Soliton multidimensional equations and integrable evolutions preserving Laplace's equation

The KP equation, which is an integrable nonlinear evolution equation in 2+1, i.e., two spatial and one temporal dimensions, is a physically significant generalization of the KdV equation. The question of constructing an integrable generalization of the KP equation in 3+1, has been one of the central open problems in the field of integrability. By complexifying the independent variables of the KP equation, I obtain an integrable nonlinear evolution equation in 4+2. The requirement that real initial conditions remain real under this evolution, implies that the dependent variable satisfies a nonlinear evolution equation in 3+1 coupled with Laplace's equation. A reduction of this system of equations to a single equation in 2+1 contains as particular cases certain singular integro-differential equations which appear in the theory of water waves.     

The Davey-Stewartson Equation on the Half-Plane

The Davey-Stewartson (DS) equation is a nonlinear integrable evolution equation in two spatial dimensions. It provides a multidimensional generalisation of the celebrated nonlinear Schrödinger (NLS) equation and it appears in several physical situations. The implementation of the Inverse Scattering Transform (IST) to the solution of the initial-value problem of the NLS was presented in 1972, whereas the analogous problem for the DS equation was solved in 1983. These results are based on the formulation and solution of certain classical problems in complex analysis, namely of a Riemann Hilbert problem (RH) and of either a d-bar or a non-local RH problem respectively. A method for solving the mathematically more complicated but physically more relevant case of boundary-value problems for evolution equations in one spatial dimension, like the NLS, was finally presented in 1997, after interjecting several novel ideas to the panoply of the IST methodology. Here, this method is further extended so that it can be applied to evolution equations in two spatial dimensions, like the DS equation. This novel extension involves several new steps, including the formulation of a d-bar problem for a sectionally non-analytic function, i.e. for a function which has different non-analytic representations in different domains of the complex plane. This, in addition to the computation of a d-bar derivative, also requires the computation of the relevant jumps across the different domains. This latter step has certain similarities (but is more complicated) with the corresponding step for those initial-value problems in two dimensions which can be solved via a non-local RH problem, like KPI.     

Properties of supersymmetric integrable systems of KP type

The recently proposed supersymmetric extensions of reduced Kadomtsev-Petviashvili (KP) integrable hierarchies in N = 1, 2 superspace are shown to contain in the purely bosonic limit new types of ordinary non-supersymmetric integrable systems. The latter are coupled systems of several multi-component non-linear Schr?dinger-like hierarchies whose basic nonlinear evolution equations contain additional quintic and higher-derivative nonlinear terms. Also, we obtain the N = 2 supersymmetric extension of Toda chain model as Darboux-B?cklund orbit of the simplest reduced N = 2 super-KP hierarchy and find its explicit solution. Received 13 September 2001 Published online 2 October 2002 RID="a" ID="a"e-mail: nissimov@inrne.bas.bg RID="b" ID="b"e-mail: svetlana@inrne.bas.bg     

(4,3) and (5,2) Nonlinear Evolution Equations of (2+1)-Dimensional KP Hierarchy

The Kadometsev-Petviashvili (KP) equation is generalized to a class of equations of the (2+1)-dimensional KP hierarchy. The (4,3) and (5,2) integrable evolution equations with high nonlinearity are given in detail.     

Novel methods for finding general forms of new multi-soliton solutions to (1+1)-dimensional KdV equation and (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation

In this paper, two novel methods used to solve (1+1) and (2+1)-dimensional completely integrable equations are proposed. The methods are applied to handle the KdV and Kadomtsev–Petviashvili (KP) equations with variable coefficients, and the general forms of new multi-soliton solutions are formally obtained, respectively. In addition, the new multi-soliton solution is suitable to two different type KP equations. Comparing with the Hirota’s method, the results show that new methods are straightforward handling the KdV and KP equations without conjecturing the transformation and good in dealing the equations with variable coefficients.     

