Line Soliton Solutions Research Articles

Lax pairs informed neural networks solving integrable systems

Lax pairs are one of the most important features of integrable system. In this work, we propose the Lax pairs informed neural networks (LPNNs) tailored for the integrable systems with Lax pairs by designing novel network architectures and loss functions, comprising LPNN-v1 and LPNN-v2. The most noteworthy advantage of LPNN-v1 is that it can transform the solving of complex integrable systems into the solving of relatively simple Lax pairs, and it not only efficiently solves data-driven localized wave solutions, but also obtains spectral parameter and corresponding spectral function in spectral problems of the integrable systems. On the basis of LPNN-v1, we additionally incorporate the compatibility condition/zero curvature equation of Lax pairs in LPNN-v2, its major advantage is the ability to solve and explore high-accuracy data-driven localized wave solutions and associated spectral problems for integrable systems with Lax pairs. The numerical experiments focus on studying abundant localized wave solutions for very important and representative integrable systems with Lax pairs, including the soliton solution of the Korteweg-de Vries (KdV) equation and modified KdV equation, rogue wave solution of the nonlinear Schrödinger equation, kink solution of the sine-Gordon equation, non-smooth peakon solution of the Camassa-Holm equation and pulse solution of the short pulse equation, as well as the line-soliton solution of Kadomtsev-Petviashvili equation and lump solution of high-dimensional KdV equation. The innovation of this work lies in the pioneering integration of Lax pairs informed of integrable systems into deep neural networks, thereby presenting a fresh methodology and pathway for investigating data-driven localized wave solutions and spectral problems.

On the line-soliton solutions of a coupled modified Kadomtsev–Petviashvili system in two-layer shallow water

In this paper, the line-soliton solutions of a coupled modified Kadomtsev–Petviashvili (MKP) equation are studied. Based on the tau-functions and Grassmannian Gr(N,M), we present and study the coupled modified Kadomtsev–Petviashvili equation and its coupled soliton solutions v and vˆ which can describe two layers shallow water. By giving examples and theoretical analysis, we give a general classification of the structures of coupled soliton solutions v and vˆ. At the same time, we describe the solutions v, vˆ by Le-diagram, chord diagram and analyze the influence of parameter ϵ on the solutions v, vˆ.

Dynamics of Rational and Lump-Soliton Solutions to the Reverse Space-Time Nonlocal Hirota-Maccari System

We mainly construct lump-soliton solutions of the (2 + 1)-dimensional reverse space-time Hirota-Maccari (HM) equation by using the KP hierarchy reduction method. Meanwhile, with the help of a long wave limit, rational solutions to nonlocal HM equation are studied. According to the appropriate parameter selections, these solutions can be divided into two types: line soliton solutions and lump-soliton solutions. Moreover, we obtain one-lump, two-lump and W-type soliton to the nonlocal HM equation. These new lump-soliton solutions expand the structure of nonlocal nonlinear systems and aid in the comprehension of physical phenomena.

Collapse dynamics for two-dimensional space-time nonlocal nonlinear Schrödinger equations

The question of collapse (blow-up) in finite time is investigated for the two-dimensional (non-integrable) space-time nonlocal nonlinear Schrödinger equations. Starting from the two-dimensional extension of the well known AKNS q,r system, three different cases are considered: (i) partial and full parity-time (PT) symmetric, (ii) reverse-time (RT) symmetric, and (iii) general q,r system. Through extensive numerical experiments, it is shown that collapse of Gaussian initial conditions depends on the value of its quasi-power. The collapse dynamics (or lack thereof) strongly depends on whether the nonlocality is in space or time. A so-called quasi-variance identity is derived and its relationship to blow-up is discussed. Numerical simulations reveal that this quantity reaching zero in finite time does not (in general) guarantee collapse. An alternative approach to the study of wave collapse is presented via the study of transverse instability of line soliton solutions. In particular, the linear stability problem for perturbed solitons is formulated for the nonlocal RT and PT symmetric nonlinear Schrödinger (NLS) equations. Through a combination of numerical and analytical approaches, the stability spectrum for some stationary one soliton solutions is found. Direct numerical simulations agree with the linear stability analysis which predicts filamentation and subsequent blow-up.

