since $U^{\varepsilon,\delta}$ and $u_{c}$ are
related through the scaling

$U^{\varepsilon,\delta}(\theta)=u_{c}\left(\frac{\theta}{\sqrt{\delta}}\right)\
,$ (50)

the graph of $U^{\varepsilon,\delta}$ is obtained from the graph of $u_{c}$ by
scaling down in the axis direction by a factor $\sqrt{\delta}$. Accordingly, a
structure of length scale $L$ in the graph of $u_{c}$ will have length scale
$\sqrt{\delta}L$ in the graph of $U^{\varepsilon,\delta}$.

We have seen that $u_{c}(\tau)\to u_{\pm}$ as $\tau\to\pm\infty$, and from the
analysis leading to (47) the region of oscillations at the high energy regime
is of order $\frac{1}{c}$. In the regime of small energies the length scale of
oscillations is again of order $\frac{1}{c}$. When we transfer these length
scales at the level of the original variables, they both become of size
$O(\sqrt{\delta}\frac{1}{c})=O(\delta/\varepsilon)$ and they will shrink to
zero provided that $\delta=o(\varepsilon)$. This finishes the proof of the
main theorem.

###  3.4. Is $\delta=o(\varepsilon)$ optimal ?

The range $\delta=o(\varepsilon)$ is optimal for the linearized system
associated with (17). The linearized equation around $u_{+}$ takes the form

$\frac{d^{2}\tilde{u}}{d\tau^{2}}+c\frac{d\tilde{u}}{d\tau}+\alpha\tilde{u}=0\
,$

with $c=\tfrac{s\varepsilon}{\sqrt{\delta}}$, $\alpha=\phi(u_{+})>0$ and
$\tilde{u}=u-u_{+}$. The characteristic polynomial for the linear differential
equation has complex eigenvalues when $c\ll 1$ which are
$\rho_{\pm}=-\frac{c}{2}\pm\frac{i}{2}\sqrt{|4\alpha-c^{2}|}$, and its
solution is

$\tilde{u}(\tau)=Ae^{-\tfrac{c}{2}\tau}\cos(\omega\tau+\beta)\ ,$

where $A$ is an amplitude, $\omega=\sqrt{\alpha-\tfrac{c^{2}}{4}}$ the
frequency and $\beta$ a phase shift. When expressing the solution in terms of
the original variables we have

$u^{\varepsilon,\delta}\left(\frac{x-st}{\sqrt{\delta}}\right)-u_{+}=Ae^{-\tfrac{\varepsilon}{\delta}\tfrac{s}{2}(x-st)}\cos\left(\left(\sqrt{\alpha-\tfrac{s^{2}\varepsilon^{2}}{4\delta}}\right)\frac{x-st}{\sqrt{\delta}}+\beta\right)\
.$

In the regime of moderate dispersion $\delta=o(\varepsilon)$ the right side
converges to zero as $\delta,\varepsilon\to 0$ and the traveling wave
converges to a shock.

In the moderate dispersion regime $\delta=o(\varepsilon)$, the convergence to
a shock wave is in a strong sense as often expected in shock wave theory. In
the complementary region $\varepsilon=o(\delta)$ the solution of the
linearized equation oscillates vigorously around the constant state $u_{+}$.
Such an oscillatory tail converges to a constant state in a weak sense since
the average of the oscillations around the constant $u_{+}$ cancel out. This
behavior characterizes only the linearized problem and it is not clear if it
will persist for the nonlinear problem. Weak convergence could conceivably
give a meaning on how the limiting state $u_{+}$ is achieved even in a regime
of strong dispersion.

##  4\. Dispersive Shocks in Quantum Hydrodynamics with Viscosity

We consider the one dimensional quantum hydrodynamics system (QHD) with
artificial viscosity

$\displaystyle\rho_{t}+j_{x}$ $\displaystyle=0\ ,$ (51) $\displaystyle
j_{t}+\left(\frac{j^{2}}{\rho}+\rho^{\gamma}\right)_{x}$
$\displaystyle=\varepsilon
j_{xx}+\delta\rho\left(\frac{\sqrt{\rho}_{xx}}{\sqrt{\rho}}\right)_{x}\ ,$

where $\rho$ is the density, $u$ the fluid velocity, $p(\rho)$ the pressure,
and $j$ the fluid momentum, $j=\rho u$. The Bohm’s potential, also known as
Quantum potential, represents a dispersive term and artificial viscosity is
also introduced in this system modeling effects of dissipation. Our objective
is to show that the combined effect of diffusion and dispersion leads in the
moderate dispersion regime to a dispersive shock wave with oscillatory tails.
Traveling wave solutions of (51) have been studied in [15] in a weak
dispersion regime $\delta=\varepsilon^{2}$, where existence of traveling waves
and convergence to a shock when $\delta=\varepsilon^{2}\to 0$ is proved.

At first sight, the nonlinearities and dispersion in the system (51) appear
more complex than in the system (6), but casting the problem in the right
variables will transform the traveling wave analysis to examining a
Hamiltonian system with weak friction (18).

###  4.1. Shocks in gas dynamics

The system of isentropic gas dynamics

$\displaystyle\rho_{t}+(\rho u)_{x}$ $\displaystyle=0\ ,$ (52)
$\displaystyle(\rho u)_{t}+(\rho u^{2}+p(\rho))_{x}$ $\displaystyle=0\ ,$

is hyperbolic when $p^{\prime}(\rho)>0$ and has eigenvalues
$\lambda_{\pm}=u\pm\sqrt{p^{\prime}(\rho)}$ and corresponding right
eigenvectors $r_{\pm}=\big{(}\pm\rho,\sqrt{p^{\prime}(\rho)}\big{)}^{T}$.
Under the condition

$\big{(}\rho^{2}p^{\prime}(\rho)\big{)}^{\prime}>0\ ,$ (53)

both characteristic fields are genuinely nonlinear
$r_{\pm}\cdot\nabla\lambda_{\pm}>0$.

