LLump chains in the KP-I equation
Charles Lester a , Andrey Gelash b,c , Dmitry Zakharov d , Vladimir Zakharov b,e a Department of Physics, University of Arizona, Tucson, AZ b Skolkovo Institute of Science and Technology, Moscow, Russia c Institute of Automation and Electrometry SB RAS, Novosibirsk, Russia d Department of Mathematics, Central Michigan University, Mount Pleasant, MI e 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 formof the τ -function. The basic solution is a linear periodic chain of lumps propagating with distinct group and wavevelocities. More generally, our solutions are evolving linear arrangements of lump chains, and can be viewed as theKP-I analogues of the family of line-soliton solutions of KP-II. However, the linear arrangements that we construct forKP-I are more general, and allow degenerate configurations such as parallel or superimposed lump chains. We alsoconstruct solutions describing interactions between lump chains and individual lumps, and discuss the relationshipbetween 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 + uu x + u xxx ] x = − α 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 ofresearch papers and several monographs [2, 16, 18, 25, 26]. The KP-I and KP-II forms of the equation are physicallydistinct and correspond to α = − α =
1, respectively.The KP-I and KP-II equations are universal models describing weakly nonlinear waves in media with dispersionof velocity. However, from a mathematical point of view they are quite distinct. They have numerous physicalapplications, 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 (see [11, 22, 44]). However, KP-IIis completely integrable, while KP-I, in general, is not (see the paper [40] for the analysis of the di ff erence betweenthe two equations).Both versions of the KP equation are solvable using the inverse scattering method. The KP equation is the com-patibility condition for an overdetermined linear system α Ψ y + Ψ xx + u Ψ = , Ψ t + Ψ xxx + u Ψ x + (3 u x + α w ) Ψ = , w x + u y = . (2)The Lax representation for KP was found independently by Zakharov and Shabat [41] and Dryuma [10]. For KP-I wehave α = i and Equation (2) is a non-stationary one-dimensional Schr¨odinger equation with the potential − u , while α = ff erence 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 local-ized solitons known as lumps . A lump solution of KP-I was first constructed numerically by Petviashvili [31], whodeveloped 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 Preprint submitted to Elsevier February 16, 2021 a r X i v : . [ n li n . S I] F e b onstructed using the Hirota transform. Krichever [19, 20] showed that the dynamics of the lumps in KP-I is con-trolled by the Calogero–Moser system. Lumps with distinct asymptotic velocities retain their velocities and phasesafter scattering, but lumps with the same velocity undergo anomalous scattering, and may form bound states knownas multilumps [12, 14, 27, 28, 29, 34]. Lump and multilump solutions of KP-I were described in the framework of theinverse scattering method in [1, 35].Unlike the KP-I equation, KP-II is not known to have spatially localized solutions, nor does it have nonsingularrational 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 evolvingpolyhedral arrangements [3, 4, 5, 6, 8] that are described by an elaborate combinatorial theory (see [17] and themonograph [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 atan arbitrary angle to the line. Lump chains can interact by splitting or merging, and the large-scale structure oflump chain solutions of KP-I resembles that of the line-soliton solutions of KP-II. However, lump chains may havedegenerate 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 otherchains. 