Lie Algebras and Integrable Systems

A 3? 3 matrix Lie algebra is first introduced, its subalgebras and the generated Lie algebras are obtained, respectively. Applications of a few Lie subalgebras give rise to two integrable nonlinear hierarchies of evolution equations from their reductions we obtain the nonlinear Schrödinger equations, the mKdV equations, the Broer-Kaup (BK) equation and its generalized equation, etc. The linear and nonlinear integrable couplings of one integrable hierarchy presented in the paper are worked out by casting a 3? 3 Lie subalgebra into a 2? 2 matrix Lie algebra. Finally, we discuss the elliptic variable solutions of a generalized BK equation.     

Multiple soliton solutions for the integrable couplings of the KdV and the KP equations

In this work, we study the nonlinear integrable couplings of the KdV and the Kadomtsev-Petviashvili (KP) equations. The simplified Hirota’s method will be used for this study. We show that these couplings possess multiple soliton solutions the same as the multiple soliton solutions of the KdV and the KP equations, but differ only in the coefficients of the transformation used. This difference exhibits soliton solutions for some equations and anti-soliton solutions for others.     

On generalized Lax equation of the Lax triple of KP Hierarchy

In terms of the operator Nambu 3-bracket and the Lax pair (L, B_n) of the KP hierarchy, we propose the generalized Lax equation with respect to the Lax triple (L, B_n, B_m). The intriguing results are that we derive the KP equation and another integrable equation in the KP hierarchy from the generalized Lax equation with the different Lax triples (L, B_n, B_m). Furthermore we derive some no integrable evolution equations and present their single soliton solutions.     

A self-dual Yang-Mills hierarchy and its reductions to integrable systems in 1+1 and 2+1 dimensions

The self-dual Yang-Mills equations play a central role in the study of integrable systems. In this paper we develop a formalism for deriving a four dimensional integrable hierarchy of commuting nonlinear flows containing the self-dual Yang-Mills flow as the first member. We show that upon appropriate reduction and suitable choice of gauge group it produces virtually all well known hierarchies of soliton equations in 1+1 and 2+1 dimensions and can be considered as a universal integrable hierarchy. Prototypical examples of reductions to classical soliton equations are presented and related issues such as recursion operators, symmetries, and conservation laws are discussed.     

Supersymmetric Integrable Systems in (2 + 1) Dimensions and Their Backlund Transformation

Nonlinear integrable systems in (2 + 1)dimensions which are supersymmetric are generated in twodifferent ways. In one approach the homogeneous spacesof super-Lie algebra are used, and in the other we use a different technique of extending thedimension of the system. The two sets of equations turnout to be different. The methodologies ofDarbux–Backlund transformation and gaugetransformation are used to generate the Backlund transformations ofthese equations. An important result of our analysis isthe existence of purely fermionic nonlinear systems in(2 + 1) dimensions.     

Soliton,breather, and rogue wave solutions for solving the nonlinear Schrödinger equation using a deep learning method with physical constraints

The nonlinear Schro¨dinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particular in high dimensions,lots of methods are proposed to effectively obtain different kinds of solutions,such as neural networks among others.Recently,a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation’s dynamical behaviors from spatiotemporal data directly.Compared with traditional neural networks,this method can obtain remarkably accurate solution with extraordinarily less data.Meanwhile,this method also provides a better physical explanation and generalization.In this paper,based on the above method,we present an improved deep learning method to recover the soliton solutions,breather solution,and rogue wave solutions of the nonlinear Schro¨dinger equation.In particular,the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time.Moreover,the effects of different numbers of initial points sampled,collocation points sampled,network layers,neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions.Numerical experiments show that the dynamical behaviors of soliton solutions,breather solution,and rogue wave solutions of the integrable nonlinear Schro¨dinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.     