Shock waves are discontinuous solutions of (52) connecting two states
$(\rho_{-},u_{-})$ to $(\rho_{+},u_{+})$. They have been studied extensively,
e.g. [24, Ch 18]. Shocks are constructed by solving the Rankine-Hugoniot
conditions

$\displaystyle-s[\rho]+[\rho u]$ $\displaystyle=0$ (54) $\displaystyle-s[\rho
u]+[\rho u^{2}+p]$ $\displaystyle=0$

where $s$ is the shock speed, and we use the usual notation
$[\rho]=\rho_{+}-\rho_{-}$ etc. Introduce $m=\rho u-su$ and note that

$[m]=0\quad\text{and}\quad m[u]+[p]=0\ .$

This implies that $m$ stays constant across the shock

$\rho_{+}(u_{+}-s)=\rho_{-}(u_{-}-s)=:m\,,\quad
m=-\frac{p_{+}-p_{-}}{u_{+}-u_{-}}\ ,$ (55)

where $p_{+}=p(\rho_{+})$, $p_{-}=p(\rho_{-})$ and $m$ is computed by

$m^{2}=-\frac{p_{+}-p_{-}}{\frac{1}{\rho_{+}}-\frac{1}{\rho_{-}}}\ .$ (56)

A 1-shock associated to the $\lambda_{-}$ characteristic speed satisfies the
Lax shock condition when

$u_{+}-\sqrt{p^{\prime}(\rho_{+})}<s<u_{-}-\sqrt{p^{\prime}(\rho_{+})}\ .$

If (53) is satisfied using (55) one checks

$\sqrt{\rho_{+}^{2}p^{\prime}(\rho_{+})}>m>\sqrt{\rho_{-}^{2}p^{\prime}(\rho_{-})}>0\
,$

and we deduce that for a Lax admissible 1-shock we have

$\rho_{+}>\rho_{-}\,,\quad m>0\,,\quad s<u_{+}<u_{-}\ .$

A 2-shock associated to the $\lambda_{+}$ characteristic speed satisfies the
Lax shock condition when

$u_{+}+\sqrt{p^{\prime}(\rho_{+})}<s<u_{-}+\sqrt{p^{\prime}(\rho_{+})}\ .$

In a similar way, using (53), we deduce

$\rho_{+}<\rho_{-}\,,\quad m<0\,,\quad u_{+}<u_{-}<s\ .$

The systems (7) and (52) are equivalent by the transformation
$\hat{y}(\cdot,t):x\to y$ from Lagrangian to Eulerian coordinates, [24].
Indeed, using $y=\hat{y}(x,t)$, $v=\frac{\partial\hat{y}}{\partial t}$ and
$w=\frac{\partial\hat{y}}{\partial x}$, the equations

$w_{t}=v_{x}\,,\quad v_{t}=\sigma(w)_{x}\ ,$

express respectively the equality of mixed partial derivatives and the balance
of momentum in Lagrangian coordinates. The balance of mass takes the form
$\rho(\hat{y}(x,t),t)=\frac{1}{w}(x,t)$ and may be viewed as defining the
density. The velocity in Eulerian coordinates $u(y,t)$ relates to the
Lagrangian velocity $v(x,t)$ through $v(x,t)=\hat{u}(\hat{y}(x,t),t)$. Using
these formulas one checks that $(\rho,u)(y,t)$ satisfy the system (52) with
the identification for the pressure

$p(\rho)=-\sigma\left(\frac{1}{\rho}\right)\,,\quad w=\frac{1}{\rho}\ .$ (57)

The Rankine-Hugoniot conditions, the equations for the shock curves, and the
Lax shock-admissibility conditions in Lagrangian and Eulerian coordinates
transform to each other. In particular, the condition (53) in Eulerian
coordinates corresponds to the condition $\sigma^{\prime\prime}(w)<0$ for the
Lagrangian counterpart.

###  4.2. Reduction of traveling waves to a Hamiltonian system with friction

Consider (51) and introduce for its solution
$(\rho^{\varepsilon,\delta},j^{\varepsilon,\delta})$ the ansatz of traveling
waves,

$\displaystyle\rho^{\varepsilon,\delta}(x,t)=\rho\left((x-st)/\sqrt{\delta}\right)=\rho(\tau)\,,$
$\displaystyle
j^{\varepsilon,\delta}(x,t)=j\left((x-st)/\sqrt{\delta}\right)=j(\tau)\,,$

where $j=\rho u$, $\tau=(x-st)/\sqrt{\delta}$ and (with a slight abuse of
notation) we retain the notation $(\rho(\tau),j(\tau))$ for the solution of
the traveling wave equations

$\displaystyle-s\rho^{\prime}+j^{\prime}$ $\displaystyle=0\ ,$ (58)
$\displaystyle-sj^{\prime}+\left(\frac{j^{2}}{\rho}+p(\rho)\right)^{\prime}$
$\displaystyle=\frac{\varepsilon}{\sqrt{\delta}}j^{\prime\prime}+\rho\left(\frac{(\sqrt{\rho})^{\prime\prime}}{\sqrt{\rho}}\right)^{\prime}\
.$

Next, we fix $(\rho_{-},u_{-})$, $(\rho_{+},u_{+})$ and $s$ that satisfy the
Rankine-Hugoniot conditions (54). The first equation in (58) gives

$\rho_{-}(u_{-}-s)=\rho(u-s)=m\ ,$ (59)

where the constant mass flux (relative to the shock) $m$ is computed via (56).
Using the well known formula