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 havenever been considered before.We construct solutions of KP-I using the Grammian form of the τ -function. This form was derived using thedressing method in [41], and using Sato theory in [24], and is perhaps less known than the Wronskian form. Thedressing method was first used to solve the KdV equation in the pioneering paper [33], and was generalized andapplied 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 intermediatestage 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 τ -function The purpose of this paper is to study a family of solutions of the KP-I equation that can be constructed using theGrammian form of the τ -function, which we now recall [24, 25, 30]. Fix a positive integer M , which we call the rank of the solution. Let ψ j = ψ + j ( x , y , t ) for j = , . . . , M be a linearly independent set of solutions to the linear system i ∂ y ψ + ∂ x ψ = , ∂ t ψ + ∂ x ψ = , (3)and similarly let ψ − j ( x , y , t ) be solutions to the conjugate system i ∂ y ψ − ∂ x ψ = , ∂ t ψ + ∂ x ψ = . Assume that all ψ ± j lie in L (( −∞ , x ]) with respect to the variable x for any x , and let c jk be an arbitrary constant M × M -matrix. Then the function u ( x , y , t ) = ∂ x log τ, τ ( x , y , t ) = det (cid:104) c jk + (cid:104) ψ + j , ψ − k (cid:105) (cid:105) , (cid:104) ψ + j , ψ − k (cid:105) = (cid:90) x −∞ ψ + j ( x (cid:48) , y , t ) ψ − k ( x (cid:48) , y , t ) dx (cid:48) (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 ψ − j = ψ j .It is customary to choose c jk = δ jk to ensure that the solution (4) is non-singular; we call solutions of KP-I obtainedin 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 ψ j are linearly independent, then the reduced τ -function (4) is the determinant of a Grammatrix, and hence the solution is nonsingular. We discuss the relationship between regular and reduced solutions of2P-I in Section 4, for now we note that the latter can be obtained from the former by setting ψ + j = C ψ j , where C is areal constant, and taking the limit C → + ∞ . It would also be interesting to consider solutions where the matrix c jk isnonzero but does not have maximal rank, however this is beyond the scope of our paper.In this paper, we restrict our attention to functions ψ j with finite spectral support. Fix a positive integer N , calledthe order of the solution, and fix distinct eigenvalues λ , . . . , λ N with positive real parts. Denote φ ( x , y , t , λ ) = λ x + i λ y − λ t , and let p s ( x , y , t , λ ) denote the polynomial (homogeneous of degree s in x , y , and t ) defined by p s ( x , y , t , λ ) = e − φ ( x , y , t ,λ ) ∂ s λ e φ ( x , y , t ,λ ) , so that for example p = , p = x + i λ y − λ t , p = p + iy − λ t , . . . Any function of the form ∂ s λ e φ = p s e φ is a solution of (3).We now consider solutions of KP-I given by the tau-function (4), where the eigenfunctions ψ j are given by ψ j ( x , y , t ) = N (cid:88) n = S (cid:88) s = C jns p s ( x , y , t , λ jn ) e φ ( x , y , t ,λ jn ) . (5)The highest degree S of a polynomial p s that occurs in any of the ψ j is called the depth of the solution. Thecomplex constants C jns are required to satisfy a non-degeneracy condition to ensure that the functions ψ j are linearlyindependent. 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 genericsolution.1. Line-solitons.
The simplest solution of KP-I, called a line-soliton , is the regular solution obtained from (4)and (5) for M = N =
1, and S =
0, in other words by setting ψ ( x , y , t ) = Ce φ ( x , y , t ,λ ) . This solution is atranslation-invariant traveling wave, and a similar solution exists for KP-II. However, unlike the KP-II case, aline-soliton solution of KP-I is unstable (see [30, 37]).2. Rational solutions: lumps and multi-lumps.
A distinguishing feature of the KP-I equation is the existence ofrational, spatially localized solutions, which are not known for KP-II. Consider the solution of KP-I given by (4)and (5), where each function ψ j is a polynomial multiple of a single exponential e φ ( x , y , t ,λ j ) (the eigenvalues λ j corresponding to the ψ j may or may not be distinct). In this case the integral (cid:104) ψ + j , ψ − k (cid:105) occurring in (4) isa polynomial multiple of e ( λ j + λ k ) x . In the regular case (when c jk = δ jk ), the τ -function is a sum of distinctexponentials. However, in the reduced case (when c jk = τ -function is a polynomial multiple of a singleexponential term exp (cid:80) ( λ j + λ 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 λ 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 boundedor undergo anomalous scattering, while the collections of lumps corresponding to di ff erent λ j undergo normalscattering without phase shifts. Multilump solutions of KP-I were obtained in a number of papers (see forexample [12, 14, 19, 20, 27, 28, 29, 34]). The most general Grammian form of the multilump solutions of KP-Iwas considered in [9].3. Lump chains.
