Positons, negatons and complexitons of the mKdV equation with non-uniformity terms

The N-soliton solution of the mKdV equation with non-uniformity terms is obtained through Hirota method and Wronskian technique. We can also derive its positons, negatons and complexitons by a matrix extension of the Wronskian formulation.     

Soliton solutions, Bäcklund transformation and Wronskian solutions for the extended (2+1)-dimensional Konopelchenko-Dubrovsky equations in fluid mechanics

This paper is to investigate the extended (2+1)-dimensional Konopelchenko-Dubrovsky equations, which can be applied to describing some phenomena in the stratified shear flow, the internal and shallow-water waves and plasmas. Bilinear-form equations are transformed from the original equations and N-soliton solutions are derived via symbolic computation. Bilinear-form Bäcklund transformation and single-soliton solution are obtained and illustrated. Wronskian solutions are constructed from the Bäcklund transformation and single-soliton solution.     

Multi-component Wronskian solution to the Kadomtsev-Petviashvili equation

It is known that the Kadomtsev-Petviashvili (KP) equation can be decomposed into the first two members of the coupled Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy by the binary non-linearization of Lax pairs. In this paper, we construct the N-th iterated Darboux transformation (DT) for the second- and third-order m-coupled AKNS systems. By using together the N-th iterated DT and Cramer’s rule, we find that the KPII equation has the unreduced multi-component Wronskian solution and the KPI equation admits a reduced multi-component Wronskian solution. In particular, based on the unreduced and reduced two-component Wronskians, we obtain two families of fully-resonant line-soliton solutions which contain arbitrary numbers of asymptotic solitons as y → ?∞ to the KPII equation, and the ordinary N-soliton solution to the KPI equation. In addition, we find that the KPI line solitons propagating in parallel can exhibit the bound state at the moment of collision.     

The interaction processes of the N-soliton solutions for an extended generalization of Vakhnenko equation

The N-soliton solutions for an extended generalization of Vakhnenko equation are calculated by applying the improved Hirota method and the variable transformations. The N-soliton solutions can be expressed explicitly. Furthermore, the interaction processes are discussed for the N-soliton solutions.     

Bäcklund transformation,superposition formulae and N-soliton solutions for the perturbed Korteweg–de Vries equation

With symbolic computation, under investigation in this paper is the perturbed Korteweg–de Vries equation for the nonlocal solitary waves and arrays of wave crests. Via the Hirota method, the bilinear form, Bäcklund transformation and superposition formulae are obtained. N-soliton solutions in terms of the Wronskian are constructed. Asymptotic analysis is used to analyze the collision dynamics, and figures are plotted to illustrate the influence of the perturbation. We find that the perturbation affects the propagation velocities of the solitons, but does not affect the amplitudes and widths of the solitons. Besides, the solitonic collisions turn out to be elastic.     

N-soliton solutions,Bäcklund transformation and conservation laws for the integro-differential nonlinear Schröbinger equation from the isotropic inhomogeneous Heisenberg spin magnetic chain

Under investigation in this paper is an integro-differential nonlinear Schröbinger (IDNLS) equation, which is equivalent to the spin evolution equation of a classical in-homogeneous Heisenberg magnetic chain in the continuum limit. Based on the Hirota method, the bilinear form and N-soliton solution for the IDNLS equation are derived with the help of symbolic computation. Moreover, N-soliton solution for the IDNLS equation is expressed in terms of the double Wronskian and testified through the direct substitution into the bilinear form. Besides, the bilinear Bäcklund transformation and infinitely many conservation laws are also obtained for the IDNLS equation. Propagation characteristics and interaction behaviors of the solitons are discussed by analysis of such physical quantities as the soliton amplitude, width, velocity and initial phase. Interactions of the solitons are proved to be elastic through the asymptotic analysis. Effect of inhomogeneity on the interaction of the solitons is studied graphically.     