Symmetries, conservation laws, and line soliton solutions of a two-dimensional generalized KdV equation with $ p $-power

Symmetries, conservation laws, and line soliton solutions of a two-dimensional generalized KdV equation with $ p $-power

Line-soliton, lump and interaction solutions to the (2+1)-dimensional Hirota-Satsuma-Ito equation with time-dependent via Hirota bilinear forms

In this paper, we consider the (2+1)-dimensional generalized Hirota-Satsuma-Ito equation with time-dependent, which has applications in describing the propagation of shallow water waves. Based on the bilinear formalism and with the aid of symbolic computation, we obtain line-soliton, lump, one-lump-one-stripe and one-lump-one-soliton using various ansatze’s function. To realize dynamics, we use diverse plots to analyze these solutions. Obtained solutions are reliable in the mathematical physics and engineering.

Magnetic lump motion in saturated ferromagnetic films.

In this paper, we study in detail the nonlinear propagation of a magnetic soliton in a ferromagnetic film. The sample is magnetized to saturation by an external field perpendicular to film plane. A generalized (2+1)-dimensional short-wave asymptotic model is derived. The bilinearlike forms of this equation are constructed and exact magnetic line soliton solutions are exhibited. It is observed that a series of stable lumps can be generated by an unstable magnetic soliton under Gaussian disturbance. Such magnetic lumps are highly stable and can maintain their shapes and velocities during evolution or collision. The interaction between lump and magnetic solitons, as well as the interaction between two lumps, are numerically investigated. We further discuss the nonlinear motion of lumps in ferrites with Gilbert damping and inhomogeneous exchange effects. The results show that the Gilbert-damping effects make the amplitude and velocity of the magnetic lump decay exponentially during propagation. And the shock waves are generated from a lump when quenching the strength of inhomogeneous exchange.

Symmetry Analysis, Exact Solutions and Conservation Laws of a Benjamin–Bona–Mahony–Burgers Equation in 2+1-Dimensions

The Benjamin–Bona–Mahony equation describes the unidirectional propagation of small-amplitude long waves on the surface of water in a channel. In this paper, we consider a family of generalized Benjamin–Bona–Mahony–Burgers equations depending on three arbitrary constants and an arbitrary function G(u). We study this family from the standpoint of the theory of symmetry reductions of partial differential equations. Firstly, we obtain the Lie point symmetries admitted by the considered family. Moreover, taking into account the admitted point symmetries, we perform symmetry reductions. In particular, for G′(u)≠0, we construct an optimal system of one-dimensional subalgebras for each maximal Lie algebra and deduce the corresponding (1+1)-dimensional nonlinear third-order partial differential equations. Then, we apply Kudryashov’s method to look for exact solutions of the nonlinear differential equation. We also determine line soliton solutions of the family of equations in a particular case. Lastly, through the multipliers method, we have constructed low-order conservation laws admitted by the family of equations.

Rogue waves and lumps on the nonzero background in the ‐symmetric nonlocal Maccari system

AbstractIn this paper, the ‐symmetric version of the Maccari system is introduced, which can be regarded as a two‐dimensional generalization of the defocusing nonlocal nonlinear Schrödinger equation. Various exact solutions of the nonlocal Maccari system are obtained by means of the Hirota bilinear method, long‐wave limit, and Kadomtsev–Petviashvili (KP) hierarchy method. Bilinear forms of the nonlocal Maccari system are derived for the first time. Simultaneously, a new nonlocal Davey‐Stewartson–type equation is derived. Solutions for breathers and breathers on top of periodic line waves are obtained through the bilinear form of the nonlocal Maccari system. Hyperbolic line rogue wave (RW) solutions and semirational ones, composed of hyperbolic line RW and periodic line waves are also derived in the long‐wave limit. The semirational solutions exhibit a unique dynamical behavior. Additionally, general line soliton solutions on constant background are generated by restricting different tau‐functions of the KP hierarchy, combined with the Hirota bilinear method. These solutions exhibit elastic collisions, some of which have never been reported before in nonlocal systems. Additionally, the semirational solutions, namely, (i) fusion of line solitons and lumps into line solitons and (ii) fission of line solitons into lumps and line solitons, are put forward in terms of the KP hierarchy. These novel semirational solutions reduce to ‐lump solutions of the nonlocal Maccari system with appropriate parameters. Finally, different characteristics of exact solutions for the nonlocal Maccari system are summarized. These new results enrich the structure of waves in nonlocal nonlinear systems, and help to understand new physical phenomena.