In this paper, we are mostly concerned with reduced solutions of depth S =
0, in other wordswhen each function ψ j is a linear combination of exponentials. In order for the solution to be non-singular, werequire N ≥ 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 generaldistinct from the velocity of the individual lumps). The time evolution of the underlying linear arrangementsupporting the lumps is very similar to that of the line-soliton solutions of KP-II (see [16]). However, the lineararrangements that can occur for lump chains are more general than those of KP-II line-solitons, and allow for3arious degenerate configurations such as parallel or superimposed lump chains. The regular solution of KP-Iof depth S = M = S = Lumps and lump chains.
In Section 4, we also construct an example of a reduced solution of depth S > S >
3. Lump chains: reduced solutions of depth S = In this section, we consider solutions of KP-I given by (4) that are reduced ( c jk = ψ -functions are given by (5) with depth S =
0, in other words are sums of pure exponential terms with no polynomialmultiples. We mostly focus on solutions of rank M =
1, in other words having the form u ( x , y , t ) = ∂ x log τ, τ ( x , y , t ) = (cid:90) x −∞ | ψ ( z , t , y ) | dz , (6)where the ψ -function is a sum of N exponentials. We will see that such a solution is an arrangement of linear lumpchains , with the individual lumps moving with constant velocity along the chains, and the entire assembly evolvingwith time. Such lump chain solutions of KP-I bear a strong resemblance to the well-known line-soliton solutions ofKP-II, which are the subject of an elaborate combinatorial theory (see [16]).The ψ -function defining a lump chain solution of rank M = N is defined by 2 N complex parameters.It is convenient to introduce them as follows. Let λ n = a n + ib n , θ n = ρ n + i ϕ n , n = , . . . , N , be complex constants, where we assume that 0 < a ≤ a ≤ · · · ≤ a N and that λ n (cid:44) λ m for n (cid:44) m . Define the functions Φ n ( x , y , t ) = λ n x + i λ n y − λ n t + θ n , then the function ψ ( x , y , t ) = N (cid:88) n = (cid:112) a n e Φ n ( x , y , t ) satisfies the linear system (3). Plugging ψ into (6), we obtain the following formula for the τ -function: τ ( x , y , t ) = N (cid:88) n = e F n + N − (cid:88) n = N (cid:88) m = n + µ nm e F n + F m cos( G n − G m − ϕ nm ) , (7)where we have denoted F n ( x , y , t ) = Re Φ n ( x , y , t ) , G n ( x , y , t ) = Im Φ n ( x , y , t ) , and the constants µ nm and ϕ nm are given by µ nm = (cid:114) a n a m ( a n + a m ) + ( b n − b m ) , ϕ nm = tan − (cid:32) b n − b m a n + a m (cid:33) . 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 = , where A nm = Re( λ n − λ m ) , B nm = − Im( λ n − λ m ) , C nm = − λ n − λ m ) , D nm = Re( θ n − θ m ) . (8)We now describe the structure of the corresponding solution u ( x , y , t ) for N ≥ N = θ n multiplies τ by a real constant, and hence does not change u ,therefore the solution is in fact determined by 2 N − .1. Lump chain of rank M = and order N = . The solution of KP-I with τ -function (7) of rank M = N = N ≥
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 = τ -function (7) can be simplifiedby factoring out the exponential term e F + F . The corresponding solution of KP-I is given by u ( x , y , t ) = ∂ ∂ x log (cid:2) cosh( F ) + µ cos( G − G − ϕ ) (cid:3) . (9)The arguments of the cosh and cos functions are linear: F = A x + B y + C t + D , G − G − ϕ = α x + β y + γ t + δ , where we have denoted α = Im( λ − λ ) , β = Re( λ − λ ) , γ = − λ − λ ) , δ = Im( θ − θ ) . Introduce the quantities X = B γ − C β A β − B α , Y = C α − A γ A β − B α , then the non-degeneracy condition 0 < a ≤ a implies that the vector ( X , Y ) is nonzero, and that the denominators donot vanish. The functions F = F − F and G − G , and therefore u , satisfy the di ff erential equation f t + X f x + Y f y = u ( x , y , t ) = u ( x − Xt , y − Yt ) is a traveling wave with the velocity vector ( X , Y ). Furthermore, u satisfies thestationary Boussinesq equation (cid:104) − Xu x − Yu y + uu x + u xxx (cid:105) x = u yy . For a fixed moment of time t , the solution (9) is localized near the line F = x , y )-plane. Indeed, away fromthe line, the argument of cosh has a large absolute value, so the τ -function has a single dominant exponential term andhence u is exponentially small. The normal direction vector U = ( A , B ) of the line F = X , Y ). In particular, if a = a then the line F = x -axis, which cannot happen for a line-soliton of KP-II. The line F = V = − C A + B ( A , B ) , (10)and is stationary if C = C (cid:44) F = F =