$\rho\left(\frac{\sqrt{\rho}_{xx}}{\sqrt{\rho}}\right)_{x}=\frac{1}{2}\left(\rho(ln\rho)_{xx}\right)_{x}\
,$

for the Bohm potential, we integrate (58) and obtain

$\displaystyle-s(\rho-\rho_{-})+(\rho u-\rho_{-}u_{-})$ $\displaystyle=0\ ,$
(60) $\displaystyle-s^{2}\rho^{\prime}+\big{(}\rho
u^{2}-\rho_{-}u_{-}^{2}+p(\rho-p(\rho_{-})\big{)}$
$\displaystyle=\frac{s\varepsilon}{\sqrt{\delta}}\rho^{\prime\prime}+\frac{1}{2}\left(\rho(ln\rho)^{\prime\prime}\right)^{\prime}\
.$

In turn, setting $c=\frac{s\varepsilon}{\sqrt{\delta}}$ and using (54)–(56) we
arrive at

$\frac{1}{2}\rho\left(\ln\rho\right)^{\prime\prime}+c\rho^{\prime}-\left(p(\rho)-p(\rho_{-})+m^{2}\left(\frac{1}{\rho}-\frac{1}{\rho_{-}}\right)\right)=0\
.$ (61)

Setting $x=\ln\rho$ we see that $x(\tau)$ solves the equation

$x^{\prime\prime}+2cx^{\prime}+\psi(x)=0\ ,$ (62)

where the function $\psi(x)$ can be expressed in the following equivalent
forms

$\displaystyle\psi(x)$
$\displaystyle=-\frac{2}{\rho}\left(p(\rho)-p(\rho_{-})+m^{2}\left(\frac{1}{\rho}-\frac{1}{\rho_{-}}\right)\right)\Bigg{|}_{\rho=e^{x}}$
$\displaystyle=2w\left(\sigma(w)-\sigma(w_{-})-m^{2}(w-w_{-})\right)\bigg{|}_{w=e^{-x}}\
,$ (63)

where we used the changes of variables $\rho=e^{x}$ for the first identity,
and the formula (57) and $w_{-}=\frac{1}{\rho_{-}}$ for the second. This
allows to write $\psi(x)$ in the form $\psi(x)=g(e^{x})=f(e^{-x})$ where

$f(w)=2w\big{(}\sigma(w)-\sigma(w_{-})-m^{2}(w-w_{-})\big{)}\ .$

The potential function $\Psi(x)$ is defined up to an arbitrary constant via

$\displaystyle\Psi(x):=-2\int_{w_{-}}^{w}\big{(}\sigma(z)-\sigma(w_{-})-m^{2}(z-w_{-})\big{)}\,dz\Bigg{|}_{w=e^{-x}}+\text{\em
Const.}$ $\displaystyle\frac{d\Psi}{dx}(x)=f(e^{-x})=\psi(x)\ .$

###  4.3. Convergence from oscillating traveling waves to shocks

We established that the traveling wave problem (60) reduces to solving (62)
for $x(\tau)$ and then defining

$\rho(\tau)=e^{x(\tau)}\,,\quad u(\tau)=s+\frac{m}{\rho(\tau)}$

The aim is to apply Proposition 6 to (62). The hypotheses can in principle be
checked for the following reasons: The genuine nonlinearity hypothesis for
$\sigma(w)$ suggests that $f(w)$ has a sign between the roots
$f(w_{-})=f(w_{+})=0$. Moreover,

$\frac{df}{dw}(w)=2\big{(}\sigma(w)-\sigma(w_{-})-m^{2}(w-w_{-})\big{)}+2w(\sigma^{\prime}(w)-m^{2})$
$\frac{df}{dw}(w_{\pm})=2w_{\pm}(\sigma^{\prime}(w_{\pm})-m^{2})\,,$

and since $w>0$ the sign of $\frac{df}{dw}(w_{\pm})$ amounts to the Lax shock
conditions.

We present the details for a 2-shock that satisfies the Lax conditions. Then
$\rho_{+}<\rho_{-}$, $m<0$ and $u_{+}<u_{-}<s$. The system (51) is invariant
under the change of variables

$\hat{x}=x-\kappa
t\,\quad\hat{u}=u+\kappa\,,\quad\hat{\rho}=\rho\,,\quad\mbox{for
$\kappa\in\mathbb{R}$}.$

Therefore by a change of variables we may assume $u_{+}=0<u_{-}<s$. This
amounts to observing the flow from a coordinate system moving with the
velocity of the fluid at $\infty$. The values $\rho_{+}<\rho_{-}$, $m<0$
remain unchanged.

Next, we employ the change of variables $x=\ln\rho$ and proceed to verify the
hypotheses of Proposition 6 for the arrangement $x_{s}<x_{+}<x_{-}$. Then
$x(\tau)$ satisfies (62) with $\psi(x)$ given by (63). Note that
$w_{-}=\tfrac{1}{\rho_{-}}<\tfrac{1}{\rho_{+}}=w_{+}$. Since
$\sigma^{\prime\prime}(w)<0$ we have

$f(w)=2w\big{(}\sigma(w)-\sigma(w_{-})-m^{2}(w-w_{-})\big{)}\,,\quad
w\in(w_{-},w_{+})\ ,$

and $f(w)>0$ on $(0,\infty)-(w_{-},w_{+})$. Moreover,

$\frac{df}{dw}(w_{-})=2w_{-}(\sigma^{\prime}(w_{-})-m^{2})>0\quad\text{and}\quad\frac{df}{dw}(w_{+})=2w_{+}(\sigma^{\prime}(w_{+})-m^{2})<0\,.$

All hypotheses of Proposition 6 are fulfilled for the arrangement
$x_{s}<x_{+}<x_{-}$ with a maximum at $x_{-}$ and minimum at $x_{+}$.
Proceeding as in section 3.3 we have:

######  Theorem 7.

Let $p(\rho)$ satisfy (53) and suppose that $s$, $(\rho_{-},u_{-})$,
$(\rho_{+},u_{+})$ define a 1-shock (or a 2-shock) that satisfies the Lax
shock conditions. There exist a unique (up to shifts) traveling wave solution
$(\rho^{\varepsilon,\delta},(\rho u)^{\varepsilon,\delta})(x-st)$ to the
system (51) that connects state $(\rho_{-},u_{-})$ to $(\rho_{+},u_{+})$. When
the shift is appropriately selected, the traveling wave converges strongly as
$\varepsilon$, $\delta\to 0$ with $\delta=o(\varepsilon)$ to the Lax-shock
solution of (52).