A KAM Theorem with Applications to Partial Differential Equations of Higher Dimensions

The existence of lower dimensional KAM tori is shown for a class of nearly integrable Hamiltonian systems of infinite dimensions where the second Melnikov’s conditions are completely eliminated and the algebraic structure of the normal frequencies are not needed. As a consequence, it is proved that there exist many invariant tori and thus quasi-periodic solutions for nonlinear wave equations, Schrödinger equations and other equations of any spatial dimensions.     

Self-Consistent Sources for Integrable Equations Via Deformations of Binary Darboux Transformations

We reveal the origin and structure of self-consistent source extensions of integrable equations from the perspective of binary Darboux transformations. They arise via a deformation of the potential that is central in this method. As examples, we obtain in particular matrix versions of self-consistent source extensions of the KdV, Boussinesq, sine-Gordon, nonlinear Schrödinger, KP, Davey–Stewartson, two-dimensional Toda lattice and discrete KP equation. We also recover a (2+1)-dimensional version of the Yajima–Oikawa system from a deformation of the pKP hierarchy. By construction, these systems are accompanied by a hetero binary Darboux transformation, which generates solutions of such a system from a solution of the source-free system and additionally solutions of an associated linear system and its adjoint. The essence of all this is encoded in universal equations in the framework of bidifferential calculus.     

拟Wrongsky行列式与扩张的KP及KdV可积序列

将Sato关于KP和KdV可积序列的理论推广到矩阵Lax算子的情形,得到一批我们称之为扩张的KP和KdV序列的新可积方程族。从构造过程可以得知,这些新可积方程族的解可以用我们所称的拟Wrongsky行列式来表达,而这种拟Wrongsky行列式正是不久前我们研究2阶扩张的广义Toda方程的解时得到的。 关键词：     

Kinetic equation for a dense soliton gas

We propose a general method to derive kinetic equations for dense soliton gases in physical systems described by integrable nonlinear wave equations. The kinetic equation describes evolution of the spectral distribution function of solitons due to soliton-soliton collisions. Owing to complete integrability of the soliton equations, only pairwise soliton interactions contribute to the solution, and the evolution reduces to a transport of the eigenvalues of the associated spectral problem with the corresponding soliton velocities modified by the collisions. The proposed general procedure of the derivation of the kinetic equation is illustrated by the examples of the Korteweg-de Vries and nonlinear Schr?dinger (NLS) equations. As a simple physical example, we construct an explicit solution for the case of interaction of two cold NLS soliton gases.     

Two new integrable Kadomtsev–Petviashvili equations with time-dependent coefficients: multiple real and complex soliton solutions

ABSTRACT

In this work, we develop two new integrable Kadomtsev–Petviashvili (KP) equations with time-dependent coefficients. The integrability property of each equation is explicitly demonstrated exhibiting the Painlevé test to confirm its integrability. Moreover, each equation admits multiple real and multiple complex soliton solutions. We introduce complex forms of the simplified Hirota's method to derive multiple complex soliton solutions. These two model equations are likely to be of applicative relevance, because it may be considered an application of a large class of nonlinear KP equations.     

Infinitely-many conservation laws for two (2+1)-dimensional nonlinear evolution equations in fluids

In this paper, a method that can be used to construct the infinitely-many conservation laws with the Lax pair is generalized from the (1+1)-dimensional nonlinear evolution equations (NLEEs) to the (2+1)-dimensional ones. Besides, we apply that method to the Kadomtsev–Petviashvili (KP) and Davey–Stewartson equations in fluids, and respectively obtain their infinitely-many conservation laws with symbolic computation. Based on that method, we can also construct the infinitely-many conservation laws for other multidimensional NLEEs possessing the Lax pairs, including the cylindrical KP, modified KP and (2+1)-dimensional Gardner equations, in fluids, plasmas, optical fibres and Bose–Einstein condensates.     