0, the phase of the solution (9) is determined by the argument of the cosine function. Thesolution is periodic along the line, and consists of a sequence of lumps (see Figure 1). The distance between twoconsecutive lumps is equal to L = π (cid:113) A + B A β − B α . (11)The individual lumps propagate with velocity vector ( X , Y ), which may be oriented arbitrarily relative the chain F =
0. To see that the individual peaks are indeed KP-I lumps, we note that the distance L between two consecutivelumps diverges as λ → λ . Setting a = a − ε, b = b − εµ, a = a + ε, b = b + εµ, θ = θ = , in the limit ε → t = µ ) the lump solution of KP-I: u ( x , y ) = ∂ ∂ x log[1 + a ( x − by ) + a y ] = a (1 − a ( x − by ) + a y )(1 + a ( x − by ) + a y ) . (12)5 igure 1: Reduced lump chain of order M = N =
2, given by Equation (9) with λ = / + i / λ = / − i / t = (a) u ( x , y ). (b) Amplitude of u ( ˜ x , y ) along the line F = The KP-I equation has infinitely many integrals of motion, the simplest being (cid:90) ∞−∞ u ( x , y , t ) dx (in general, thisintegral 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 oforder N = (cid:90) ∞−∞ u ( x , y , t ) dx = A . We call the quantity A the flux of the lump chain.We note that the integral (cid:90) ∞−∞ 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 → λ → λ . However, for the one-lump solution u ( x , y ) given by (12) the integral over the entire plane is nonzero: (cid:90) R u ( x , y ) dx ∧ dy = π a > . There is no contradiction here, since u ( x , y ) does not vanish su ffi ciently rapidly as x + y → ∞ , and this improperintegral cannot be evaluated using Fubini’s theorem.It has already been observed by a number of authors that a linear chain of lumps can occur as part of a solution ofthe KP-I equation. A chain of lumps appears in [43], and formula (9) occurs in [18] (see p. 74), but is not analyzedin 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 spatiallylocalized solitons, and constructed a lump chain for KP-I in this manner. Burtsev showed in [7] that a lump chain isunstable with respect to transverse perturbations, as is the case for a line-soliton. The development of the instabilityof the chain soliton was studied in [30] = and order N = . We now consider the reduced solutions u ( x , y , t ) of KP-I with τ -function given by (7) in the case N =
3. A genericsolution of this form consists of three lump chains meeting at a triple point, and a number of degenerate configurationsare also possible.The τ -function (7) for N = e F , e F , and e F , and three mixed terms.Introduce the normal vectors U mn = ( A mn , B mn ), where A mn and B mn are given by (8). The normal vectors satisfy U = U + U , and their collinearity is controlled by the quantity η = A B − A B = a b ( a − a ) + a b ( a − a ) + a b ( a − a ) . (13)6 igure 2: N = λ = / + i / λ = / − i / λ = / + i / ff erent moments of time.Figure 3: N = λ = / + i / λ = / − i / λ = / + i / t =
0, and with di ff erent relative phases. (a) δ = δ = (b) δ = π and δ = (c) δ = δ = π . For generic values of λ , λ , and λ we have η (cid:44)
0, and no two of the three vectors 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 pureexponential terms e F 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 F m and e F n are equal and are comparable to the mixedexponential term containing e F m + F n . The triple point is given by the equation F = F = F =
0, moves linearly with t , and passes through (0 ,
0) at t = θ m are purely imaginary.The solution itself is localized on the boundaries of the sectors. Along the boundary F mn =
0, the solution canbe approximated by an order N = U mn , which we call an [ m , n ]-chain. Depending on the values of the spectral parameters, there are twopossibilities. In the first, shown on Figure 2, the [2 , , , , A = A + A . 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 thatdetermine the solution, namely the relative phases δ mn = Im( θ m − θ n ). Figure 3 shows the solution for three di ff erentsets of values of the relative chain phases.There are additionally a number of degenerate configurations, corresponding to η =
0. In this case the vectors U , U , and U are collinear, and hence so are the lines F = F = F =
