# Shape-Driven Caging Dynamics of Hard Polygons

Vyas Ramasubramani<EMAIL_ADDRESS>Department of Chemical Engineering,
University of Michigan, Ann Arbor, MI 48109  Thi Vo<EMAIL_ADDRESS>Department
of Chemical Engineering, University of Michigan, Ann Arbor, MI 48109  Joshua
A. Anderson<EMAIL_ADDRESS>Department of Chemical Engineering, University
of Michigan, Ann Arbor, MI 48109  Sharon C. Glotzer<EMAIL_ADDRESS>Department of Chemical Engineering, University of Michigan, Ann Arbor, MI
48109  Department of Physics, University of Michigan, Ann Arbor, MI 48109
Biointerfaces Institute, University of Michigan, Ann Arbor, MI 48109

###### Abstract

The fundamentals of Brownian motion have been largely well understood since
the early 20th century, with most recent additions focusing on understanding
anomalous diffusion via rescaling of drag coefficents. That focus emphasizes
long-time dynamic behavior, but recent results indicate that additional,
secondary modes are also present at short and intermediate times in fluids of
anisotropic particles. Here, we study the dynamics of a representative family
of nearly-hard $n$-gons. Using molecular dynamics simulations, we study a
distinct form of caging only present in anisotropic systems. We show that this
caging behavior emerges in the mean squared displacement of $n$-gons at
intermediate particle volume fractions. We then develop an extended Langevin
theory directly coupling translational and rotational motion as a function of
the relative anisotropy for different $n$-gons that predicts the observed
caging behavior. Extending our theory to incorporate secondary, off-phase
cross-correlations between particles further enables the prediction of both
translational and rotational relaxation times of the system.

The seminal contributions of Einstein and Smoluchowski in the early 1900s mark
the beginning of the modern study of Brownian motion. Their works established
a relationship between the long-time diffusion constant of a particle and its
friction coefficient – commonly referred to as the Stokes-Einstein (SE)
relation – and provided a means to estimate the diffusion coefficient in a
fluid via observation of a particle’s mean squared displacement (MSD), a
crucial step in validating the atomic theory of matter [1, 2]. Their findings
opened the floodgates to follow-up works such as the Langevin equation [3] and
the incorporation of Brownian motion into formal theories of stochastic
processes [4] that have since made the topic ubiquitous in mathematics and the
sciences.

Extensions were made to generalize the theory of translational Brownian motion
to rotational diffusion via rescaling the diffusion constant [5] as well as
direct coupling between rotational and translational diffusion for ellipsoidal
particles [6, 7], culminating in the widely used Stokes-Einstein-Debye (SED)
relation [8]. Further generalizations were proposed for anisotropic particles
[9, 10], but these results remained relatively obscure outside of a few
theoretical studies [11, 12, 13, 14] until they were experimentally validated
using high resolution digital video microscopy [15]. A common assumption
employed throughout the above works is the idea that long-time diffusion of
aspherical particles is identical to that of spherical particles and dynamical
variations arising from particle anisotropy only manifest in the short-time
regime. As such, previous theoretical developments have particularly focused
on the idea of rescaling the long-time diffusion constant of an anisotropic
system, leaving the core differential form of the Langevin equation unaltered
[9, 10, 11, 12, 13, 14].

Recent findings using modern computational and high resolution experimental
imaging techniques, however, have provided evidence for the emergence of
intermediate caging regimes in the measured MSDs of anisotropic particles [16,
8, 17, 18, 19, 20, 21, 22]. Due to the presence of particle anisotropy, such
regimes are associated with particle shape and distinct from traditional
caging observed in crowded systems of spheres. These findings point at
interesting dynamics manifesting from the interplay between instantaneous
rotational and translational motions of anisotropic particles that, upon
dissection, could reveal new physics. Doing so, however, necessitates a
reconsideration of the form of the classical Langevin equation and subsequent
relaxation of the assumption that rescaling the long-time diffusion constant
for anisotropic systems is sufficient for capturing the relevant dynamical
behaviors. This is because solutions of the classical Langevin equation can
only predict a short-time ballistic and long-time diffusive behavior. As such,
any intermediate caging modes cannot be captured using the approximations
inherent in its derivation.

Previous efforts aimed at understanding the instantaneous behaviors of
dynamical systems culminated in well-established and validated theories of
time correlation functions as well as mode coupling analyses that are now
widely employed in the study of liquids [23, 24, 25, 26]. Such theories take
advantage of the projection-operator formalism and memory functions for
predicting dynamical responses across a wide range of different systems.
Examples include standard Lennard-Jones [27, 28], charged [29], molecular [30,
31], and even polymeric [32, 33] liquids. However, due to the complexity
arising from interactions intrinsic to real liquids, these approaches often
employ input from computer simulations to define the decay of the relevant
memory functions or self-consistent techniques in cases where phenomenological
simplifications can be made to approximate their functional forms [27, 28, 29,
30, 31, 32, 33]. This is particularly true for systems of anisotropic
particles where analytical treatments opt to separate translational and
rotational behaviors and resort to simulation inputs to study the effects of
coupling [34, 35, 36]. As such, simple solutions akin to those produced by the
Langevin equation remain elusive for anisotropic systems.

In this work, we study a model system of hard anisotropic particles such that
all deviations away from classical Brownian motion can be attributed to
particle shape and the effects of anisotropy on collisions. In this regard,
this system serves as an ideal testing ground for developing the extensions to
theories of Brownian motion. Previous studies have shown that hard anisotropic
particles develop statistical, effective directional entropic forces (DEFs)
upon crowding to maximize the system entropy [37, 38]. Our results indicate
that these DEFs lead to the emergence of a caging mode at intermediate
particle volume fraction that arises solely due to particle shape, distinct
from other modes of caging for systems of isotropic particles. We propose a
generalized Langevin theory that stems from simplifications of the previously
mentioned projection-operator approaches in the limit of hard, otherwise non-
interacting, anisotropic particles and show that the new theory captures the
volume fraction dependent emergence of this shape-driven caging behavior. We
further show that an extension of this theory predicts the translational and
rotational relaxation times for different shapes, including unexpected non-
monotonic behavior with respect to the number of sides of the polygon that
hints at the presence of more complex, higher-order effects.

## MSD of Hard Polygons

— We simulated nearly-hard polygons using a recently developed anisotropic
analogue to the Weeks-Chandler-Anderson (WCA) potential [39, 40]. Molecular
dynamics simulations were conducted in the NVT ensemble with a Nosé-Hoover
thermostat [41] using the HOOMD-blue simulation package [42]. We consider unit
area regular $n$-gons where $n$ ranges from $3-9$ across a wide range of
particle volume fractions $\phi$ corresponding to fluid phases for all
polygons. Each simulation contains $N=1024$ particles run using HOOMD-blue’s
reduced units of $\epsilon=1$, temperature $T^{*}=1$, and a timestep of
$10^{-4}$.

Figure 1:  The MSD of polygons at $\phi=0.68$. At this intermediate particle
volume fraction the MSD exhibits a caging regime (dashed black line, slope
$m=1.01$) prior to the onset of diffusion (dashed gray line, slope $m=0.60$)
for polygons. The solid black line is the MSD of disks. Although this caging
regime is present for all polygons, it appears at different particle volume
fractions for triangles and squares (inset) than for other polygons. The plot
is shown in the reduced distance and time units used by HOOMD-blue. The MSD
was computed using the freud analysis package [43].