The Complex Geometry of Weak Piecewise Smooth Solutions of Integrable Nonlinear PDE's¶of Shallow Water and Dym Type

An extension of the algebraic-geometric method for nonlinear integrable PDE's is shown to lead to new piecewise smooth weak solutions of a class of N-component systems of nonlinear evolution equations. This class includes, among others, equations from the Dym and shallow water equation hierarchies. The main goal of the paper is to give explicit theta-functional expressions for piecewise smooth weak solutions of these nonlinear PDE's, which are associated to nonlinear subvarieties of hyperelliptic Jacobians. The main results of the present paper are twofold. First, we exhibit some of the special features of integrable PDE's that admit piecewise smooth weak solutions, which make them different from equations whose solutions are globally meromorphic, such as the KdV equation. Second, we blend the techniques of algebraic geometry and weak solutions of PDE's to gain further insight into, and explicit formulas for, piecewise-smooth finite-gap solutions. The basic technique used to achieve these aims is rather different from earlier papers dealing with peaked solutions. First, profiles of the finite-gap piecewise smooth solutions are linked to certain finite dimensional billiard dynamical systems and ellipsoidal billiards. Second, after reducing the solution of certain finite dimensional Hamiltonian systems on Riemann surfaces to the solution of a nonstandard Jacobi inversion problem, this is resolved by introducing new parametrizations. Amongst other natural consequences of the algebraic-geometric approach, we find finite dimensional integrable Hamiltonian dynamical systems describing the motion of peaks in the finite-gap as well as the limiting (soliton) cases, and solve them exactly. The dynamics of the peaks is also obtained by using Jacobi inversion problems. Finally, we relate our method to the shock wave approach for weak solutions of wave equations by determining jump conditions at the peak location. Received: 16 February 1999 / Accepted: 10 April 2001     

Non-linear superposition in non-linear evolution equations

Special non-linear evolution equations are discussed and classified in integrable and non-integrable partial differential equations. Important methods of solution are sketched. An interesting task is the calculation of the so-called superposition functions permitting to construct from a few known solutions whole chains of further solutions. For integrable partial differential equations they are computed with the help of BÄcklund transformations providing a rather general superposition rule for special types of solutions. At present however a consistent treatment of non-linear evolution equations is only possible in 1 + 1 space and time dimensions. The extension to higher space dimensions brings many problems. An insight into the results and problems in the one-dimensional case is given together with an outlook indicating the conceptual difficulties for a treatment in higher space dimensions.Presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.     

Polaron on a one-dimensional lattice: II. A moving polaron

Continuum-limit equations for moving polarons on a one-dimensional lattice with a harmonic interaction potential between adjacent particles and a simple nonlinear potential with a cubic nonlinearity are derived for the first time; for some particular cases, their solutions are obtained. For a harmonic lattice in the continuum limit, a system of integrable nonlinear partial differential equations is derived. A one-soliton solution to this system describes a polaron moving with a constant velocity. The speed of this polaron is uniquely related to its amplitude, with its values ranging from zero to the speed of sound. For a nonlinear lattice, the resulting system of differential equations is integrable at a certain ratio of the problem parameters. The one-soliton solution to this system, as in the harmonic case, describes a polaron moving with a constant velocity. At arbitrary values of the lattice parameters, the nonlinear lattice was studied by numerical methods. It turned out that, in the entire range of parameters, the nonlinear lattice gives rise to moving polarons, with the speed of the polaron being determined by the competition between the electron-photon interaction parameter α and the nonlinearity parameter β. At α ? β, the behavior of the polaron is very close to the dynamics on the harmonic lattice. In the opposite case, the dynamic nonlinearity begins to dominate, giving rise to dynamics inherent to solitons, so that speed of the polaron can exceed the speed of sound. In a certain range of α and β, numerical calculations revealed a family of polaron-type stable solutions, the envelope of which can have several peaks. The numerical and exact analytical solutions are in very good agreement for a sufficiently large radius of the polaron, when the system of equations obtained in the continuum approximation has a solution.