Line-solitons, line-shocks, and conservation laws of a universal KP-like equation in 2+1 dimensions

A universal KP-like equation in 2+1 dimensions, which models general nonlinear wave phenomena exhibiting p-power nonlinearity, dispersion, and small transversality, is studied. Special cases include the integrable KP (Kadomtsev-Petviashvili) equation and its modified version, as well as their p-power generalizations. Two main results are obtained. First, all low-order conservation laws are derived, including ones that arise for special powers p. The conservation laws comprise momenta, energy, and Galilean-type quantities, as well as topological charges. Their physical meaning and properties are discussed. The topological charges are shown to give rise to integral constraints on initial data for the Cauchy problem. Second, all line-soliton solutions are obtained in an explicit form. A parameterization is given using the speed and the direction angle of the line-soliton, and the allowed kinematic region is determined in terms of these parameters. Basic kinematical properties of the line-solitons are also discussed. These properties differ significantly compared to those for KP line-solitons and their p-power generalizations. A line-shock solution is shown to emerge when a special limiting case of the kinematic region is considered.

Dynamics of lump–soliton solutions to the [formula omitted]-symmetric nonlocal Fokas system

We use the bilinear Kadomtsev–Petviashvili (KP) hierarchy reduction method for deriving new families of explicit lump–soliton solutions to the PT-symmetric nonlocal Fokas system. These lump–soliton solutions are semi-rational solitons that are classified into three different species under appropriate parametric restrictions: line solitons, rational lumps, and semi-rational lump–soliton solutions. There are three different fundamental lump–soliton solutions: two-line soliton, rational two-lump solution, and two-lump–two-line-soliton solution. The two-line-soliton solution displays five patterns according to their asymptotic analysis, the rational two-lump solution possesses six distinct patterns, and the two-lump–two-line-soliton solution has four different patterns. The two-lump–two-line-soliton solution displays inelastic interaction phenomena consisting of two lumps fusing into or fissioning from the two-line soliton. The multi-lump-soliton solutions illustrate the superimposition of N (N≥2) individual fundamental lump–soliton solutions: 2N-line solitons, rational 2N-lumps, and 2N-lump–2N˜-line solitons (N˜≤N). The higher-order lump–line-soliton solutions consist of rational 2n0-lump (n0≥2) solutions or 2n0-lump–two-line-soliton solutions.

Lie Symmetries and Low-Order Conservation Laws of a Family of Zakharov-Kuznetsov Equations in 2 + 1 Dimensions

In this work, we study a generalised (2+1) equation of the Zakharov–Kuznetsov (ZK)(m,n,k) equation involving three arbitrary functions. From the point of view of the Lie symmetry theory, we have derived all Lie symmetries of this equation depending on the arbitrary functions. Line soliton solutions have also been obtained. Moreover, we study the low-order conservation laws by applying the multiplier method. This family of equations is rich in Lie symmetries and conservation laws. Finally, when the equation is expressed in potential form, it admits a variational structure in the case when two of the arbitrary functions are linear. In addition, the corresponding Hamiltonian formulation is presented.

Conservation Laws and Travelling Wave Solutions for Double Dispersion Equations in (1+1) and (2+1) Dimensions

In this article, we investigate two types of double dispersion equations in two different dimensions, which arise in several physical applications. Double dispersion equations are derived to describe long nonlinear wave evolution in a thin hyperelastic rod. Firstly, we obtain conservation laws for both these equations. To do this, we employ the multiplier method, which is an efficient method to derive conservation laws as it does not require the PDEs to admit a variational principle. Secondly, we obtain travelling waves and line travelling waves for these two equations. In this process, the conservation laws are used to obtain a triple reduction. Finally, a line soliton solution is found for the double dispersion equation in two dimensions.

Coupled sine-Gordon systems in living cellular structure

In this paper, we present a coupled sine-Gordon (CSG) system, which describes the dynamics of two deoxyribonucleic acid (DNA) twisting strands in open configuration. We investigate the associated linear eigenvalue problem of the CSG system and obtain generalized formulae for multiple soliton solutions by employing the Darboux transformation (DT). Explicit expressions of one- and two-kink and line soliton solutions, single breather and first-order degenerate solutions are obtained by using particular solutions to the associated linear eigenvalue problem. We also present the dynamics of the different solutions.

LUMP AND MIXED ROGUE-SOLITON SOLUTIONS TO THE 2+1 DIMENSIONAL ABLOWITZ-KAUP-NEWELL-SEGUR EQUATION

In this paper, the 2+1 dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equation which obtained from the potential Boiti-Leon-Manna-Pempi nelli (pBLMP) equation, is introduced. Through the bilinear method and ansatz technique, the rational solutions consisting of rogue wave and lump soliton solutions are constructed, where we discuss the condition of guaranteeing the positiveness and analyticity of the lump solutions. The collection of a quadratic function with an exponential function describing rationalexponential solutions is proved, the interaction consisting of one lump and one soliton with fission and fusion phenomena. The second kind of interaction comprises the line rogue wave and soliton solution, which is inelastic. With the usage of the extended homoclinic test approach, the homoclinic breather-wave solution is derived. The characteristics of these various solutions are exhibited and illustrated graphically.