0. Depending on the valuesof the λ m , for fixed t , either the ( x , y )-plane consists of half-planes in which e F and e F are dominant, or there is an7dditional intermediate strip in which e F is dominant. For generic λ , λ , and λ (satisfying η = , , , , Figure 4: Two parallel lump chains merging into one: N = λ = / + i / λ = / − i / λ = / + i / ff erent moments of time. (a) t = − . (b) t = − . (c) t = − . (d) t = . (e) t = . (a) t = . Imposing the additional condition A C − A C =
0, we obtain a further degeneration: the three lines F = F =
0, and F = N = F = F =
0, and F = t . Thethree lump chains merge into a complex, periodic or quasi-periodic chain supported along the common line, whichpropagates linearly (see Figure 5). Figure 5:
Quasi-periodic lump chain of order M = N = λ = + i / λ = / − i / λ = . + . i , at times t = − . t = .
0, and t = .
0. Inset in (a) shows amplitude along the quasi-periodic lump chain. = and order N ≥ . We now discuss the general form of the solution (7) for arbitrary N , which is determined by the spectral parameters λ n . The τ -function is a sum of purely exponential terms e F n and mixed exponentials e F n + F m with trigonometricmultipliers. For fixed t , the shape of the solution is determined by the relative values of the F n : if one exponential e F n is dominant in the τ -function, then the solution u is exponentially small (a mixed term containing e F n + F m cannotbe the only dominant term). Hence the ( x , y )-plane is divided into finitely many polygonal regions, in the interior ofwhich a single term F n is dominant. In fact, a given term F n may in general fail to be dominant anywhere, so theremay be fewer than N regions, and there may be only two. On the boundary of F nm = F n − F m = F n = F m and all other F k are significantly smaller, the solution can be approximated by an order N = U nm = ( A nm , B nm ), which we callan [ n , m ]-chain. Hence the ( x , y )-plane decomposes into finitely many polygonal regions with lump chains along theboundaries, and the entire arrangement evolves linearly with t . The structure of the lump chains closely resembles thearrangement 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 genericexamples of order N =
4, and discuss the possible degenerate behavior. The first example, shown on Figure 6, may becalled an H -configuration. It consists two triple points that are separated by a lump chain bridge. The bridge contractsand disappears at t =
0, and the triple points scatter along a di ff erent bridge. A similar configuration appears in theKP-II equation (see [3, 16]). Figure 6: H-shaped arrangement of chains of order M = N =
4, with eigenvalues λ = / + i / λ = / − i / λ = / + i / λ = / + i / t = − . t = t = . The second example, shown on Figure 7, has three triple points bounding a finite triangular region. The regionshrinks and disappears at t =
0, and the solution henceforth resembles a solution of order N =
3. This configurationcan be reversed in time, with a triangular region appearing out of a triple point. We stress that both these examplesare generic, in other words the structure of the lump chains does not change under small perturbations of the λ n . Figure 7: Triangular arrangement of chains of order M = N =
4, with eigenvalues λ = λ = / + i √ / λ = / − i √ / λ = / √ − i / (2 √ t = − . t = − . t = . The reader may recognize that the structure of lump chain solutions of KP-I is very similar to the structure ofline-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 lumpchains of rank M = N . 9 .3.1. Generic solutions: the number of chains at infinity and forbidden configurations. We first consider the case when the eigenvalues λ n are su ffi ciently generic. A natural first question is to determinethe linear configurations of chains that may occur, in particular, the number of lump chains extending to infinity. Anorder N line-soliton of KP-II always has N solitons extending to infinity, but a generic order N solution of KP-I mayhave anywhere between 3 and N infinite lump chains. Similarly, certain configurations of lines are forbidden for KP-IIline-solitons but may occur for KP-I lump chain solutions. For example, the solution given on Figure 7 has N = Various degenerate configurations may be achieved by imposing appropriate conditions on the eigenvalues λ n .The triple points where lump chains meet may be stationary