While the MSDs of these shapes all collapse onto the classical limit for
dilute fluids (Section I, SI), the MSDs at intermediate particle volume
fractions clearly show a plateau indicating caging behavior that depends on
the number of sides of the polygon, appearing around $\phi=0.68$ for $n\geq 5$
and $\phi=0.75$ for triangles and squares (see Fig. 1). Although hard disk
fluids also exhibit various forms of caging, the particle volume fraction at
which we observe this caging transition is evidently a function of particle
shape, suggesting that this particular caging behavior is influenced by
rotation-translation coupling, a general feature of anisotropic systems. The
projection operator formalism [26] can be used to develop a generalized
Langevin equation that accounts for this coupling, yielding an equation we can
solve for an MSD that directly embeds rotation-translation coupling. We start
with the standard equation of motion of observables $\mathbf{A}$:

$\frac{\partial}{\partial
t}\mathbf{A}(t)-i\mathbf{\Omega}\cdot\mathbf{A}+\int_{0}^{t}\mathbf{M}(t-s)\cdot\mathbf{A}(s)ds=\mathbf{f}(t)$
(1)

where $\mathbf{\Omega}$ is the frequency matrix, $\mathbf{M}$ is the memory
kernel, and $\mathbf{f}$ is the random noise [26]. The classical Langevin
equation results from focusing on the linear velocity as the observable of
interest. However, anisotropic particles require a multivariate observable of
position ($\mathbf{r}$), velocity ($\mathbf{v}$), and angular velocity
($\mathbf{l}$), such that
$\mathbf{A}=\left[\mathbf{r},\mathbf{v},\mathbf{l}_{p}\right]$, where
$\mathbf{l}_{p}$ is the projected form of $\mathbf{l}$, ensuring
orthogonality. Applying the projection framework to this set of observables
gives a system of three, coupled differential equations (Section II, SI). To
parallel the approach employed in the derivation of the classical Langevin
equation, we focus only on the velocity equation:

$m\dot{\mathbf{v}}(t)+\xi\mathbf{v}(t)+\gamma_{c}\left(\alpha/I_{n}\right)^{-1/2}\mathbf{v}^{2}(t)=f(t)$
(2)

where $I_{n}$ is the moment of inertia of any given regular $n$-gon, $f$ is
the fluctuating force, $\alpha$ is a proportionality constant relating linear
and angular velocities, $\xi$ is a friction coefficient, and $\gamma_{c}$ is a
translational-to-rotational coupling constant that encodes a relationship
between translational and rotational friction.

In deriving this expression we have focused on the real components (i.e.
$i\mathbf{\Omega}\cdot\mathbf{A}=0$) and assumed a constant memory kernel. The
same approximations were used for the classical Langevin equation; therefore,
Eq. 2 can be interpreted as an anisotropic generalization of the classical
Langevin equation. In addition, to eliminate terms in angular velocity, we
derived a relationship between the linear and rotational momenta of polygons
(Section II, SI), which results in the unfamiliar term in $\mathbf{v}^{2}$. In
the limit of $\gamma_{c}\rightarrow 0$ (i.e. no rotation-translation
coupling), we recover the classical Langevin equation. Like in the classical
Langevin equation, most of these terms can be evaluated analytically within
the framework of this theory (see SI). While individual constants can be
analytically computed, no exact solution exists for Eq. 2; however, an
approximation can be obtained via Taylor expansion about the intermediate time
regime to give the following functional form for the MSD of polygons (Section
III, SI):

$\begin{split}\langle
r^{2}(t)\rangle=\frac{1}{4\gamma^{2}}\Bigg{\\{}B_{1}+B_{2}e^{-\xi
t}\Bigg{[}e^{-\xi
t}+\left(\frac{1}{\xi^{2}e^{2}}\right)t+\frac{2}{\xi^{2}}\Bigg{]}\Bigg{\\}}^{2}+\left(\frac{B_{3}}{\xi}\right)t+\frac{1}{2}\frac{B_{3}^{2}\gamma^{2}}{\xi^{4}}\hskip
25.60747pt\\\
B_{1}=4\left[\frac{B_{3}^{2}\gamma^{4}}{\xi^{4}}\right]^{2}+B_{2}\left[1+\frac{2}{\gamma^{2}}\right],\hskip
5.69054ptB_{2}=-\frac{1}{2}\left[\frac{\xi^{7}}{B_{3}^{3}\gamma^{6}}\right]\left[2\xi+\frac{2}{\xi}-\frac{1}{\xi^{2}e^{2}}\right]^{-1},\hskip
8.53581ptB_{3}\sim\frac{\langle mv^{2}(t)\rangle}{m}\end{split}$ (3)

where we have defined $\gamma=\gamma_{c}\left(\alpha/I_{n}\right)^{-1/2}$ and
$\langle r^{2}(t)\rangle\equiv\langle\left(r(t)-r(t_{o})\right)^{2}\rangle$.
Equation 3 contains the usual linear term at long times, but expanding the
exponential yields a secondary linear regime; combining both gives

$\langle
r^{2}(t)\rangle\sim\Bigg{\\{}\frac{B_{3}}{\xi}+\frac{\left(e^{-2}-2\xi\right)\left(2\xi^{-1}-B_{1}B_{2}^{-1}\right)}{2B_{2}^{2}\gamma^{2}\xi^{2}}\Bigg{\\}}t$
(4)

The first term on the right hand side of Eq. 4 is analogous to SE and the
second term results from coupling between rotation and translation. Therefore,
the long-time behavior of Eq. 3 is interpreted as perturbations about the SE
limit due to intermediate rotational-translational coupling, producing a net
“average” diffusion constant. Phenomenologically, the long-time limit
converges to the commonly employed approach of rescaling the long-time
diffusion constant. Plugging $B_{1}$ and $B_{2}$ into Eq. 4, the solution also
converge to the SE limit for $\gamma\rightarrow 0$. At short times, the
exponential terms vanish and the classical $t^{2}$ ballistic solution is
recovered. At intermediate times, however, the terms involving decaying
exponentials become non-negligible, providing the source for the observed
caging regime. As shown in Fig. 2, our theory predicts the shape-driven caging
behavior observed in the simulations across different shapes, including its
dependence on particle volume fraction (Fig. III.1, SI).

Figure 2:  Comparison of measured MSD from Fig. 1 with theory prediction at
$\phi=0.68$; all systems are in a fluid phase. Data points are from
simulation, lines indicate theory predictions. MSDs are artificially offset to
clearly show comparison between theory and simulation.

## Estimating Relaxation Times

— To gain deeper insight into the dynamical behavior of this family of
$n$-gons, we extracted the relaxation times for translation $\tau_{trans}$ and
rotation $\tau_{rot}$ from simulation (data points in Fig. 3). Inspection of
the data points in Fig. 3 reveals that $\tau_{trans}/\tau_{rot}$ drops off
more quickly for $3$-, $4$-, and $6$-gons than for other polygons, with the
data for the $6$-gons crossing that of the $5$-gons. One possible explanation
lies in the emergence of strong DEFs between polygons upon increased crowding.
Previous computational studies have shown that, while DEFs emerge in all
crowded systems of polygons, only $3$-, $4$-, and $6$-gons exhibit strong DEFs
while still in the disordered fluid phase [37, 38, 44]. Strong DEFs lead to
orientational alignment between particles that then drive the observed rapid
arrest of rotational relative to translational dynamics. Additionally, the
emergence of strong DEFs induces effective attractions between particles,
resulting in long-lived local motifs, which can be interpreted in terms of
caged dynamics [45, 32]. Our theoretical treatment thus far does not take
caging into account, and so not surprisingly is unable to properly explain the
data (Fig. III.2, SI). This motivates us to reintroduce the frequency term
$i\mathbf{\Omega}\cdot\mathbf{A}$ into the governing equations. This term
measures the off-phase cross-correlations between particles and captures local
fluctuations arising from caging. Such cross-correlations are of secondary
importance for MSD predictions since Eq. 3 is interested in the emergent, bulk
effect of rotation-translation coupling, and thus the simpler expression is
sufficient to reproduce the data. However, dissecting the interplay between
$\tau_{trans}$ and $\tau_{rot}$ seeks to decouple rotational and translational
motions, necessitating the reintroduction of relevant frequency terms. Doing
so results in the following modification of Eq. 2