As an illustration, we present in Figure 4 a numerical solution to (61) for
$p(\rho)=\rho^{\gamma}$ with $\gamma=1.4$, between states $\rho_{-}=1.5$,
$\rho_{+}=1$, with the parameter $c=0.02$.

Figure 4. (a) Solution $x=\log(\rho)$; (b) Phase portrait ($x$,$u$) ($c=0.02$,
$\gamma=1.4$, $\rho_{-}=1.5$, $\rho_{+}=1$)

##  5\. Undular bores in a Boussinesq-Peregrine system

Undular bores are structures observed on free surface flows that propagate
mainly in one direction. They have been described in a weakly dispersive and
weakly nonlinear asymptotic regime by the Korteweg-de Vries-Burgers (KdVB)
equation. Recently, it was shown that Peregrine’s system [19] with weak
dissipation can also describe undular bores as traveling wave solutions with
high accuracy [5]. Note that Peregrine’s theory was initiated for the study of
solitary waves and also for the study of undular bores [18].

To illustrate consider a dissipative Peregrine-Boussinesq system written in
nondimensional unscaled form

$\displaystyle\eta_{t}+u_{x}+(\eta u)_{x}=0\ ,$ (64) $\displaystyle
u_{t}+\eta_{x}+uu_{x}-\delta u_{xxt}-\varepsilon u_{xx}=0\ ,$

where $\eta$ denotes the free-surface elevation above the rest position
$\eta=0$, $u$ is the horizontal velocity of the fluid evaluated at some depth
$\theta$ above the horizontal bottom located at depth $\theta=-1$, while
$\delta,\varepsilon>0$. This system is a dispersive/dissipative extension of
the nonlinear shallow-water wave equations (also known as St. Venant
equations). The latter form a system of hyperbolic conservation laws derived
as a low-order approximation of the Euler equations for water wave theory
[25].

Here, we will consider traveling wave solutions of (64) propagating with speed
$s>1$ to the right. The symmetry $s\to-s$, $\eta\to\eta$, $u\to-u$ implies
that anything true for traveling waves with $s>1$ is also valid for traveling
waves with $s<-1$. The existence of traveling wave solutions to (64) was
established in [5] describe undular bores when $\varepsilon^{2}<4\delta
s\alpha(s)$ and regularized shock waves when $\varepsilon^{2}\geq 4\delta
s\alpha(s)$, where
$\alpha(s)=\frac{s-\sqrt{s^{2}+8}}{2}+\frac{4s}{(s-\sqrt{s^{2}+8})^{2}}$. Here
we verify that if $\delta=o(\varepsilon)$ as $\varepsilon\to 0$, even in the
regime $\varepsilon^{2}<4\delta s\alpha(s)$, these traveling waves tend to a
classical shock waves of the shallow water equations.

In order to apply the previous theory, consider the ansatz

$\eta^{\varepsilon,\delta}(x,t)=-\eta(\tau),\quad
u^{\varepsilon,\delta}(x,t)=u(\tau),\quad\tau=-\frac{x-st}{\sqrt{s\delta}}\ .$

and assume for simplicity that $\lim_{\tau\to-\infty}(\eta,u)=(0,0)$ and
$\lim_{\tau\to+\infty}(\eta,u)=(\eta_{+},u_{+})$. The Rankine-Hugoniot
conditions dictate (see [5])

$u_{+}=\frac{3s-\sqrt{s^{2}+8}}{2}<s\ .$

After integration over $(-\infty,\tau)$ the system (64) yields

$-s\eta-u+\eta u=0,\quad
su+\eta-\frac{1}{2}u^{2}-u^{\prime\prime}-\frac{\varepsilon}{\sqrt{s\delta}}u^{\prime}=0\
,$ (65)

Eliminating the unknown $\eta$ in (65) we obtain the second-order equation

$u^{\prime\prime}+cu^{\prime}+\phi(u)=0\ ,$ (66)

where $c=\varepsilon/\sqrt{s\delta}$ and

$\phi(u)=-su+\frac{u}{s-u}+\frac{1}{2}u^{2}\ .$ (67)

The potential energy

$\Phi(u)=\int_{0}^{u}\phi(z)dz=\frac{1}{6}u^{3}-\frac{s}{2}u^{2}-u+s\ln\frac{s}{s-u}\
,$

has an inflection point at $u_{c}=s-\sqrt[3]{s}$ a maximum at $(0,0)$ and a
minimum $(u_{+},\Phi(u_{+}))$, with $u_{+}>u_{c}=s-\sqrt[3]{s}$. One checks
that it satisfies (32)–(33), and that
$\Phi^{\prime\prime}(u_{-})=\phi^{\prime}(0)=(1-s)(1+s)/s<0$ for $s>1$, while
$\Phi^{\prime\prime}(u_{+})=\frac{(u_{+}-s)^{3}+s}{(u_{+}-s)^{2}}>0$ holds
since $u_{+}>s-\sqrt[3]{s}$ for $s>1$. The graph of $\Phi(u)$ is depicted in
Figure 5.