relative to one another, and may even coincide for alltimes, producing stable quadruple points and points of higher multiplicity. A solution may have sets of parallel lumpchains, in which case the number of chains at infinity may be greater than the order N . Finally, lump chains maycoincide, producing quasiperiodic chains of higher order. ≥ . The structure of reduced solutions of KP-I of depth S = M ≥ M = τ -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 decompositionevolves linearly with time. Polygonal regions may appear and disappear at certain moments of time. The boundariesof the polygons support lump chains, and the total flux of the lump chains arriving at a given vertex is equal to theflux of the chains that are leaving. In degenerate cases, there may be coinciding polygonal boundaries supportingquasiperiodic superpositions of lump chains. We give a single example of such a solution with rank M = N = Figure 8: The time evolution of a rank 2 order 4 solution with eigenvalues λ = / + i / λ = / − i / λ = / + i / λ = / + i / λ = / + i / λ = / − i / λ = / + i / λ = / + i /
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 positivedepth. We first consider regular solutions, and for simplicity restrict our attention to rank N =
1. The τ -function ofsuch a solution is nearly identical to that of the reduced solution (7), and has the form τ ( x , y , t ) = + N (cid:88) n = e F n + N − (cid:88) n = N (cid:88) m = n + µ nm e F n + F m cos( G n − G m − ϕ nm ) . (14)10s 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 τ -function is the constant 1. Since a n = Re λ n >
0, this region contains, for a given fixed y , all points ( x , y ) with su ffi ciently large negative x . Inside this regionthe τ -function is approximately constant, and the solution u is exponentially small. At the boundary of this region,the two dominant terms in the τ -function are the 1 and one of the exponentials e F n . Hence the boundary of the regionwhere 1 dominates is a line-soliton of KP-I, instead of a lump chain. In other words, the solution consists of an infiniteline-soliton of KP-I coupled with an arrangement of lump chains (see Figure 9). Figure 9: Regular solution of rank M = N =
3, with eigenvalues λ = / + i / λ = / − i /
4, and λ = / + i /
8, at di ff erentmoments of time. It is possible to degenerate a regular solution to a reduced solution by replacing the 1 in Equation (14) with an ε and taking the limit ε →
0. The line-soliton occurs on the boundary of the region where the ε is the dominant term,and this region moves in the negative x -direction as ε →
0. In the limit, the line-soliton disappears to infinity, and weare left with a solution consisting entirely of lump chains. Therefore, the limiting procedure that produces reducedsolutions out of regular solutions has the e ff ect of removing the line-soliton and isolating the lump chain structure. Figure 10: A lump chain radiates an individual lump, which propagates away. λ = / λ = / We also briefly consider the structure of reduced solutions of depth S >
0. Consider again the general form ofthe τ -function (4), where c jk = ψ j are given by (5). As discussed in Section 2, the τ -function is rational ifeach ψ j is a polynomial multiple of a single exponential term e φ ( x , y , t ,λ j ) . The corresponding solution is localized in the( x , y )-plane and represents the normal (if all λ j are distinct) or anomalous scattering of lumps, or even bound states oflumps. We now consider what happens in general, when each ψ j is a multiple of several exponentials. For su ffi cientlylarge x and y , the polynomial terms are negligible compared to the exponentials, and the ψ j can be assumed to be11urely exponential. Hence the solution can be assumed to have depth S = x , y )-plane, however, the polynomial terms in the ψ j produce individuallumps. Hence, the overall structure of the solution is an arrangement of lump chains interacting with finitely manyindividual 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 givea single example of such a solution, consisting of a lump chain emitting an individual lump. We note that the localnumber of lumps is conserved: two lumps from the chain meet and scatter, with one lump propagating away and theother 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 ψ -function given by ψ ( x , y , t ) = e ( − t + x + iy ) ( − t + x + iy + + e ( − t + x + iy ) .