$m\dot{\mathbf{v}}(t)+\left[\xi+\eta\left(\alpha/I_{n}\right)^{-1/4}\right]\mathbf{v}(t)+\gamma_{c}\left(\alpha/I_{n}\right)^{-1/2}\mathbf{v}^{2}(t)=f(t)$
(5)

where $\eta$ is a term encapsulating the projection of a polygon’s rotational
mode onto its positional mode (Section II, SI). The solution to Eq. 5 is
analogous to Eq. 3 with $\xi$ replaced with
$\xi_{\omega}\sim\xi+\eta\left(\alpha/I_{n}\right)^{-1/4}$. To derive explicit
expressions for $\tau_{trans}$ and $\tau_{rot}$ we employ the following
scaling arguments. We define $\tau_{rot}$ as the time when the the rotation-
translation coupling terms ($B_{1}$ and $B_{2}$) are dominant and
$\tau_{trans}$ as the time when the standard diffusive term $B_{3}$ dominates
in the solution to Eq. 5 (Section III, SI). The functional forms for the
relaxation times are:

$\tau_{trans}=\xi_{\omega}^{-1}\mathcal{W}\left(\frac{B_{1}B_{2}^{-1}-1+\left(\xi_{\omega}^{-1}-\xi_{\omega}^{-2}\right)\left(e^{-2}+2\right)}{\gamma
B_{2}^{2}B_{3}\xi_{\omega}^{-2}}\right)$ (6)

$\tau_{rot}=\xi_{\omega}^{-1}\mathcal{W}\left(\frac{B_{1}B_{2}^{-1}-1+\left(\xi_{\omega}^{-1}-\xi_{\omega}^{-2}\right)\left(e^{-2}+2\right)}{\left(e^{-2}-2\xi_{\omega}\right)\left(2\xi_{\omega}^{-1}-B_{1}B_{2}^{-1}\right)\xi_{\omega}^{-3}}\right)$
(7)

where $\mathcal{W}(x)$ defines the Lambert W-function [46].

Figure 3:  Ratio of translational to rotational relaxation time as a function
of particle volume fraction for different $n$-gons.
$\tau_{trans}\tau_{rot}^{-1}$ as predicted from Eqs. 6 and 7 (solid lines)
overlaid with simulation (data points).

Computing $\tau_{trans}/\tau_{rot}$ using Eqs. 6 and 7 for all $n$-gons across
different particle volume fractions shows good agreement between theory and
simulation (Fig. 3), suggesting that frequency effects can serve as a first-
order perturbative correction for relaxation behavior within the secondary
regime. Comparisons between theory and simulation for $\tau_{rot}$ and
$\tau_{trans}$ individually are shown in Fig. III.3 and III.4 of the SI,
respectively. The resulting estimate of $\tau_{trans}/\tau_{rot}$ captures the
expected limiting behaviors of the system reasonably well:
$\tau_{trans}/\tau_{rot}$ is very large at low $\phi$ because velocity
correlations die off slowly due to the low frequency of collisions.
Conversely, $\tau_{trans}/\tau_{rot}$ decays to zero at high $\phi$ because
rotations in this system become arrested (drastically augmenting $\tau_{rot}$)
while velocities are effectively random due to the large number of local
collisions. The theory even correctly predicts the crossing of the data for
$5$-gons and $6$-gons around $\phi=0.74$ observed in the simulation data.
Without the frequency term, no crossing is predicted, as shown in Fig III.2 in
the SI.

It is worthwhile noting here that there appears to be a consistent slight
overestimation of $\tau_{trans}/\tau_{rot}$ in the low $\phi$ limit prior to
the onset of caging ($\phi\leq 0.6$). We suspect that this results from our
choice to approximate the memory kernels ($\mathbf{M}$) in Eq. 1 as constants
rather than a convolutional integral with their corresponding position,
velocity, or angular velocity terms. Doing so underestimates the effects of
cross-correlations and results in an augmented $\tau_{trans}$. Addressing
these secondary effects involves dissecting the full memory kernel
$\mathbf{M}$ which goes beyond our current aim of developing an analytical
extension to the classical Langevin equation.

The observed separation of timescales for simple systems of regular polygons
points at the presence of rich, unexplored dynamical behavior within the
family of anisotropic shapes. Even more complex behaviors may lie in wait upon
extension to 3-dimensional systems of polyhedra. Proper understanding and
elucidation of such hidden interactions not only enrich our understanding of
anisotropic dynamics, but also provide crucial insights needed for future
assembly engineering applications of shaped building blocks.

This research was supported in part by the National Science Foundation,
Division of Materials Research Award # DMR 1808342 and by the Department of
the Navy, Office of Naval Research under ONR award number N00014-18-1-2497.
V.R. acknowledges the 2019-2020 J. Robert Beyster Computational Innovation
Graduate Fellowship from the College of Engineering, University of Michigan.
This research was supported in part through computational resources and
services supported by Advanced Research Computing at the University of
Michigan, Ann Arbor. This work used the Extreme Science and Engineering
Discovery Environment (XSEDE) [47], which is supported by National Science
Foundation grant number ACI-1548562; XSEDE award DMR 140129. This research
used resources of the Oak Ridge Leadership Computing Facility, which is a DOE
Office of Science User Facility supported under Contract DE-AC05-00OR22725.
Hardware provided by NVIDIA Corp. is gratefully acknowledged.

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# Lump chains in the KP-I equation

Charles Lester  Andrey Gelash  Dmitry Zakharov  Vladimir Zakharov  Department
of Physics, University of Arizona, Tucson, AZ  Skolkovo Institute of Science
and Technology, Moscow, Russia  Institute of Automation and Electrometry SB
RAS, Novosibirsk, Russia  Department of Mathematics, Central Michigan
University, Mount Pleasant, MI  Department of Mathematics, University of
Arizona, Tucson, AZ

###### Abstract

We construct a broad class of solutions of the KP-I equation by using a
reduced version of the Grammian form of the $\tau$-function. The basic
solution is a linear periodic chain of lumps propagating with distinct group
and wave velocities. More generally, our solutions are evolving linear
arrangements of lump chains, and can be viewed as the KP-I analogues of the
family of line-soliton solutions of KP-II. However, the linear arrangements
that we construct for KP-I are more general, and allow degenerate
configurations such as parallel or superimposed lump chains. We also construct
solutions describing interactions between lump chains and individual lumps,
and discuss the relationship between the solutions obtained using the reduced
and regular Grammian forms.

###### keywords:

Grammian form , lump solutions , line-solitons , tau-function

##  1 Introduction

The Kadomtsev–Petviashvili equation

$[u_{t}+6uu_{x}+u_{xxx}]_{x}=-3\alpha^{2}u_{yy}$ (1)

was derived in [15], and was first mentioned by its current name in [41]. The
KP equation is the subject of hundreds of research papers and several
monographs [2, 16, 18, 25, 26]. The KP-I and KP-II forms of the equation are
physically distinct and correspond to $\alpha^{2}=-1$ and $\alpha^{2}=1$,
respectively.

The KP-I and KP-II equations are universal models describing weakly nonlinear
waves in media with dispersion of velocity. However, from a mathematical point
of view they are quite distinct. They have numerous physical applications,
such as the theory of shallow water waves (see, for instance, the monographs
[2, 16]) and plasma physics (Kadomtsev and Petviashvili were both renowned
plasma physicists). Both KP-I and KP-II are Hamiltonian systems. The Cauchy
problem for both equations is uniquely solvable for initial data in $L^{1}$
(see [11, 22, 44]). However, KP-II is completely integrable, while KP-I, in
general, is not (see the paper [40] for the analysis of the difference between
the two equations).

Both versions of the KP equation are solvable using the inverse scattering
method. The KP equation is the compatibility condition for an overdetermined
linear system

$\alpha\Psi_{y}+\Psi_{xx}+u\Psi=0,\quad\Psi_{t}+4\Psi_{xxx}+6u\Psi_{x}+(3u_{x}+3\alpha
w)\Psi=0,\quad w_{x}+u_{y}=0.$ (2)

The Lax representation for KP was found independently by Zakharov and Shabat
[41] and Dryuma [10]. For KP-I we have $\alpha=i$ and Equation (2) is a non-
stationary one-dimensional Schrödinger equation with the potential $-u$, while
$\alpha=1$ corresponds to KP-II, and the linear problem is a heat equation
with a source term. This fact alone reveals the substantial underlying
difference between the theories of KP-I and KP-II.

