Line Solitons Research Articles

Variable separation approach by means of the mapping method

A mapping method is used to obtain some new types of variable separation excitations, with two arbitrary functions, of the (2+1)-dimensional Broer–Kaup–Kupershmidt (BKK) system. From the variable separation solution and by selecting appropriate functions, a class of the (2+1)-dimensional non-propagating solitary waves are found. Moreover, two types of chaotic patterns (chaotic line soliton pattern and chaotic–chaotic pattern) are discussed.

A modified predictor–corrector scheme for the two-dimensional sine-Gordon equation

A three-time level finite-difference scheme based on a fourth order in time and second order in space approximation has been proposed for the numerical solution of the nonlinear two-dimensional sine-Gordon equation. The method, which is analysed for local truncation error and stability, leads to the solution of a nonlinear system. To avoid solving it, a predictor–corrector scheme using as predictor a second-order explicit scheme is proposed. The procedure of the corrector has been modified by considering as known the already evaluated corrected values instead of the predictor ones. This modified scheme has been tested on the line and circular ring soliton and the numerical experiments have proved that there is an improvement in the accuracy over the standard predictor–corrector implementation.

A third order numerical scheme for the two-dimensional sine-Gordon equation

A rational approximant of third order, which is applied to a three-time level recurrence relation, is used to transform the two-dimensional sine-Gordon (SG) equation into a second-order initial-value problem. The resulting nonlinear finite-difference scheme, which is analyzed for stability, is solved by an appropriate predictor–corrector (P–C) scheme, in which the predictor is an explicit one of second order. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behavior of the proposed P–C/MPC schemes is tested numerically on the line and ring solitons known from the bibliography, regarding SG equation and conclusions for both the mentioned schemes regarding the undamped and the damped problem are derived.

Reflection and reconnection interactions of resonant dromions

The interactions between dromions have still not yet been understood very well. In this short paper, two new types of dromion interactions, dromion reflection and reconnection, are reported for the generalized Broer–Kaup system as a representative example. Dromion reflection is when a dromion behaves like a ball reflected by a wall. Essentially, it is a dromion interacting with an invisible ghost wall caused by a ghost line soliton. While dromion reconnection interaction is the scenario where the bounded dromions may be opened and then reconnected during the interaction. These two novel interaction phenomena are quite universal in high dimensions.

Elastic and inelastic line-soliton solutions of the Kadomtsev–Petviashvili II equation

The Kadomtsev–Petviashvili II (KPII) equation admits a large variety of multi-soliton solutions which exhibit both elastic as well as inelastic types of interactions. This work investigates a general class of multi-solitons which were not previously studied, and which do not in general conserve the number of line solitons after interaction. The incoming and outgoing line solitons for these solutions are explicitly characterized by analyzing the τ -function generating such solutions. A special family of N-soliton solutions is also considered in this article. These solutions are characterized by elastic soliton interactions, in the sense that amplitude and directions of the individual line solitons as y → ∞ are the same as those of the individual line solitons as y → − ∞ . It is shown that the solution space of these elastic N-soliton solutions can be classified into ( 2 N − 1 ) ! ! disjoint sectors which are characterized in terms of the amplitudes and directions of the N line solitons.

New periodic wave solutions for the shallow water equations and the generalized Klein–Gordon equation

The periodic wave solutions and the corresponding solitary solutions for the shallow water equations and the generalized Klein–Gordon equation are obtained by means of mapping method. The solutions obtained in this paper include as well the shock wave solution, complex line period, complex line soliton and rational solutions. Moreover, the obtained solutions are degenerated in terms of hyperbolic function solutions and trigonometric function solutions when the modulus m of the Jacobi elliptic function is driven to 1 and 0, respectively. The previously known periodic and solitary wave solutions are recovered. Many new results are presented.

The solution of the two-dimensional sine-Gordon equation using the method of lines

The method of lines is used to transform the initial/boundary-value problem associated with the two-dimensional sine-Gordon equation in two space variables into a second-order initial-value problem. The finite-difference methods are developed by replacing the matrix-exponential term in a recurrence relation with rational approximants. The resulting finite-difference methods are analyzed for local truncation error, stability and convergence. To avoid solving the nonlinear system a predictor–corrector scheme using the explicit method as predictor and the implicit as corrector is applied. Numerical solutions for cases involving the most known from the bibliography line and ring solitons are given.