5. Summary and conclusion
We have constructed a new family of lump chain solutions of the KP-I equations using the Grammian form ofthe τ -function. A simple lump chain consists of an infinite line of equally spaced lumps. The lumps propagate withequal velocity, which is in general distinct from the group velocity of the line. The general solution consists of anevolving polyhedral arrangement of lump chains. At a point where three or more lump chains meet, the individuallumps from the incoming chains are redistributed along the outgoing chains, with the number of lumps being locallyconserved. 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: parallelchains, chains of equal velocity, quasiperiodic superimposed chains, stable points of high multiplicity, and forbiddenpolyhedral 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 individuallumps that are emitted and / or absorbed by the lump chains. A detailed classification of the solutions of KP-I that maybe obtained by the Grammian method is an interesting and di ffi cult 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).
References [1] M. J. Ablowitz, S. Chakravarty, A. D. Trubatch, and J. Villarroel. A novel class of solutions of the non-stationary Schr¨odinger and theKadomtsev-–Petviashvili I equations.
Physics Letters A
J. Fluid Mech., v. 87, pp. 17–31, 1978.[4] G. Biondini. Line soliton interactions of the Kadomtsev–Petviashvili equation.
Physical review letters
Journal of mathematical physics
Journal of Physics A: Mathematical and General
Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki
88, pp.1609-1615, 1985.[8] S. Chakravarty and Y. Kodama. Classification of the line-soliton solutions of KPII.
Journal of Physics A: Mathematical and Theoretical
Theoretical and Mathematical Physics
Journal of Experimental and TheoreticalPhysics Letters
19, p.387, 1974.[11] A. S. Fokas and M. J. Ablowitz. On the inverse scattering of the time-dependent Schr¨odinger equation and the associated Kadomtsev–Petviashvili equation.
Stud. Appl. Math.
69, no. 3, 211-228, 1983.[12] K. A. Gorshkov, D. E. Pelinovsky, and Yu. A. Stepanyants. Normal and anomalous scattering, formation and decay of bound states oftwo-dimensional solitons described by the Kadomtsev–Petviashvili equation.
JETP
13] K. A. Gorshkov, L. A. Ostrovsky, and Yu. A. Stepanyants. Dynamics of soliton chains: From simple to complex and chaotic motions. In:
Long-Range Interactions, Stochasticity and Fractional Dynamics, pp. 177–218, eds. Albert C.J. Luo and Valentin Afraimovich, Springer,2010.[14] W. Hu, W. Huang, Z. Lu, and Y. Stepanyants. Interaction of multi-lumps within the Kadomtsev—Petviashvili equation.
Wave Motion
77, pp.243-256, 2018.[15] B. B. Kadomtsev, V. I. Petviashvili. On the stability of solitary waves in weakly dispersing media.