The KP-I equation has a rich family of rational solutions, describing the
interactions of stable, spatially localized solitons known as _lumps_. A lump
solution of KP-I was first constructed numerically by Petviashvili [31], who
developed an original method for numerically constructing stationary solutions
for a wide class of nonlinear PDEs. Lumps and their interactions were first
studied analytically in [23], and received their name in [32], where they were
constructed using the Hirota transform. Krichever [19, 20] showed that the
dynamics of the lumps in KP-I is controlled by the Calogero–Moser system.
Lumps with distinct asymptotic velocities retain their velocities and phases
after scattering, but lumps with the same velocity undergo anomalous
scattering, and may form bound states known as multilumps [12, 14, 27, 28, 29,
34]. Lump and multilump solutions of KP-I were described in the framework of
the inverse scattering method in [1, 35].

Unlike the KP-I equation, KP-II is not known to have spatially localized
solutions, nor does it have nonsingular rational solutions. Instead, the KP-II
equation has an interesting family of _line-soliton_ solutions. An individual
line-soliton is a translation-invariant traveling wave. When several line-
solitons interact, they form complicated evolving polyhedral arrangements [3,
4, 5, 6, 8] that are described by an elaborate combinatorial theory (see [17]
and the monograph [16]). Line-soliton solutions also exist for KP-I but are
unstable with respect to transverse perturbations; this was shown in the
original paper [15] for large perturbations and in [30, 37] for all scales.
For stability of three-dimensional solitons, see [21].

The goal of this paper is to describe a new class of solutions of the KP-I
equation, which we call _lump chains_. A simple chain consists of lumps evenly
spaced along a line, with the lumps propagating with a single velocity at an
arbitrary angle to the line. Lump chains can interact by splitting or merging,
and the large-scale structure of lump chain solutions of KP-I resembles that
of the line-soliton solutions of KP-II. However, lump chains may have
degenerate behavior that does not occur with line-solitons, such as parallel
and superimposed chains. More generally, we construct solutions consisting of
lump chains that emit individual lumps, which may be then absorbed by other
chains. Solutions of KP-I containing a periodic chain of lumps have been
described by a number of authors [7, 13, 18, 30, 36, 43]. However, to the best
of our knowledge, solutions consisting of several interacting lump chains have
never been considered before.

We construct solutions of KP-I using the Grammian form of the $\tau$-function.
This form was derived using the dressing method in [41], and using Sato theory
in [24], and is perhaps less known than the Wronskian form. The dressing
method was first used to solve the KdV equation in the pioneering paper [33],
and was generalized and applied to the KP-II equation in [41]. A more modern
treatment can be found in the papers [38, 39, 42].

As we have noted, individual lump solutions of KP-I are stable, while line-
solitons and lump chains are unstable. In [30] it was shown that a line-
soliton can emit a lump chain, hence the latter should be considered as an
intermediate stage of the instability development. In the long run, a line-
soliton transforms into an expanding cloud of lumps, which can be treated as a
model of integrable turbulence.

##  2 The Grammian form of the $\tau$-function

The purpose of this paper is to study a family of solutions of the KP-I
equation that can be constructed using the Grammian form of the
$\tau$-function, which we now recall [24, 25, 30]. Fix a positive integer $M$,
which we call the _rank_ of the solution. Let $\psi_{j}=\psi^{+}_{j}(x,y,t)$
for $j=1,\ldots,M$ be a linearly independent set of solutions to the linear
system

$i\partial_{y}\psi+\partial_{x}^{2}\psi=0,\quad\partial_{t}\psi+4\partial_{x}^{3}\psi=0,$
(3)

and similarly let $\psi^{-}_{j}(x,y,t)$ be solutions to the conjugate system

$i\partial_{y}\psi-\partial_{x}^{2}\psi=0,\quad\partial_{t}\psi+4\partial_{x}^{3}\psi=0.$

Assume that all $\psi^{\pm}_{j}$ lie in $L^{2}((-\infty,x_{0}])$ with respect
to the variable $x$ for any $x_{0}$, and let $c_{jk}$ be an arbitrary constant
$M\times M$-matrix. Then the function

$u(x,y,t)=2\partial_{x}^{2}\log\tau,\quad\tau(x,y,t)=\det\left[c_{jk}+\langle\psi^{+}_{j},\psi^{-}_{k}\rangle\right],\quad\langle\psi^{+}_{j},\psi^{-}_{k}\rangle=\int_{-\infty}^{x}\psi_{j}^{+}(x^{\prime},y,t)\psi_{k}^{-}(x^{\prime},y,t)dx^{\prime}$
(4)

is a solution of the KP-I equation (1). To obtain real-valued solutions, we
let $c_{jk}$ be real-valued, and we set $\psi^{-}_{j}=\overline{\psi}_{j}$.

It is customary to choose $c_{jk}=\delta_{jk}$ to ensure that the solution (4)
is non-singular; we call solutions of KP-I obtained in this way _regular_. In
this paper, however, we are more interested in the case $c_{jk}=0$; we call
such solutions _reduced_. We note that if the solutions $\psi_{j}$ are
linearly independent, then the reduced $\tau$-function (4) is the determinant
of a Gram matrix, and hence the solution is nonsingular. We discuss the
relationship between regular and reduced solutions of KP-I in Section 4, for
now we note that the latter can be obtained from the former by setting
$\psi^{+}_{j}=C\psi_{j}$, where $C$ is a real constant, and taking the limit
$C\to+\infty$. It would also be interesting to consider solutions where the
matrix $c_{jk}$ is nonzero but does not have maximal rank, however this is
beyond the scope of our paper.

In this paper, we restrict our attention to functions $\psi_{j}$ with finite
spectral support. Fix a positive integer $N$, called the _order_ of the
solution, and fix distinct eigenvalues $\lambda_{1},\ldots,\lambda_{N}$ with
positive real parts. Denote

$\phi(x,y,t,\lambda)=\lambda x+i\lambda^{2}y-4\lambda^{3}t,$

and let $p_{s}(x,y,t,\lambda)$ denote the polynomial (homogeneous of degree
$s$ in $x$, $y$, and $t$) defined by

$p_{s}(x,y,t,\lambda)=e^{-\phi(x,y,t,\lambda)}\partial_{\lambda}^{s}e^{\phi(x,y,t,\lambda)},$

so that for example

$p_{0}=1,\quad p_{1}=x+2i\lambda y-12\lambda^{2}t,\quad
p_{2}=p_{1}^{2}+2iy-24\lambda t,\quad\ldots$

Any function of the form $\partial_{\lambda}^{s}e^{\phi}=p_{s}e^{\phi}$ is a
solution of (3).

We now consider solutions of KP-I given by the tau-function (4), where the
eigenfunctions $\psi_{j}$ are given by

$\psi_{j}(x,y,t)=\sum_{n=1}^{N}\sum_{s=0}^{S}C_{jns}p_{s}(x,y,t,\lambda_{jn})e^{\phi(x,y,t,\lambda_{jn})}.$
(5)

The highest degree $S$ of a polynomial $p_{s}$ that occurs in any of the
$\psi_{j}$ is called the _depth_ of the solution. The complex constants
$C_{jns}$ are required to satisfy a non-degeneracy condition to ensure that
the functions $\psi_{j}$ are linearly independent. We do not spell out this
condition, and instead verify it in each particular example.

An exhaustive classification of the solutions of KP-I obtained in this manner
is far beyond the scope of this paper. Instead, our goal is to describe
several interesting families of solutions that illuminate the behavior of the
generic solution.

  1. 1.

_Line-solitons._ The simplest solution of KP-I, called a _line-soliton_ , is
the regular solution obtained from (4) and (5) for $M=1$, $N=1$, and $S=0$, in
other words by setting $\psi(x,y,t)=Ce^{\phi(x,y,t,\lambda)}$. This solution
is a translation-invariant traveling wave, and a similar solution exists for
KP-II. However, unlike the KP-II case, a line-soliton solution of KP-I is
unstable (see [30, 37]).

  2. 2.