Dynamics of the generalized (3 + 1)-dimensional nonlinear Schröbinger equation in cosmic plasmas

Under investigation in this paper is a generalized (3 + 1)-dimensional nonlinear Schröbinger equation with the variable coefficients, which governs the nonlinear dynamics of the ion-acoustic envelope solitons in the magnetized electron-positron-ion plasma with two-electron temperatures in space or astrophysics. Bilinear forms and Bäcklund transformations are derived through the Bell polynomials. N-soliton solutions are constructed in the form of the double Wronskian determinant and the N-th order polynomials in N exponentials. Shape and motion of one soliton have been graphically analyzed, as well as the interactions of two and three solitons. When β(t) and γ(t) are both the periodic functions of the reduced time t, where γ(t) is the loss (gain) coefficient, and β(t) means the combined effects of the transverse perturbation and magnetic field, the shape and motion of one soliton as well as the interactions of two or three solitons will occur periodically. All the interactions can be elastic with certain coefficients.     

A new representation of N-soliton solution and limiting solutions for the fifth order KdV equation

A new representation of N-soliton solution of the fifth order KdV equation is obtained by using Bäcklund transformation method. It is shown that the new representation of N-soliton solution is in agreement with Hirota’s expression. Some novel soliton solutions are derived by performing an appropriate limiting procedure on the known soliton solutions.     

Bäcklund transformation in bilinear form for a higher-order nonlinear Schrödinger equation

Bäcklund transformation in bilinear form is presented for a higher-order nonlinear Schrödinger equation, which describes the propagation of ultrashort light pulses in optical fibers. With symbolic computation and starting from the Bäcklund transformation, the analytical soliton solution is obtained from a trivial solution and the inverse scattering transform scheme is also derived. Furthermore, the N$N$-soliton solution in double Wronskian form is given, and the value of the arbitrary constant appearing in the Bäcklund transformation is determined for a transformation between the (N−1)$(N-1)$ and N$N$-soliton solutions. The results obtained from the Bäcklund transformation might be valuable in optical communications.     

Bäcklund transformation and multi-soliton solutions for a (2 + 1)-dimensional Korteweg-de Vries system via symbolic computation

In this paper, the (2 + 1)-dimensional Korteweg-de Vries system is symbolically investigated. By the bilinear method, the N-soliton solution is presented. Then, based on the Bäcklund transformation in bilinear form, a new Bäcklund transformation is obtained and new representation of the N-soliton solution is derived. A class of novel multi-soliton solutions are obtained by the new Bäcklund transformation and the availability of symbolic computation is demonstrated.     

Gauge transformation, elastic and inelastic interactions for the Whitham-Broer-Kaup shallow-water model

Whitham-Broer-Kaup (WBK) model is a model for the dispersive long wave in shallow water. With symbolic computation, gauge transformation between the WBK model and a parameter Ablowitz-Kaup-Newell-Segur (AKNS) system is hereby constructed. By selecting seeds, we derive two sorts of multi-soliton solutions for the WBK model via a N-fold Darboux transformation (DT) of the parameter AKNS system, which are expressed in terms of the Vandermonde-like and double Wronskian determinants, respectively. Different from the bilinear way, the double Wronskian solutions can be obtained via the N-fold DT with a linear algebraic system and matrix differential equation solved. A novel inelastic interaction is graphically discussed, in which the soliton complexes are formed after the collision. Our results could be helpful for interpreting certain shallow-water-wave phenomena.     

Integrable discretizations of the Dym equation

Integrable discretizations of the complex and real Dym equations are proposed. N-soliton solutions for both semi-discrete and fully discrete analogues of the complex and real Dym equations are also presented.     

Darboux transformation and N-soliton solutions for a more general set of coupled integrable dispersionless system

The Darboux transformation and Lax pair of a more general set of coupled integrable dispersionless system are derived. By Darboux transformation, N-soliton solutions for the coupled integrable dispersionless system are obtained. In particular, the multi-soliton solutions are shown through some figures.     

