Lump chains in the KP-I equation

The Kadomtsev–Petviashvili equation

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

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.

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

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

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

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

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

so that for example

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

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.

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

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

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

then the function

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

where we have denoted

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

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

where

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.

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

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).

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.

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

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.

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