_Rational solutions: lumps and multi-lumps._ A distinguishing feature of the
KP-I equation is the existence of rational, spatially localized solutions,
which are not known for KP-II. Consider the solution of KP-I given by (4) and
(5), where each function $\psi_{j}$ is a polynomial multiple of a single
exponential $e^{\phi(x,y,t,\lambda_{j})}$ (the eigenvalues $\lambda_{j}$
corresponding to the $\psi_{j}$ may or may not be distinct). In this case the
integral $\langle\psi^{+}_{j},\psi^{-}_{k}\rangle$ occurring in (4) is a
polynomial multiple of $e^{(\lambda_{j}+\overline{\lambda}_{k})x}$. In the
regular case (when $c_{jk}=\delta_{jk}$), the $\tau$-function is a sum of
distinct exponentials. However, in the reduced case (when $c_{jk}=0$), the
$\tau$-function is a polynomial multiple of a single exponential term
$\exp\sum(\lambda_{j}+\overline{\lambda}_{j})x$, and the exponential
disappears when taking the second logarithmic derivative. Therefore, the
corresponding solution $u$ is a rational function of $x$, $y$, and $t$. These
are the so-called _lump_ and _multi-lump_ solutions of KP-I. Corresponding to
each distinct eigenvalue $\lambda_{j}$ there is a lump, or, more generally, a
collection of lumps, whose number is related to the depth $S$. The lumps in
each collection are either bounded or undergo anomalous scattering, while the
collections of lumps corresponding to different $\lambda_{j}$ undergo normal
scattering without phase shifts. Multilump solutions of KP-I were obtained in
a number of papers (see for example [12, 14, 19, 20, 27, 28, 29, 34]). The
most general Grammian form of the multilump solutions of KP-I was considered
in [9].

  3. 3.

_Lump chains._ In this paper, we are mostly concerned with reduced solutions
of depth $S=0$, in other words when each function $\psi_{j}$ is a linear
combination of exponentials. In order for the solution to be non-singular, we
require $N\geq M$. As we will see, the corresponding reduced solution $u$ of
KP-I is an arrangement of _lump chains_ , which are sequences of lumps moving
along parallel trajectories (the group velocity of the chain is in general
distinct from the velocity of the individual lumps). The time evolution of the
underlying linear arrangement supporting the lumps is very similar to that of
the line-soliton solutions of KP-II (see [16]). However, the linear
arrangements that can occur for lump chains are more general than those of KP-
II line-solitons, and allow for various degenerate configurations such as
parallel or superimposed lump chains. The regular solution of KP-I of depth
$S=0$ and rank $M=1$ consists of a linear arrangement of lump chains
interacting with a single line-soliton of KP-I. We give a detailed description
of certain families of reduced lump chain solutions in Section 3, and we give
a single example of a regular solution of depth $S=0$ in Section 4.

  4. 4.

_Lumps and lump chains._ In Section 4, we also construct an example of a
reduced solution of depth $S>0$ that is not rational. The solution consists of
a chain of lumps that emits, at a certain moment of time, a single lump which
propagates away from the chain. We conjecture that the general reduced
solution of KP-I with depth $S>0$ consists of an arrangement of lump chains,
with an additional number of individual lumps being either emitted or
reabsorbed by the chains. The general regular solution consists of such an
arrangement, and additional line-solitons.

##  3 Lump chains: reduced solutions of depth $S=0$.

In this section, we consider solutions of KP-I given by (4) that are reduced
($c_{jk}=0$), and where the auxiliary $\psi$-functions are given by (5) with
depth $S=0$, in other words are sums of pure exponential terms with no
polynomial multiples. We mostly focus on solutions of rank $M=1$, in other
words having the form

$u(x,y,t)=2\partial_{x}^{2}\log\tau,\quad\tau(x,y,t)=\int_{-\infty}^{x}|\psi(z,t,y)|^{2}\,dz,$
(6)

where the $\psi$-function is a sum of $N$ exponentials. We will see that such
a solution is an arrangement of linear _lump chains_ , with the individual
lumps moving with constant velocity along the chains, and the entire assembly
evolving with time. Such lump chain solutions of KP-I bear a strong
resemblance to the well-known line-soliton solutions of KP-II, which are the
subject of an elaborate combinatorial theory (see [16]).

The $\psi$-function defining a lump chain solution of rank $M=1$ and order $N$
is defined by $2N$ complex parameters. It is convenient to introduce them as
follows. Let

$\lambda_{n}=a_{n}+ib_{n},\quad\theta_{n}=\rho_{n}+i\varphi_{n},\quad
n=1,\ldots,N,$

be complex constants, where we assume that $0<a_{1}\leq a_{2}\leq\cdots\leq
a_{N}$ and that $\lambda_{n}\neq\lambda_{m}$ for $n\neq m$. Define the
functions

$\Phi_{n}(x,y,t)=\lambda_{n}x+i\lambda_{n}^{2}y-4\lambda_{n}^{3}t+\theta_{n},$

then the function

$\psi(x,y,t)=\sum_{n=1}^{N}\sqrt{2a_{n}}e^{\Phi_{n}(x,y,t)}$

satisfies the linear system (3). Plugging $\psi$ into (6), we obtain the
following formula for the $\tau$-function:

$\tau(x,y,t)=\sum_{n=1}^{N}e^{2F_{n}}+\sum_{n=1}^{N-1}\sum_{m=n+1}^{N}2\mu_{nm}e^{F_{n}+F_{m}}\cos(G_{n}-G_{m}-\varphi_{nm}),$
(7)

where we have denoted

$F_{n}(x,y,t)=\operatorname{Re}\Phi_{n}(x,y,t),\quad
G_{n}(x,y,t)=\operatorname{Im}\Phi_{n}(x,y,t),$

and the constants $\mu_{nm}$ and $\varphi_{nm}$ are given by

$\mu_{nm}=2\sqrt{\frac{a_{n}a_{m}}{(a_{n}+a_{m})^{2}+(b_{n}-b_{m})^{2}}},\quad\varphi_{nm}=\tan^{-1}\left(\frac{b_{n}-b_{m}}{a_{n}+a_{m}}\right).$

The large-scale structure of the solution is governed by the linear functions

$F_{nm}(x,y,t)=F_{n}(x,y,t)-F_{m}(x,y,t)=A_{nm}x+B_{nm}y+C_{nm}t+D_{nm}=0,$

where

$A_{nm}=\operatorname{Re}(\lambda_{n}-\lambda_{m}),\quad
B_{nm}=-\operatorname{Im}(\lambda_{n}^{2}-\lambda_{m}^{2}),\quad
C_{nm}=-4\operatorname{Re}(\lambda_{n}^{3}-\lambda_{m}^{3}),\quad
D_{nm}=\operatorname{Re}(\theta_{n}-\theta_{m}).$ (8)

We now describe the structure of the corresponding solution $u(x,y,t)$ for
$N\geq 2$ (the solution is trivial for $N=1$). We note that adding a common
complex constant to the $\theta_{n}$ multiplies $\tau$ by a real constant, and
hence does not change $u$, therefore the solution is in fact determined by
$2N-1$ complex parameters.