Figure 5. The potential $\Phi(u)$ for $s=2$

Proposition 6 in Section 3 may be applied directly to (66) with
$c=\varepsilon/\sqrt{s\delta}$. It shows that traveling wave solutions of (64)
tend to entropic shocks of the shallow water wave equations when
$\delta=o(\varepsilon)$ as $\varepsilon,\delta\to 0$. Figure 6 shows the
convergence of a dissipative-dispersive shock wave to a classical shock wave
obtained numerically by taking $\delta=\varepsilon^{1.5}$ and $s=2$ as
$\varepsilon\to 0$. We observe that as $\delta$ becomes smaller the interval
where the oscillations are extended becomes smaller as well. The wave-front
becomes steeper tending in the limit to a shock. In all cases the quantity
$\varepsilon^{2}-4\delta s\alpha(s)$ remained negative even if it was very
small.

Figure 6. Convergence of a dissipative-dispersive shock wave to a classical
shock wave of the shallow water wave equations when $\delta=o(\varepsilon)$ as
$\delta,\varepsilon\to 0$. (The horizontal axis scales vary between images
while the maximum of the traveling waves is at $\tau=0$.)

## Acknowledgments

DM thanks KAUST for their hospitality during a visit when this work was
initiated.

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# Note on the Margolus-Levitin quantum speed limit for arbitrary fidelity

Krzysztof<EMAIL_ADDRESS>Katarzyna
<EMAIL_ADDRESS>Piotr
<EMAIL_ADDRESS>
( 1Department of Computer Science, Faculty of Physics and Applied Informatics
University of Lodz, 149/153 Pomorska St., 90-236 Lodz, Poland
2Department of Statistical Methods, Faculty of Economics and Sociology
University of Lodz, 41/43 Rewolucji 1905 St., 90-214 Lodz, Poland)

###### Abstract

For vanishing fidelity between initial and final states two important quantum
speed limits, the Mandelstam-Tamm limit (involving energy dispersion) and
Margolus-Levitin one (involving excitation energy expectation value) have been
derived. While the generalization of the former limit to the case of arbitrary
fidelity is straightforward, the relevant generalization of the latter, given
in the seminal paper by Giovanetti et al (_Phys. Rev._ A67 (2003), 052109) was
based on the conjectured equality of lower and upper bounds on the right hand
side of generalized Margolus-Levitin inequality, verified numerically up to
seven digits. Only recently there appear two proofs of the conjecture. We
provide below a very elementary new proof, based on the simplest tools from
differential calculus. Thus the generalized Margolus-Levitin speed limit can
be derived much in the spirit of the original one valid for vanishing
fidelity.

##  I Inroduction

In recent decades much effort has been devoted to the study of the so called
quantum speed limits, i.e. a lower bounds on time it takes quantum system to
evolve in some definite way. They have been intensively studied in recent
years since this problem is of great practical importance in quantum
technologies. Many forms of quantum speed limits have been derived which
involve various quantities: energy, fidelity, purity, entropy etc. and concern
closed (isolated) and open systems (see [1, 2, 3] for reviews and [4, 5], for
more recent references). In what follows we will be considering the case of
pure states of isolated systems. The prototype of quantum speed limit is the
famous Mandelstam-Tamm [6] relation which gives the lower bound on time
necessary for the quantum system to evolve from an initial state to the
orthogonal one. It reads

$t\geq\frac{\pi}{2\Delta E},$ (1)

where

$\Delta
E\equiv\sqrt{\left<\left(\hat{H}-\big{<}\hat{H}\big{>}_{0}\right)^{2}\right>_{0}}$
(2)

is the energy dispersion in the initial state ($\big{<}\ldots\big{>}_{0}$
denotes here the expectation value in the initial state). Actually, a more
general result can be derived [6, 7, 8]: for

$\delta\equiv\left|\left<\text{initial state}|\text{final
state}\right>\right|^{2}$ (3)

being the fidelity between the initial and final states, the evolution time is
bounded by

$t\geq\frac{\arccos\sqrt{\delta}}{\Delta E}.$ (4)

Margolus and Levitin [9] derived an alternative speed limit yielding lower
bound for the orthogonalization time in terms of expectation value of energy
in initial state. It reads

$t\geq\frac{\pi}{2\left<H-E_{0}\right>},$ (5)

with $E_{0}$ being the ground state energy of initial state. The natural
question arises whether the inequality (5) may be generalized to the case of
arbitrary fidelity between the initial and the final state. Giovanetti et al.
[10] have shown that the relevant bound takes the form

$t\geq\frac{\alpha(\delta)}{\left<H-E_{0}\right>},$ (6)

with $\alpha(\delta)$ being some function of fidelity. Although they did not
provided the closed analytical formula for $\alpha$, the upper and lower
bounds for the latter were given. Actually, these bounds agree numerically to
7 significant figures leading to the conjecture that, in fact, they coincide.
If this is the case the exact formula for $\alpha$ is at our disposal. Its
form is sketched on Fig. 1

Figure 1: The function $\alpha(\delta)$ entering eq. (6).

Quite recently, Hörnedal and Sönnerborn [11] provided a rigorous proof, based
on symplectic geometry, that the exact formula for $\alpha$ coincides with the
upper bound on it, derived in [10]. They rely heavily on symplectic geometric
interpretation which suggests a specific character of Margolus-Levitin bound.
The more advanced tools they use allow for drawing more detailed conclusions.
Alternative proof of the coincidence of Giovannetti et al. upper and lower
bounds on $\alpha$ has been given in [12]. It is, however, more involved and
uses the arguments which seem to deviate significantly from the clear
reasoning presented in Ref. [10].

In the present note we give a straightforward proof of the equality of upper
and lower bounds on $\alpha$, derived in the paper by Giovannetti et al. [10].