Chaos, solitons and fractals in (2+1)-dimensional KdV system derived from a periodic wave solution

Abstract With the help of an extended mapping method and a linear variable separation method, new types of variable separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with two arbitrary functions for (2 + 1)-dimensional Korteweg–de Vries system (KdV) are derived. Usually, in terms of solitary wave solutions and rational function solutions, one can find some important localized excitations. However, based on the derived periodic wave solution in this paper, we find that some novel and significant localized coherent excitations such as dromions, peakons, stochastic fractal patterns, regular fractal patterns, chaotic line soliton patterns as well as chaotic patterns exist in the KdV system as considering appropriate boundary conditions and/or initial qualifications.

Singular periodic soliton solutions and resonance for the Kadomtsev–Petviashvili equation☆

Exact periodic soliton solutions of the Kadomtsev–Petviashvili (KP) equation are obtained using the two-soliton and generalized Hirota methods. Singular and non-singular phenomenons of various periodic soliton solutions are studied. The resonance interaction between y -periodic solitons and line solitons is investigated.

Interaction of a weakly relativistic soliton in the magnetized plasma

The results of analytical investigations on the properties of two-dimensional (2D) magnetosonic solitons with a relativistic amplitude are presented in the presence of an applied static and uniform magnetic field B⃑0. In the weakly relativistic limit, a Kadomtsev-Petviashvili (KP) equation is derived for the 2D relativistic magnetosonic soliton by a reductive perturbation method. The N line-soliton solution of the KP equation is presented. The interaction law of the line solitons is studied by figure analysis, which shows that the elastic and inelastic interactions of the line solitons are all possible and determined by the Wronskian determinant.

Localized coherent structures of Ishimori equation I through Hirota’s bilinearization method: Time dependent/stationary boundaries

Abstract Ishimori equation is a (2 + 1) dimensional generalization of the (1 + 1) dimensional integrable classical continuous Heisenberg ferromagnetic spin equation. The richness of the coherent structures admitted by Ishimori equation I such as dromion, lump and rationally–exponentially localized solutions, have been demonstrated in the literature through ∂ ¯ technique and binary Darboux transformation method. To our knowledge Hirota’s method had been adopted to construct only the vortex solutions of Ishimori equation II. For the first time, the various types of localized coherent structures mentioned above have been constructed in this paper for the Ishimori equation I using the Hirota’s direct method. In particular we have brought out the significance of boundaries and arbitrary functions in generating all these types of localized structures and proved that the absence of such boundaries leads only to line soliton solutions.

Soliton solutions of the Kadomtsev-Petviashvili II equation

We study a general class of line-soliton solutions of the Kadomtsev-Petviashvili II (KPII) equation by investigating the Wronskian form of its tau-function. We show that, in addition to the previously known line soliton solutions of KPII, this class also contains a large variety of multisoliton solutions, many of which exhibit nontrivial spatial interaction patterns. We also show that, in general, such solutions consist of unequal numbers of incoming and outgoing line solitons. From the asymptotic analysis of the tau function, we explicitly characterize the incoming and outgoing line solitons of this class of solutions. We illustrate these results by discussing several examples.

Fractal and chaotic patterns of Nizhnik–Novikov–Veselov system derived from a periodic wave solution

Starting from an extended mapping method and a linear variable separation method, new families of variable separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with arbitrary functions for ( 2 + 1 )-dimensional Nizhnik–Novikov–Veselov system (NNV) are derived. Usually, in terms of solitary wave solutions and rational function solutions, one can find some important localized excitations. However, based on the derived periodic wave solution in this Letter, we find that some novel and interesting localized coherent excitations such as stochastic fractal patterns, regular fractal patterns, chaotic line soliton patterns and chaotic patterns also exist in the NNV system as considering appropriate boundary conditions and/or initial qualifications.

Restudy of Structures and Interactions of Solitons in(2+1)-Dimensional Nizhnik–Novikov–VeselovEquations

Some new structures and interactions of solitons for the(2+1)-dimensional Nizhnik–Novikov–Veselov equation are revealedwith the help of the idea of the bilinear method and variableseparation approach. The solutions to describe the interactionsbetween two dromions, between a line soliton and a y-periodicsoliton, and between two y-periodic solitons are included in ourresults. Detailed behaviors of interaction are illustrated bothanalytically and in graphically. Our analysis shows that theinteraction properties between two solitons are related to theform of interaction constant. The form of interaction constant andthe dispersion relationship are related to the form of the seedsolution {u0,v0,w0} in Bäcklund transformation.