###  3.1 Lump chain of rank $M=1$ and order $N=2$.

The solution of KP-I with $\tau$-function (7) of rank $M=1$ and order $N=2$ is
the basic building block for solutions of order $N\geq 3$, so we study it in
detail. This solution is a linear traveling wave consisting of an infinite
chain of lumps, and is analogous to the simple line-soliton solution of KP-II.
In the $N=2$ case, the $\tau$-function (7) can be simplified by factoring out
the exponential term $e^{F_{1}+F_{2}}$. The corresponding solution of KP-I is
given by

$u(x,y,t)=2\frac{\partial^{2}}{\partial
x^{2}}\log\left[\cosh(F_{21})+\mu_{12}\cos(G_{2}-G_{1}-\varphi_{21})\right].$
(9)

The arguments of the $\cosh$ and $\cos$ functions are linear:

$F_{21}=A_{21}x+B_{21}y+C_{21}t+D_{21},\quad
G_{2}-G_{1}-\varphi_{21}=\alpha_{21}x+\beta_{21}y+\gamma_{21}t+\delta_{21},$

where we have denoted

$\alpha_{21}=\operatorname{Im}(\lambda_{2}-\lambda_{1}),\quad\beta_{21}=\operatorname{Re}(\lambda_{2}^{2}-\lambda_{1}^{2}),\quad\gamma_{21}=-4\operatorname{Im}(\lambda_{2}^{3}-\lambda_{1}^{3}),\quad\delta_{21}=\operatorname{Im}(\theta_{2}-\theta_{1}).$

Introduce the quantities

$X=\frac{B_{21}\gamma_{21}-C_{21}\beta_{21}}{A_{21}\beta_{21}-B_{21}\alpha_{21}},\quad
Y=\frac{C_{21}\alpha_{21}-A_{21}\gamma_{21}}{A_{21}\beta_{21}-B_{21}\alpha_{21}},$

then the non-degeneracy condition $0<a_{1}\leq a_{2}$ implies that the vector
$(X,Y)$ is nonzero, and that the denominators do not vanish. The functions
$F_{21}=F_{2}-F_{1}$ and $G_{2}-G_{1}$, and therefore $u$, satisfy the
differential equation $f_{t}+Xf_{x}+Yf_{y}=0$, hence $u(x,y,t)=u(x-Xt,y-Yt)$
is a traveling wave with the velocity vector $(X,Y)$. Furthermore, $u$
satisfies the stationary Boussinesq equation

$\left[-Xu_{x}-Yu_{y}+6uu_{x}+u_{xxx}\right]_{x}=3u_{yy}.$

For a fixed moment of time $t$, the solution (9) is localized near the line
$F_{21}=0$ in the $(x,y)$-plane. Indeed, away from the line, the argument of
$\cosh$ has a large absolute value, so the $\tau$-function has a single
dominant exponential term and hence $u$ is exponentially small. The normal
direction vector $\mathbf{U}_{21}=(A_{21},B_{21})$ of the line $F_{21}=0$ may
be an arbitrary nonzero vector, and is not in general parallel to the velocity
$(X,Y)$. In particular, if $a_{1}=a_{2}$ then the line $F_{21}=0$ is parallel
to the $x$-axis, which cannot happen for a line-soliton of KP-II. The line
$F_{21}=0$ propagates with the wave vector

$\mathbf{V}_{21}=-\frac{C_{21}}{A_{21}^{2}+B_{21}^{2}}(A_{21},B_{21}),$ (10)

and is stationary if $C_{21}=0$ (which also cannot happen for a KP-II line-
soliton). Note, however, that $C_{21}\neq 0$ if the line $F_{21}=0$ is
vertical.

Along the line $F_{21}=0$, the phase of the solution (9) is determined by the
argument of the cosine function. The solution is periodic along the line, and
consists of a sequence of lumps (see Figure 1). The distance between two
consecutive lumps is equal to

$L_{21}=2\pi\frac{\sqrt{A_{21}^{2}+B_{21}^{2}}}{A_{21}\beta_{21}-B_{21}\alpha_{21}}.$
(11)

The individual lumps propagate with velocity vector $(X,Y)$, which may be
oriented arbitrarily relative the chain $F_{21}=0$. To see that the individual
peaks are indeed KP-I lumps, we note that the distance $L_{21}$ between two
consecutive lumps diverges as $\lambda_{2}\to\lambda_{1}$. Setting

$a_{1}=a-\varepsilon,\quad b_{1}=b-\varepsilon\mu,\quad
a_{2}=a+\varepsilon,\quad
b_{2}=b+\varepsilon\mu,\quad\theta_{1}=\theta_{2}=0,$

in the limit $\varepsilon\to 0$ we obtain (for $t=0$ and arbitrary $\mu$) the
lump solution of KP-I:

$u(x,y)=2\frac{\partial^{2}}{\partial
x^{2}}\log[1+a^{2}(x-by)^{2}+a^{4}y^{2}]=\frac{4a^{2}(1-a^{2}(x-by)^{2}+a^{4}y^{2})}{(1+a^{2}(x-by)^{2}+a^{4}y^{2})^{2}}.$
(12)

The KP-I equation has infinitely many integrals of motion, the simplest being
$\displaystyle\int_{-\infty}^{\infty}u(x,y,t)dx$ (in general, this integral is
a linear function of $y$, but for our solutions it is in fact constant). It is
easy to verify that for a lump chain of order $N=2$ we have

$\frac{1}{4}\int_{-\infty}^{\infty}u(x,y,t)\,dx=A_{21}.$

We call the quantity $A_{21}$ the _flux_ of the lump chain.

We note that the integral $\displaystyle\int_{-\infty}^{\infty}u(x,y,t)\,dx$
is equal to zero for a one-lump solution (12), since $u(x,y)$ is the
$x$-derivative of a rational function that vanishes at infinity. This agrees
with the limiting procedure, since $A_{21}\to 0$ as
$\lambda_{2}\to\lambda_{1}$. However, for the one-lump solution $u(x,y)$ given
by (12) the integral over the entire plane is nonzero:

$\int_{\mathbb{R}^{2}}u(x,y)\,dx\wedge dy=\frac{4\pi}{a}>0.$

There is no contradiction here, since $u(x,y)$ does not vanish sufficiently
rapidly as $x^{2}+y^{2}\to\infty$, and this improper integral cannot be
evaluated using Fubini’s theorem.

Figure 1: Reduced lump chain of order $M=1$ and rank $N=2$, given by Equation
(9) with $\lambda_{1}=1/2+i/2$ and $\lambda_{2}=3/8-i/4$ at $t=0$. (a) $2$D
profile of $u(x,y)$. (b) Amplitude of $u(\tilde{x},y)$ along the line
$F_{12}=0$.

It has already been observed by a number of authors that a linear chain of
lumps can occur as part of a solution of the KP-I equation. A chain of lumps
appears in [43], and formula (9) occurs in [18] (see p. 74), but is not
analyzed in detail. In [30], chains of lumps parallel to the $y$-axis are
shown to result from the decay of an unstable line-soliton. Zaitsev [36]
developed a procedure for constructing stationary wave solutions of integrable
systems out of spatially localized solitons, and constructed a lump chain for
KP-I in this manner. Burtsev showed in [7] that a lump chain is unstable with
respect to transverse perturbations, as is the case for a line-soliton. The
development of the instability of the chain soliton was studied in [30]

###  3.2 Lump chains of rank $M=1$ and order $N=3$.

We now consider the reduced solutions $u(x,y,t)$ of KP-I with $\tau$-function
given by (7) in the case $N=3$. A generic solution of this form consists of
three lump chains meeting at a triple point, and a number of degenerate
configurations are also possible.

The $\tau$-function (7) for $N=3$ consists of three purely exponential terms
$e^{2F_{1}}$, $e^{2F_{2}}$, and $e^{2F_{3}}$, and three mixed terms. Introduce
the normal vectors $\mathbf{U}_{mn}=(A_{mn},B_{mn})$, where $A_{mn}$ and
$B_{mn}$ are given by (8). The normal vectors satisfy
$\mathbf{U}_{31}=\mathbf{U}_{21}+\mathbf{U}_{32}$, and their collinearity is
controlled by the quantity

$\eta_{123}=A_{21}B_{31}-A_{31}B_{21}=a_{1}b_{1}(a_{2}-a_{3})+a_{2}b_{2}(a_{3}-a_{1})+a_{3}b_{3}(a_{1}-a_{2}).$
(13)

For generic values of $\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$ we have
$\eta_{123}\neq 0$, and no two of the three vectors $\mathbf{U}_{mn}$ are
collinear. In this case, the $(x,y)$-plane is partitioned, for fixed $t$, into
three sectors meeting at a triple point. In each sector, one of the pure
exponential terms $e^{2F_{m}}$ is dominant, and the solution $u$ is
exponentially small. Along the boundary of two sectors, given by the equation
$F_{mn}=0$, two of the exponentials $e^{2F_{m}}$ and $e^{2F_{n}}$ are equal
and are comparable to the mixed exponential term containing $e^{F_{m}+F_{n}}$.
The triple point is given by the equation $F_{1}=F_{2}=F_{3}=0$, moves
linearly with $t$, and passes through $(0,0)$ at $t=0$ if the phases
$\theta_{m}$ are purely imaginary.