##  II Lower and upper bounds coincide

Giovanetti et al. [10] derived the lower and upper bounds

$m(\delta)\leq\alpha(\delta)\leq M(\delta)$ (7)

on the function $\alpha(\delta)$ as follows. The upper bound results by
considering the particular family of two-level states parametrized by one real
variable $\xi\in[0,1]$,

$\left|\Omega_{\xi}\right>=\sqrt{1-\xi^{2}}\left|0\right>+\xi\left|E_{0}\right>$
(8)

where $\left|0\right>$ and $\left|E_{0}\right>$ are Hamiltonian eigenstates of
energy 0 and $E_{0}$, respectively. By solving explicitly the dynamics one
easily finds the evolution time as a function of fidelity between the initial
and final state. Finally, by minimizing the latter function over $\xi\in[0,1]$
one finds the relevant upper bound. It reads

$M(\delta)=\frac{2}{\pi}\,\underset{z^{2}\leq\delta}{\min}\left(\left(\frac{1+z}{2}\right)\arccos\left(\frac{2\delta-1-z^{2}}{1-z^{2}}\right)\right);$
(9)

actually, the bound presented in [10] has a slightly different, but
equivalent, form; we prefer the expression used in [11]. The lower bound is
obtained [10] by considering the family of inequalities

$\cos x+q\sin x\geq 1-ax,$ (10)

for $x\geq 0$, $q\geq 0$. They are optimized (in the sense of choosing $a$ the
smallest real number such that the inequality (10) holds for all real $x\geq
0$) by bounding the left hand side by the linear function tangent to it and
equal to 1 for $x=0$ (cf. Fig. 2).

Figure 2: Configuration leading to optimal inequality (10).

Denoting by $y$ the $x$-coordinate of the tangent point one finds

$\displaystyle\cos y+q\sin y$ $\displaystyle=1-ay,$ (11) $\displaystyle-\sin
y+q\cos y$ $\displaystyle=-a,$ (12)

with $y$ being restricted to the interval

$y\in\left[\pi-\arctan\bigg{(}\frac{1}{q}\bigg{)},\pi+\arctan(q)\right].$ (13)

Eqs. (11)-(13) define implicitly the function $a=a(q)$. Once this function is
defined, the lower bound on $\alpha$ reads [10]

$m(\delta)=\frac{2}{\pi}\,\underset{0\leq\theta<2\pi}{\min}\left[\underset{q\geq
0}{\max}\left(\frac{1-\sqrt{\delta}(\cos\theta-q\sin\theta)}{a(q)}\right)\right].$
(14)

Our aim here is to give a simple, straightforward proof of the equality

$m(\delta)=M(\delta).$ (15)

We start with simplifying the notation in eq. (14) defining the lower bound.
To this end let us put

$\displaystyle\rho\equiv\rho(\theta,\delta)\equiv
1-\sqrt{\delta}\cos\theta\geq 0,$ (16)
$\displaystyle\sigma\equiv\sigma(\theta,\delta)\equiv\sqrt{\delta}\sin\theta,$
(17) $\displaystyle F(q,\theta,\delta)\equiv\frac{\rho+\sigma q}{a(q)}.$ (18)

Then eq. (14) takes the form

$m(\delta)=\frac{2}{\pi}\,\underset{0\leq\theta<2\pi}{\min}\left(\underset{q\geq
0}{\max}F(q,\theta,\delta)\right).$ (19)

So, in order to prove the equality (15) our only task is to analyze carefully
the behaviour of $F(q,\theta,\delta)$ in the domain $q\geq 0$,
$0\leq\theta<2\pi$, $0\leq\delta\leq 1$. Let us note that the main difficulty
comes here from the denominator on the right hand side of eq. (18); the
function $a(q)$ is defined only implicitly by solving eqs. (11)$\div$(13). The
main, yet very simple, idea to deal with this problem is to parametrize the
solutions to (11) and (12) in terms of the variable $y$. By solving them for
$q$ and $a$ we find

$\displaystyle q=\frac{1-\cos y-y\sin y}{\sin y-y\cos y},$ (20) $\displaystyle
a=\frac{1-\cos y}{\sin y-y\cos y},\quad y>0.$ (21)

The function $a=a(q)$ is now given in parametrized form, $y$ being the
relevant parameter.

Before entering the detailed analysis of the function $a(q)$ let us make the
following useful remarks. With $\delta$ fixed and $\theta$ varying in the
interval $[0,2\pi)$ eqs. (16), (17) define a circle of radius $\sqrt{\delta}$,
centered at $(0,1)$ in the $\rho-\sigma$ plane:

$(\rho-1)^{2}+\sigma^{2}=\delta.$ (22)

It appears to be convenient to define the angle $\varphi$ by

$\cos\varphi\equiv\frac{\sigma}{\sqrt{\rho^{2}+\sigma^{2}}},\quad\sin\varphi\equiv\frac{\rho}{\sqrt{\rho^{2}+\sigma^{2}}}\geq
0;$ (23)

then $\varphi$ belongs to some closed subinterval of $[0,\pi]$. The relevant
geometry is depicted on Fig. 3

Figure 3: The set of points $(\rho,\sigma)$ defined by eqs. (16), (17).