Conservation laws and line soliton solutions of a family of modified KP equations

A family of modified Kadomtsev-Petviashvili equations (mKP) in 2+1 dimensions is studied. This family includes the integrable mKP equation when the coefficients of the nonlinear terms and the transverse dispersion term satisfy an algebraic condition. The explicit line soliton solution and all conservation laws of low order are derived for all equations in the family and compared to their counterparts in the integrable case.

Nonlinear dusty magnetosonic waves in a strongly coupled dusty plasma

The nonlinear propagation of magnetosonic waves in a magnetized strongly coupled dusty plasma consisting of inertialess electrons and ions as well as strongly coupled inertial charged dust particles is presented. A generalized viscoelastic hydrodynamic model for the strongly coupled dust particles and a quantum hydrodynamic model for electrons and ions are considered. In the kinetic regime, we derive a modified Kadomstev-Petviashvili (KP) equation for nonlinear magnetosonic waves of which the amplitude changes slowly with time due to the effect of a small amount of dust viscosity. The approximate analytical solutions of the modified KP equations are obtained with the help of a steady state line-soliton solution of the second type KP equation in a frame with a constant velocity. The dispersion relationship in the kinetic regime shows that the viscosity is no longer a dissipative effect.

Formula omitted]-symmetric nonlocal Davey–Stewartson I equation: Soliton solutions with nonzero background

Various solutions to the PT-symmetric nonlocal Davey–Stewartson (DS) I equation with nonzero boundary condition are derived by constraining different tau functions of the Kadomtsev–Petviashvili hierarchy combined with the Hirota bilinear method. From the first type of tau functions of the nonlocal DS I equation, we construct: (a) general line soliton solutions sitting on either a constant background or on a background of periodic line waves and (b) general lump-type soliton solutions. We find two generic types of line solitons that we call usual line solitons and new-found ones. The usual line solitons exhibit elastic collisions, whereas the new-found ones, in the evolution process, change their waveforms from an antidark (dark) shape to a dark (antidark) one. The general lump-type soliton solutions describe the interaction between 2N-line solitons and 2N-lumps, which give rise to different dynamical scenarios: (i) fusion of line solitons and lumps into line solitons, (ii) fission of line solitons into lumps and line solitons, and (iii) a combination of fusion and fission processes. By constraining another type of tau functions combined with the long wave limit method, periodic line waves, rogue waves, and semi-rational solutions to the nonlocal DS I equation are obtained in terms of determinants whose matrix elements have simple algebraic expressions. Finally, different types of general solutions of the nonlocal nonlinear Schrödinger equation, namely general higher-order breathers and mixed solutions consisting of higher-order breathers and rogue waves sitting on either a constant background or on a background of periodic line waves are obtained as reductions of the corresponding solutions of the nonlocal DS I equation.

Solitary waves of the two-dimensional Camassa–Holm—nonlinear Schrödinger equation

In this work, we study solitary waves in a -dimensional variant of the defocusing nonlinear Schrödinger (NLS) equation, the so-called Camassa–Holm-NLS (CH-NLS) equation. We use asymptotic multiscale expansion methods to reduce this model to a Kadomtsev–Petviashvili (KP) equation. The KP model includes both the KP-I and KP-II versions, which possess line and lump soliton solutions. Using KP solitons, we construct approximate solitary wave solutions on top of the stable continuous-wave solution of the original CH-NLS model, which are found to be of both the dark and anti-dark type. We also use direct numerical simulations to investigate the validity of the approximate solutions, study their evolution, as well as their head-on collisions.

Conservation Laws, Symmetries, and Line Soliton Solutions of Generalized KP and Boussinesq Equations with p-Power Nonlinearities in Two Dimensions

Nonlinear generalizations of integrable equations in one dimension, such as the KdV and Boussinesq equations with $p$-power nonlinearities, arise in many physical applications and are interesting in analysis due to critical behaviour. This paper studies analogous nonlinear $p$-power generalizations of the integrable KP equation and the Boussinesq equation in two dimensions. Several results are obtained. First, for all $p\neq 0$, a Hamiltonian formulation of both generalized equations is given. Second, all Lie symmetries are derived, including any that exist for special powers $p\neq0$. Third, Noether's theorem is applied to obtain the conservation laws arising from the Lie symmetries that are variational. Finally, explicit line soliton solutions are derived for all powers $p>0$, and some of their properties are discussed.