The solution itself is localized on the boundaries of the sectors. Along the
boundary $F_{mn}=0$, the solution can be approximated by an order $N=2$
solution described in Subsection 3.1. In other words, it is a lump chain with
normal vector $\mathbf{U}_{mn}$, which we call an $[m,n]$-chain. Depending on
the values of the spectral parameters, there are two possibilities. In the
first, shown on Figure 2, the $[2,1]$\- and $[3,2]$-lump chains meet at the
triple point. The individual lumps from the two chains interlace one by one
and form the new $[3,1]$-chain. Conversely, the lumps on the $[3,1]$-chain may
split at the triple point into two new chains. In either case, individual
lumps are preserved, and the fluxes of the three chains satisfy the local
conservation law $A_{31}=A_{21}+A_{32}$. In addition to the orientation of the
chains, the position of the triple point, and the velocities of the lumps
along the chains, there are two free parameters that determine the solution,
namely the relative phases
$\delta_{mn}=\operatorname{Im}(\theta_{m}-\theta_{n})$. Figure 3 shows the
solution for three different sets of values of the relative chain phases.

Figure 2:  $N=3$ lump chain with $\lambda_{1}=1/2+i/2$, $\lambda_{2}=3/8-i/4$
and $\lambda_{3}=1/4+i/8$ at different moments of time. Figure 3:  $N=3$ lump
chain with $\lambda_{1}=1/2+i/2$, $\lambda_{2}=3/8-i/4$ and
$\lambda_{3}=1/4+i/8$ at $t=0$, and with different relative phases. (a)
$\delta_{12}=0$ and $\delta_{13}=0$ (b) $\delta_{12}=\pi$ and $\delta_{13}=0$
(c) $\delta_{12}=0$ and $\delta_{13}=\pi$.

There are additionally a number of degenerate configurations, corresponding to
$\eta_{123}=0$. In this case the vectors $\mathbf{U}_{21}$, $\mathbf{U}_{31}$,
and $\mathbf{U}_{32}$ are collinear, and hence so are the lines $F_{21}=0$,
$F_{31}=0$ and $F_{32}=0$. Depending on the values of the $\lambda_{m}$, for
fixed $t$, either the $(x,y)$-plane consists of half-planes in which
$e^{2F_{1}}$ and $e^{2F_{3}}$ are dominant, or there is an additional
intermediate strip in which $e^{2F_{2}}$ is dominant. For generic
$\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$ (satisfying $\eta_{123}=0$),
the solution may consist of two parallel $[2,1]$\- and $[3,2]$-chains merging
into the $[3,1]$-chain, as shown on Figure 4. The opposite case is also
possible: a single $[3,1]$-chain may, at a certain moment of time, split into
two lump chains, both parallel to the original chain. A similar splitting
process was observed in [30].

Figure 4: Two parallel lump chains merging into one: $N$=3,
$\lambda_{1}=1/2+i/2$, $\lambda_{2}=3/8-i/4$ and $\lambda_{3}=3/7+i/8$ at
different moments of time. (a) $t=-15.0$ (b) $t=-10.0$ (c) $t=-5.0$ (d)
$t=0.0$ (e) $t=10.0$ (a) $t=20.0$

Imposing the additional condition $A_{21}C_{31}-A_{31}C_{21}=0$, we obtain a
further degeneration: the three lines $F_{21}=0$, $F_{31}=0$, and $F_{32}=0$
that can support the chains move are not only parallel, but move with equal
velocity. In this case, the solution consists either of two parallel lump
chains propagating at a fixed distance, or of a single lump chain (in the
latter case, the solution may be visually indistinguishable from a lump chain
of order $N=2$).

Finally, it is possible that the three lines $F_{21}=0$, $F_{31}=0$, and
$F_{32}=0$ are the same for all values of $t$. The three lump chains merge
into a complex, periodic or quasi-periodic chain supported along the common
line, which propagates linearly (see Figure 5).

Figure 5: Quasi-periodic lump chain of order $M=1$ and rank $N=3$,
$\lambda_{1}=1+i/2$, $\lambda_{2}=1/4-i/4$ and $\lambda_{3}=0.443+0.186i$, at
times $t=-20.0$, $t=0.0$, and $t=20.0$. Inset in (a) shows amplitude along the
quasi-periodic lump chain.

###  3.3 Lump chains of rank $M=1$ and order $N\geq 4$.

We now discuss the general form of the solution (7) for arbitrary $N$, which
is determined by the spectral parameters $\lambda_{n}$. The $\tau$-function is
a sum of purely exponential terms $e^{2F_{n}}$ and mixed exponentials
$e^{F_{n}+F_{m}}$ with trigonometric multipliers. For fixed $t$, the shape of
the solution is determined by the relative values of the $F_{n}$: if one
exponential $e^{2F_{n}}$ is dominant in the $\tau$-function, then the solution
$u$ is exponentially small (a mixed term containing $e^{F_{n}+F_{m}}$ cannot
be the only dominant term). Hence the $(x,y)$-plane is divided into finitely
many polygonal regions, in the interior of which a single term $F_{n}$ is
dominant. In fact, a given term $F_{n}$ may in general fail to be dominant
anywhere, so there may be fewer than $N$ regions, and there may be only two.
On the boundary of $F_{nm}=F_{n}-F_{m}=0$ of two regions, where $F_{n}=F_{m}$
and all other $F_{k}$ are significantly smaller, the solution can be
approximated by an order $N=2$ solution described in Subsection 3.1. In other
words, it is a lump chain with normal vector
$\mathbf{U}_{nm}=(A_{nm},B_{nm})$, which we call an $[n,m]$-chain. Hence the
$(x,y)$-plane decomposes into finitely many polygonal regions with lump chains
along the boundaries, and the entire arrangement evolves linearly with $t$.
The structure of the lump chains closely resembles the arrangement of line-
solitons in KP-II (see [3, 16]).

We do not develop a general theory describing the line structure of the
solutions. Instead, we give two generic examples of order $N=4$, and discuss
the possible degenerate behavior. The first example, shown on Figure 6, may be
called an $H$-configuration. It consists two triple points that are separated
by a lump chain bridge. The bridge contracts and disappears at $t=0$, and the
triple points scatter along a different bridge. A similar configuration
appears in the KP-II equation (see [3, 16]).

Figure 6: H-shaped arrangement of chains of order $M=1$ and rank $N=4$, with
eigenvalues $\lambda_{1}=17/32+i/8$, $\lambda_{2}=3/8-i/8$,
$\lambda_{3}=11/32+i/5$, $\lambda_{4}=3/32+i/4$ and times $t=-20.0$, $t=0$ and
$t=20.0$.

The second example, shown on Figure 7, has three triple points bounding a
finite triangular region. The region shrinks and disappears at $t=0$, and the
solution henceforth resembles a solution of order $N=3$. This configuration
can be reversed in time, with a triangular region appearing out of a triple
point. We stress that both these examples are generic, in other words the
structure of the lump chains does not change under small perturbations of the
$\lambda_{n}$.

Figure 7: Triangular arrangement of chains of order $M=1$ and rank $N=4$, with
eigenvalues $\lambda_{1}=1$, $\lambda_{2}=1/2+i\sqrt{3}/2$,
$\lambda_{3}=1/2-i\sqrt{3}/2$, $\lambda_{4}=1/\sqrt{3}-i/(2\sqrt{3})$, and
times $t=-2.0$, $t=-1.0$ and $t=0.0$.

The reader may recognize that the structure of lump chain solutions of KP-I is
very similar to the structure of line-soliton solutions of KP-II. We point out
that the line structure in the KP-I case may in fact be more complex.
Specifically, the following kinds of behavior, all of them forbidden for KP-II
line-solitons, can occur for KP-I lump chains of rank $M=1$ and order $N$.