Coming back to the analysis of the function $a=a(q)$, defined implicitly by
eqs. (20), (21), let us note that eq. (20) implies

$\frac{dq}{dy}=\frac{y(y-\sin y)}{(\sin y-y\cos y)^{2}}>0,$ (24)

i.e. $q$ is a monotonically growing function of $y$ wherever it is defined. We
are interested in the solutions corresponding to $0\leq q<\infty$. It is easy
to check that, due to the constraint (13), it is sufficient to consider the
interval $y_{-}\leq y\leq y_{+}$, where $y_{\pm}$ are determined by the
conditions

$\displaystyle 1-\cos y_{-}-y_{-}\sin y_{-}$
$\displaystyle=0,\quad\frac{\pi}{2}<y_{-}<\pi,$ (25) $\displaystyle\sin
y_{+}-y_{+}\cos y_{+}$ $\displaystyle=0,\quad\pi<y_{+}<\frac{3}{2}\pi.$ (26)

Numerically, $y_{-}=2.3311$, $y_{+}=4.4934$. In the interval $[y_{-},y_{+})$
$q(y)$ is monotonically growing from 0 to $\infty$. Consequently, it is
invertible. Inserting the function $y=y(q)$ into eq. (21) one obtains the
relevant fuction $a=a(q)$. The functions $q=q(y)$, $a=a(y)$ on the interval
$[y_{-},y_{+})$ are sketched on Fig. 4 and 5, respectively.

Figure 4: The function $q=q(y)$. Figure 5: The function $a=a(y).$

In order to analyze the function $a=a(q)$ note that

$\frac{da}{dy}=\frac{\sin y(\sin y-y)}{(\sin y-y\cos y)^{2}}$ (27)

and

$\frac{da}{dq}=\frac{\nicefrac{{da}}{{dy}}}{\nicefrac{{dq}}{{dy}}}=-\frac{\sin
y}{y}.$ (28)

In order to find the explicit form of $m(\delta)$ (cf. eq. (19)) one has to
find first the maximum of $F(q,\theta,\delta)$ for $q\in[0,\infty)$, $\theta$
and $\delta$ fixed; so we are looking for $\underset{q\geq
0}{\max}\,F(q,\theta,\delta)$.

According to the discussion above one can, equivalently, consider $F$ as the
function of $y$ on the interval $[y_{-},y_{+})$. It reads

$F(q(y),\theta,\delta)=\frac{\rho(\sin y-y\cos y)+\sigma(1-\cos y-y\sin
y)}{1-\cos y}$ (29)

or, using (16), (17) and (23),

$F(q(y),\theta,\delta)=\sqrt{\rho^{2}+\sigma^{2}}\left(\frac{\cos\varphi-\cos(\varphi+y)-y\sin(\varphi+y)}{1-\cos
y}\right).$ (30)

So we have to find the maximal value of $F(q(y),\theta,\delta)$ on the
interval $[y_{-},y_{+})$. As always, the maximum may arise either at the
internal point or the end point of the interval. In order to compute the
relevant maximal value we find first the derivative

$\frac{dF(q(y),\theta,\delta)}{dy}=\sqrt{\rho^{2}+\sigma^{2}}\cdot\frac{(y-\sin
y)(\cos\varphi-\cos(\varphi+y))}{(1-\cos y)^{2}}.$ (31)

The maximal value can be attained in the internal point of $[y_{-},y_{+})$
only provided the derivative (31) vanishes. This is possible iff

$\cos(\varphi+y)=\cos\varphi.$ (32)

Keeping in mind that $0\leq\varphi\leq\pi$ and $\frac{\pi}{2}<y_{-}\leq
y<y_{+}<\frac{3}{2}\pi<2\pi$ one finds the unique solution to eq. (32)

$y=2\pi-2\varphi.$ (33)

However, $y\in[y_{-},y_{+})$ yielding the following restrictions on $\varphi$

$\pi-\frac{1}{2}y_{+}<\varphi\leq\pi-\frac{1}{2}y_{-}.$ (34)

The inequalities (34) provide the necessary conditions for
$F(q(y),\theta,\delta)$ to attain the maximal value at the internal point of
the interval $[y_{-},y_{+})$. On the contrary, if
$0\leq\varphi\leq\pi-\frac{1}{2}y_{+}$ or
$\pi-\frac{1}{2}y_{-}<\varphi\leq\pi$, $\frac{dF(q(y),\theta,\delta)}{dy}$
does not vanish at the interval $[y_{-},y_{+})$ and attains its maximal value
at one of its end points.

The value of $\varphi$ depends, by virtue of eqs. (16), (17) and (23), on
$\delta$ and $\theta$. Fixing $\delta$ and varying $0\leq\theta<2\pi$ one
finds that $\varphi$ varies over some closed subinterval of $[0,\pi]$. There
are a priori two possibilities:

  * (i)

for all $\varphi$ the inequalities (34) are obeyed. This is illustrated on
Fig. 6.

Figure 6: The inequalities (34) are obeyed for all $(\rho,\sigma)$.

Then, for any $\varphi$ eq. (32) has the unique solution in the interval
$[y_{-},y_{+})$. Moreover, one easily finds

$\frac{d^{2}F(q(y),\theta,\delta)}{dy^{2}}\bigg{|}_{F^{\prime}(y)=0}=\frac{(y-\sin
y)\sin(\varphi+y)}{(1-\cos y)^{2}}.$ (35)

Now, by virtue of (33), $\varphi+y=2\pi-\varphi$ so that
$\pi+\frac{1}{2}y_{+}>\varphi+y\geq\pi+\frac{1}{2}y_{-}$ and
$\sin(\varphi+y)<0$. Therefore, $F(q(y),\theta,\delta)$ attains unique maximum
at $y=2\pi-2\varphi$. It equals

$F_{max}=\sqrt{\rho^{2}+\sigma^{2}}\left(\frac{\pi-\varphi}{\sin\varphi}\right)$
(36)

or

$F_{max}=\left(\frac{\rho^{2}+\sigma^{2}}{\rho}\right)\arccos\left(\frac{-\sigma}{\sqrt{\rho^{2}+\sigma^{2}}}\right).$
(37)

According to the equation (19) the lower bound $m(\delta)$ is obtained by
minimalizing $F_{max}$ over the points $(\rho,\sigma)$ running the circle
$\eqref{b18}$. It is obvious that the minimum is attained for $\sigma\leq 0$.
Defining