Sov. Phys. Dokl.
15, no. 6, pp. 539-541,1970.[16] Y. Kodama. Solitons in two-dimensional shallow water. SIAM, 2018.[17] Y. Kodama and L. Williams. KP solitons and total positivity for the Grassmannian.
Inventiones mathematicae
FunctionalAnalysis and Its Applications
Zap.Nauchn. Sem. LOMI, v. 84, pp. 117–130, 1979.[21] E. A. Kuznetsov, S. K. Turitsyn. Two- and three-dimensional solitons in weakly dispersive media.
Sov. Phys. JETP, v. 55, n. 5, 844–847,1982.[22] S. V. Manakov. The inverse scattering transform for the time-dependent Schr¨odinger equation and Kadomtsev–Petviashvili equation.
PhysicaD
Physics Letters A N -soliton formula for the KP equation. J. Phys. Soc. Japan
58, no. 2, pp. 412–422, 1989.[25] S. Novikov, S. V. Manakov, L. P. Pitaevskij, V. E. Zakharov. Theory of solitons. The inverse scattering method. Russian: Nauka, Moscow,1980. English: Contemporary Soviet Mathematics. New York - London: Plenum Publishing Corporation. Consultants Bureau, 1984.[26] A. R. Osborne. Nonlinear Ocean Wave and the Inverse Scattering Transform. Academic Press, 2010.[27] D. Pelinovsky. Rational solutions of the Kadomtsev–Petviashvili hierarchy and the dynamics of their poles. I. New form of a general rationalsolution.
J. Math. Phys.
35, no. 11, pp. 5820–5830, 1994.[28] D. Pelinovsky. Rational solutions of the KP hierarchy and the dynamics of their poles. II. Construction of the degenerate polynomial solutions.
Journal of Mathematical Physics
JETP Letters
57, pp. 24-28, 1993.[30] D. E. Pelinovsky, Yu. A. Stepanyants. Self-focusing instability of plane solitons and chains of two-dimensional solitons in positive-dispersionmedia.
Zh. Eksp. Teor. Fiz
Plasma Physics
2, pp. 469-472, 1976.[32] J. Satsuma, M. J. Ablowitz. Two-dimensional lumps in nonlinear dispersive systems.
Journal of Mathematical Physics
Sov. Math. Dokl.
14, pp. 1266-1270, 1973.[34] Yu. Stepanyants. Multi-lump structures in the Kadomtsev–Petviashvili equation.
Advances in Dynamics, Patterns, Cognition, pp. 307-324.Springer, Cham, 2017.[35] J. Villarroel, M. J. Ablowitz. On the discrete spectrum of the nonstationary Schr¨odinger equation and multipole lumps of the Kadomtsev–Petviashvili I equation.
Communications in mathematical physics
Soviet Physics Doklady
28, p. 720, 1983.[37] V. E. Zakharov. Instability and nonlinear oscillations of solitons.
JETP Lett.
22 (7), pp. 172-173, 1975.[38] V. E. Zakharov. On the dressing method. In:
Inverse methods in action,
Proc. Multicent. Meet., Montpellier / Fr. 1989 (Springer-Verlag, Berlin,1990), pp. 602-623, 1990.[39] V. E. Zakharov, S. V. Manakov. Construction of higher-dimensional nonlinear integrable systems and of their solutions.
Funct. Anal. Appl.
What is integrability? , Springer Ser. NonlinearDyn., 185-250 (1991).[41] V. E. Zakharov, A. B. Shabat. A scheme for integrating the nonlinear equations of mathematical physics by the method of the inversescattering problem. I.
Functional analysis and its applications
FunctionalAnalysis and Its Applications
ZhETF Pisma Redaktsiiu
39, pp.110-113, 1984.[44] X. Zhou. Inverse scattering transform for the time dependent Schr¨odinger equation with applications to the KPI equation.
Comm. Math.Phys.128, no. 3, pp. 551-564, 1990.