####  3.3.1 Generic solutions: the number of chains at infinity and forbidden
configurations.

We first consider the case when the eigenvalues $\lambda_{n}$ are sufficiently
generic. A natural first question is to determine the linear configurations of
chains that may occur, in particular, the number of lump chains extending to
infinity. An order $N$ line-soliton of KP-II always has $N$ solitons extending
to infinity, but a generic order $N$ solution of KP-I may have anywhere
between 3 and $N$ infinite lump chains. Similarly, certain configurations of
lines are forbidden for KP-II line-solitons but may occur for KP-I lump chain
solutions. For example, the solution given on Figure 7 has $N=4$ and three
infinite chains, and represents a line arrangement that cannot occur in KP-II
(see Exercise 4.6 in [16]).

####  3.3.2 Degenerate solutions: stable points, parallel chains, and higher
order chains.

Various degenerate configurations may be achieved by imposing appropriate
conditions on the eigenvalues $\lambda_{n}$. The triple points where lump
chains meet may be stationary relative to one another, and may even coincide
for all times, producing stable quadruple points and points of higher
multiplicity. A solution may have sets of parallel lump chains, in which case
the number of chains at infinity may be greater than the order $N$. Finally,
lump chains may coincide, producing quasiperiodic chains of higher order.

###  3.4 Lump chains of rank $M\geq 2$.

The structure of reduced solutions of KP-I of depth $S=0$ and higher rank
$M\geq 2$ is broadly similar to the $M=1$ case. The $\tau$-function (4) is a
sum of purely exponential terms and mixed terms involving trigonometric
multipliers. For a given moment of time $t$, the $(x,y)$-plane is partitioned
into finitely many polygons, and this decomposition evolves linearly with
time. Polygonal regions may appear and disappear at certain moments of time.
The boundaries of the polygons support lump chains, and the total flux of the
lump chains arriving at a given vertex is equal to the flux of the chains that
are leaving. In degenerate cases, there may be coinciding polygonal boundaries
supporting quasiperiodic superpositions of lump chains. We give a single
example of such a solution with rank $M=2$ and order $N=4$ in Figure 8.

Figure 8: The time evolution of a rank 2 order 4 solution with eigenvalues
$\lambda_{11}=1/2+i/8$, $\lambda_{12}=3/8-i/8$, $\lambda_{13}=1/4+i/5$,
$\lambda_{14}=1/8+i/4$, $\lambda_{21}=4/9+i/9$, $\lambda_{22}=3/9-i/9$,
$\lambda_{23}=2/9+i/3$, $\lambda_{24}=1/9+i/7$

##  4 Regular solutions and solutions of depth $S>0$: line-solitons and
individual lumps.

We now discuss the relationship between regular and reduced solutions of depth
$S=0$, and solutions of positive depth. We first consider regular solutions,
and for simplicity restrict our attention to rank $N=1$. The $\tau$-function
of such a solution is nearly identical to that of the reduced solution (7),
and has the form

$\tau(x,y,t)=1+\sum_{n=1}^{N}e^{2F_{n}}+\sum_{n=1}^{N-1}\sum_{m=n+1}^{N}2\mu_{nm}e^{F_{n}+F_{m}}\cos(G_{n}-G_{m}-\varphi_{nm}).$
(14)

As before, the $(x,y)$-plane is partitioned into polygonal regions, in each of
which one of the terms in (14) is dominant. However, there is now a new
region, on which the dominant term in the $\tau$-function is the constant 1.
Since $a_{n}=\operatorname{Re}\lambda_{n}>0$, this region contains, for a
given fixed $y$, all points $(x,y)$ with sufficiently large negative $x$.
Inside this region the $\tau$-function is approximately constant, and the
solution $u$ is exponentially small. At the boundary of this region, the two
dominant terms in the $\tau$-function are the 1 and one of the exponentials
$e^{2F_{n}}$. Hence the boundary of the region where 1 dominates is a line-
soliton of KP-I, instead of a lump chain. In other words, the solution
consists of an infinite line-soliton of KP-I coupled with an arrangement of
lump chains (see Figure 9).

Figure 9: Regular solution of rank $M=1$ and order $N=3$, with eigenvalues
$\lambda_{1}=1/2+i/2$, $\lambda_{2}=3/8-i/4$, and $\lambda_{3}=1/4+i/8$, at
different moments of time.

It is possible to degenerate a regular solution to a reduced solution by
replacing the $1$ in Equation (14) with an $\varepsilon$ and taking the limit
$\varepsilon\to 0$. The line-soliton occurs on the boundary of the region
where the $\varepsilon$ is the dominant term, and this region moves in the
negative $x$-direction as $\varepsilon\to 0$. In the limit, the line-soliton
disappears to infinity, and we are left with a solution consisting entirely of
lump chains. Therefore, the limiting procedure that produces reduced solutions
out of regular solutions has the effect of removing the line-soliton and
isolating the lump chain structure.

Figure 10: A lump chain radiates an individual lump, which propagates away.
$\lambda_{1}=1/4$, $\lambda_{2}=1/2$ with multiplicities of 1 and 2,
respectively.

We also briefly consider the structure of reduced solutions of depth $S>0$.
Consider again the general form of the $\tau$-function (4), where $c_{jk}=0$
and the $\psi_{j}$ are given by (5). As discussed in Section 2, the
$\tau$-function is rational if each $\psi_{j}$ is a polynomial multiple of a
single exponential term $e^{\phi(x,y,t,\lambda_{j})}$. The corresponding
solution is localized in the $(x,y)$-plane and represents the normal (if all
$\lambda_{j}$ are distinct) or anomalous scattering of lumps, or even bound
states of lumps. We now consider what happens in general, when each $\psi_{j}$
is a multiple of several exponentials. For sufficiently large $x$ and $y$, the
polynomial terms are negligible compared to the exponentials, and the
$\psi_{j}$ can be assumed to be purely exponential. Hence the solution can be
assumed to have depth $S=0$ and is an arrangement of lump chains, as described
in Section 3. In the finite part of the $(x,y)$-plane, however, the polynomial
terms in the $\psi_{j}$ produce individual lumps. Hence, the overall structure
of the solution is an arrangement of lump chains interacting with finitely
many individual lumps: a lump chain may emit or absorb an individual lump, and
the lumps may scatter on one another. A detailed classification of such
solutions appears to be a challenging combinatorial problem. In Figure 10, we
give a single example of such a solution, consisting of a lump chain emitting
an individual lump. We note that the local number of lumps is conserved: two
lumps from the chain meet and scatter, with one lump propagating away and the
other filling the resulting gap in the chain. Figure 10 gives an example of
such a solution with rank $M=1$, order $N=2$, and depth $S=1$, with the
$\psi$-function given by

$\psi(x,y,t)=e^{\frac{1}{4}(-2t+2x+iy)}(-3t+x+iy+1)+e^{\frac{1}{16}(-t+4x+iy)}.$

##  5 Summary and conclusion

We have constructed a new family of lump chain solutions of the KP-I equations
using the Grammian form of the $\tau$-function. A simple lump chain consists
of an infinite line of equally spaced lumps. The lumps propagate with equal
velocity, which is in general distinct from the group velocity of the line.
The general solution consists of an evolving polyhedral arrangement of lump
chains. At a point where three or more lump chains meet, the individual lumps
from the incoming chains are redistributed along the outgoing chains, with the
number of lumps being locally conserved. The linear structure of the solutions
is very similar to that of the line-soliton solutions of KP-II. However,
various degenerate configurations may occur for KP-I lump chains that cannot
occur for KP-II line-solitons: parallel chains, chains of equal velocity,
quasiperiodic superimposed chains, stable points of high multiplicity, and
forbidden polyhedral configurations. We have also constructed more general
solutions of KP-I using the Grammian method. Such solutions consist of an
arrangement of lump chains as described above, together with line-solitons and
individual lumps that are emitted and/or absorbed by the lump chains. A
detailed classification of the solutions of KP-I that may be obtained by the
Grammian method is an interesting and difficult problem, and is beyond the
scope of this paper. We plan to return to this problem in future work.

##  6 Acknowledgments

The work of the second and fourth author on the third chapter was supported by
the Russian Science Foundation (Grant No. 19-72-30028).

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