$\omega\equiv\sqrt{\delta}\cos\theta,\quad-\sqrt{\delta}\leq\omega\leq\sqrt{\delta},$
(38)

and taking into account the above remark allows us to write

$F_{max}=\left(\frac{1-2\omega+\delta}{1-\omega}\right)\arccos\left(\frac{\sqrt{\delta-\omega^{2}}}{\sqrt{1-2\omega+\delta}}\right).$
(39)

Using the identity $\arccos(2\tau^{2}-1)=2\arccos\tau$ we rewrite (39) as

$F_{max}=\left(\frac{1-2\omega+\delta}{2(1-\omega)}\right)\arccos\left(\frac{\delta-1+2\omega-2\omega^{2}}{1-2\omega+\delta}\right).$
(40)

Finally, we note that

$z=\frac{\delta-\omega}{1-\omega}$ (41)

defines one-to-one mapping of the interval $[-\sqrt{\delta},\sqrt{\delta}]$
onto itself. Eqs. (40), (41) imply

$F_{max}=\left(\frac{1+z}{2}\right)\arccos\left(\frac{2\delta-1-z^{2}}{1-z^{2}}\right)$
(42)

yielding eq. (15).

  * (ii)

for some points on the circle $\eqref{b18}$ the inequalities (34) are
violated. This illustrated on Fig. 7. For any point $(\rho,\sigma)$ belonging
either to the arc $AB$ or $CD$ the derivative
$\frac{dF(q(y),\theta,\delta)}{dy}$ is nonvanishing in the interval
$[y_{-},y_{+})$. Consider first the arc $AB$. Then
$0\leq\varphi\leq\pi-\frac{1}{2}y_{+}$ and it is easy to see that
$\frac{dF(q(y),\theta,\delta)}{dy}>0$ in the interval $[y_{-},y_{+})$.
Therefore, the maximum is attained for $y\rightarrow y_{+}$ and reads

$F_{AB}=\frac{-\sigma}{\cos y_{+}}.$ (43)

On the contrary, on the arc $CD$ $\pi-\frac{1}{2}y_{-}<\varphi\leq\pi$ and
$\frac{dF(q(y),\theta,\delta)}{dy}<0$ on the whole interval $[y_{-},y_{+})$.
Therefore, the maximum is attained for $y=y_{-}$ and reads:

$F_{CD}=\frac{\alpha}{\sin y_{-}}.$ (44)

Figure 7: Geometric setting for violating the inequalities (34).

On the remaining arcs $AC$ and $BD$ the inequalities (34) do hold and the
reasoning leading to the eq. (37) remains valid. In order to find the minimal
value of $\underset{q\geq 0}{\max}\,F(q,\theta,\delta)$ over the circle (22)
we have to compare $F_{AB}$, eq. (43), $F_{CD}$, eq. (44) and $F_{max}$, eq.
(37). To this end we define $\psi\equiv\pi-\varphi$ and parametrize the points
$(\rho,\sigma)$ by $\psi$ (in general, to any $\psi$ there correspond two
points on the circle). The relevant parametrization of the circle (22) is
presented on Fig. 8.

Figure 8: Parametrization of the circle (22).

The coordinates $(\rho,\sigma)$ of two intersection points obey the equations

$\displaystyle\rho+\sigma\tan\psi=0,$ (45)
$\displaystyle(\rho-1)^{2}+\sigma^{2}-\delta=0,$ (46)

which yield

$\displaystyle\rho=\sin^{2}\psi\pm\sin\psi\sqrt{\delta-\cos^{2}\psi},$ (47)
$\displaystyle\sigma=-\sin\psi\cos\psi\mp\cos\psi\sqrt{\delta-\cos^{2}\psi}.$
(48)

Then

$\displaystyle
F_{max}=\left(2+\frac{\delta-1}{\sin^{2}\psi\pm\sin\psi\sqrt{\delta-\cos^{2}\psi}}\right)\psi,$
(49) $\displaystyle
F_{AB}=\frac{\sin\psi\cos\psi\pm\cos\psi\sqrt{\delta-\cos^{2}\psi}}{\cos
y_{+}},$ (50) $\displaystyle
F_{CD}=\frac{\sin^{2}\psi\pm\sin\psi\sqrt{\delta-\cos^{2}\psi}}{\sin y_{-}}.$
(51)

Now, it is straightforward to check that

$F_{max}-F_{AB}=\frac{\sin\psi\pm\sqrt{\delta-\cos^{2}\psi}}{2\sin\psi}\left(2\psi-\frac{\sin
2\psi}{\cos y_{+}}\right).$ (52)

On the $AB$ arc $2\pi\geq 2\psi\geq y_{+}>\pi$ and the right hand side is
nonnegative; it vanishes for $2\psi=y_{+}$ which corresponds to the points $A$
and $B$. Moreover $F_{AB}$ takes the minimal value at $A$. As a result,
$F_{max}$ and $F_{AB}$ attain the same minimal value on the $AB$ arc.

Analogously,

$F_{max}-F_{CD}=\frac{\sin\psi\pm\sqrt{\delta-\cos^{2}\psi}}{2\sin\psi}\left(2\psi-\frac{1-\cos
2\psi}{\sin y_{-}}\right).$ (53)

On the $CD$ arc $0\leq 2\psi\leq y_{-}$ and the right hand side is
nonnegative; it vanishes for $2\psi=y_{-}$ which corresponds to the points $C$
and $D$. $F_{CD}$ takes the minimal value at $C$. Again we conclude that
$F_{max}$ and $F_{CD}$ attain the same minimal value on the $CD$ arc.

It follows from the above discussion that also in the (ii) case one obtains
the correct value of $m(\delta)$ by minimalizing $F_{max}$ on the whole circle
(18). Therefore, we can refer to the reasoning described in (i). This
concludes the proof.

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