Dressing for a Novel Integrable Generalization of the Nonlinear Schrödinger Equation

We implement the dressing method for a novel integrable generalization of the nonlinear Schrödinger equation. As an application, explicit formulas for the N-soliton solutions are derived. As a by-product of the analysis, we find a simplification of the formulas for the N-solitons of the derivative nonlinear Schrödinger equation given by Huang and Chen.     

Maple packages for computing Hirota’s bilinear equation and multisoliton solutions of nonlinear evolution equations

The Hirota method for generating Hirota’s bilinear equation and constructing soliton solutions of nonlinear evolution equations is discussed and illustrated. Two Maple programs Bilinearization and Multisoliton are presented to automatically calculate Hirota’s bilinear equations for nonlinear evolution equations and to compute their N-soliton solutions for N = 1, 2 or 3, respectively. Different kinds of examples are used to demonstrate the effectiveness of the packages.     

Symbolic computation, Bäcklund transformation and analytic N-soliton-like solution for a nonlinear Schrödinger equation with nonuniformity term from certain space/laboratory plasmas

With symbolic computation, a bilinear Bäcklund transformation is presented for a nonlinear Schrödinger equation with nonuniformity term from certain space/laboratory plasmas, and correspondingly the one-soliton-like solution is derived from the Bäcklund transformation. Simultaneously, the N-soliton-like solution in double Wronskian form is also given. Besides, the authors verify that the (N−1)- and N-soliton-like solutions satisfy the Bäcklund transformation. The results obtained in this paper might be valuable for the study of the nonuniform media.     

The negative extended KdV equation with self-consistent sources: soliton, positon and negaton

The negative extended KdV equation with self-consistent sources (eKdV⁻ESCSs) is firstly presented and the associated linear auxiliary equations are derived. The generalized binary Darboux transformation (DT) is applied to construct some new solutions of the eKdV⁻ESCSs such as singular N-soliton solution, N-soliton solution with finite amplitude, N-positon solution and N-negaton solution. The properties of these solutions are analyzed. Moreover, the interactions of two solitons, positon and negaton, positon and soliton, and two positons are discussed.     

N-lump and interaction solutions of localized waves to the (2 + 1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation arise from a model for an incompressible fluid

The present article deals with M-soliton solution and N-soliton solution of the (2 + 1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation by virtue of Hirota bilinear operator method. The obtained solutions for solving the current equation represent some localized waves including soliton, breather, lump, and their interactions, which have been investigated by the approach of the long-wave limit. Mainly, by choosing the specific parameter constraints in the M-soliton and N-soliton solutions, all cases of the one breather or one lump can be captured from the two, three, four, and five solitons. In addition, the performances of the mentioned technique, namely, the Hirota bilinear technique, are substantially powerful and absolutely reliable to search for new explicit solutions of nonlinear models. Meanwhile, the obtained solutions are extended with numerical simulation to analyze graphically, which results in localized waves and their interaction from the two-, three-, four-, and five-soliton solutions profiles. They will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics, and so on.     

Bäcklund transformations and soliton solutions for the KdV6 equation

Using Hirota technique, a Bäcklund transformation in bilinear form is obtained for the KdV6 equation. Furthermore, we present a modified Bäcklund transformation by a dependent variable transformation, it is shown that a new representation of N-soliton solution and some novel solutions to the KdV6 equation are derived by performing an appropriate limiting procedure on the known soliton solutions.     

Novel composite function solutions of the modified KdV equation

A Wronskian form expansion method is proposed to construct novel composite function solutions to the modified Korteweg-de Vries (mKdV) equation. The method takes advantage of the forms and structures of Wronskian solutions to the mKdV equation, and Wronskian entries do not satisfy linear partial differential equations. The method can be automatically carried out in computer algebra (for example, Maple).