An explicit numerical scheme for the Sine-Gordon equation in 2+1 dimensions

The paper presents an explicit finite-difference method for the numerical solution of the Sine-Gordon equation in two space variables, as it arises, for example, in rectangular large-area Josephson junction. The dispersive nonlinear partial differential equation of the system allows for soliton-type solutions, an ubiquitous phenomenon in a large-variety of physical problems. The method, which is based on fourth order rational approximants of the matrix-exponential term in a three-time level recurrence relation, after the application of finite-difference approximations, it leads finally to a second order initial value problem. Because of the existing sinus term this problem becomes nonlinear. To avoid solving the arising nonlinear system a new method based on a predictor-corrector scheme is applied. Both the nonlinear method and the predictor-corrector are analyzed for local truncation error, stability and convergence. Numerical solutions for cases involving the most known from the bibliography ring and line solitons are given. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A Series of Exact Solutions for a New (2+1)-Dimensional Calogero KdV Equation

An algebraic method is proposed to solve a new(2+1)-dimensional Calogero KdVequation and explicitly construct a series of exact solutionsincluding rational solutions, triangular solutions, exponentialsolution, line soliton solutions, and doubly periodic wave solutions.

Chaos, solitons and fractals in the nonlinear Dirac equation

By means of the asymptotic perturbation (AP) method, analytical investigation of a nonlinear Dirac equation shows the existence of interacting coherent excitations such as the dromions, lumps, ring soliton solutions and breathers as well as instanton solutions. The interaction between the localized solutions are completely elastic, because they pass through each other and preserve their shapes and velocities, the only change being a phase shift. Finally, one may obtain approximate lower-dimensional chaotic patterns such as chaotic–chaotic patterns, periodic–chaotic patterns, chaotic line soliton patterns and chaotic dromion patterns, due to the possibility of selecting appropriately some arbitrary functions. In a similar way, fractal dromion and lump patterns as well as stochastic fractal excitations can appear in the solution.

Structures and Interactions of Soliton in (2+1)-DimensionalGeneralized Nizhnik–Novikov–VeselovEquation

A variable separation approach is used to obtain exact solutionsof the (2+1)-dimensional generalized Nizhnik–Novikov–Veselovequation. Two of these exact solutions are analyzed to study theinteraction between a line soliton and a y-periodic soliton (i.e. the array of the localized structure in the y direction, whichpropagates in the x direction) and between two dromions. Theinteractions between a line soliton and a y-periodic soliton areclassified into several types according to the phase shifts due tocollision. There are two types of singular interactions. One isthe resonant interaction that generates one line soliton while theother is the extremely repulsive or long-range interaction wheretwo solitons interchange each other infinitely apart. Some newphenomena of interaction between two dromions are also reported inthis paper, and detailed behaviors of interactions are illustrated both analytically and graphically.

Young diagrams and N-soliton solutions of the KP equation

We consider $N$-soliton solutions of the KP equation, (-4u_t+u_{xxx}+6uu_x)_x+3u_{yy}=0 . An $N$-soliton solution is a solution $u(x,y,t)$ which has the same set of $N$ line soliton solutions in both asymptotics $y\to\infty$ and $y\to -\infty$. The $N$-soliton solutions include all possible resonant interactions among those line solitons. We then classify those $N$-soliton solutions by defining a pair of $N$-numbers $({\bf n}^+,{\bf n}^-)$ with ${\bf n}^{\pm}=(n_1^{\pm},...,n_N^{\pm}), n_j^{\pm}\in\{1,...,2N\}$, which labels $N$ line solitons in the solution. The classification is related to the Schubert decomposition of the Grassmann manifolds Gr$(N,2N)$, where the solution of the KP equation is defined as a torus orbit. Then the interaction pattern of $N$-soliton solution can be described by the pair of Young diagrams associated with $({\bf n}^+,{\bf n}^-)$. We also show that $N$-soliton solutions of the KdV equation obtained by the constraint $\partial u/\partial y=0$ cannot have resonant interaction.

Soliton Solutions and Interaction Between a Line Soliton and a y-Periodic Soliton in Generalized (2+1)-Dimensional Nizhnik-Novikov- Veselov Equation**The project supported by National Natural Science Foundation of China under Grant No. 10272072 and the Science Research Foundation of Huzhou Teachers College under Grant No. 200406

Using the variable separation approach, many types of exact solutions of the generalized (2+1)-dimensional Nizhnik-Novikov-Veselov equation are derived. One of the exact solutions of this model is analyzed to study the interaction between a line soliton and a -periodic soliton.