Multisoliton Solutions for the Isospectral and Nonisospectral BKP Equation with Self-Consistent Sources

The isospectral and nonisospectral BKP equation with self-consistent sources is derived to study the linear problem of the BKP system. The bilinear form of the nonisospeetral BKP equation with self-consistent sources is given and the N-soliton solutions are obtained with the Hirota method and Pfaffian technique, respectively.     

The integrability of nonisospectral and variable-coefficient KdV equation with binary Bell polynomials

Binary Bell polynomials are extended to systematically construct bilinear formalism, bilinear Bäcklund transformations, Lax pairs and infinite conservation laws of the nonisospectral and variable-coefficient KdV equation in a quick and natural way. Moreover, the infinite conservation laws are local and obtained through directly decoupling binary Bell polynomials.     

Lie Symmetry Analysis,Bäcklund Transformations and Exact Solutions to (2+1)-Dimensional Burgers' Types of Equations

This paper is concerned with the (2+1)-dimensional Burgers' and heat types of equations. All of the geometic vector fields of the equations are obtained, an optimal system of the equation is presented. Especially, the Bäcklund transformations (BTs) for the Burgers' equations are constructed based on the symmetry. Then, all of the symmetry reductions are provided in terms of the optimal system method, and the exact explicit solutions are investigated by the symmetry reductions and Bäcklund transformations.     

Bcklund Transformations and Solutions of a Generalized Kadomtsev-Petviashvili Equation

In this paper,the bilinear form of a generalized Kadomtsev-Petviashvili equation is obtained by applying the binary Bell polynomials.The N-soliton solution and one periodic wave solution are presented by use of the Hirota direct method and the Riemann theta function,respectively.And then the asymptotic analysis demonstrates one periodic wave solution can be reduced to one soliton solution.In the end,the bilinear Bcklund transformations are derived.     

Perturbation,symmetry analysis,Bäcklund and reciprocal transformation for the extended Boussinesq equation in fluid mechanics

In this work, we study a generalized double dispersion Boussinesq equation that plays a significant role in fluid mechanics, scientific fields, and ocean engineering. This equation will be reduced to the Korteweg–de Vries equation via using the perturbation analysis. We derive the corresponding vectors, symmetry reduction and explicit solutions for this equation. We readily obtain B?cklund transformation associated with truncated Painlevéexpansion. We also examine the related conservation laws of this equation via using the multiplier method. Moreover, we investigate the reciprocal B?cklund transformations of the derived conservation laws for the first time.     

Bilinear forms through the binary Bell polynomials,N solitons and Bäcklund transformations of the Boussinesq–Burgers system for the shallow water waves in a lake or near an ocean beach

Water waves are one of the most common phenomena in nature, the studies of which help energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc. Hereby, symbolic computation is performed on the Boussinesq–Burgers system for shallow water waves in a lake or near an ocean beach. For the water-wave horizontal velocity and height of the water surface above the bottom, two sets of the bilinear forms through the binary Bell polynomials and N-soliton solutions are worked out, while two auto-Bäcklund transformations are constructed together with the solitonic solutions, where N is a positive integer. Our bilinear forms, N-soliton solutions and Bäcklund transformations are different from those in the existing literature. All of our results are dependent on the water-wave dispersive power.     

Bäcklund Transformation and Multisoliton Solutions in Terms of Wronskian Determinant for (2+1)-Dimensional Breaking Soliton Equations with Symbolic Computation

In this paper, two types of the (2+1)-dimensional breaking soliton equations are investigated, which describe the interactions of the Riemann waves with the long waves. With symbolic computation, the Hirota bilinear forms and Bäcklund transformations are derived for those two systems. Furthermore, multisoliton solutions in terms of the Wronskian determinant are constructed, which are verified through the direct substitution of the solutions into the bilinear equations. Via the Wronskian technique, it is proved that theBäcklund transformations obtained are the ones between the (N-1)- and N-soliton solutions. Propagations and interactions of the kink-/bell-shaped solitons are presented. It is shown that the Riemann waves possess the solitonic properties, and maintain the amplitudes and velocities in the collisions only with some phase shifts.     

Applications of Backlund Transformations to Explicit and Exact Solutions in Nonlinear Wave. Equations

Bäcklund transformations and heat equation are used to find several families of explicit and exact solutions for the well-known Whitham-Broer-Kaup equations in shallow water and Kupershmidt equations. In result, multi-soliton solutions, rational fraction solutions and soliton-like solutions are obtained.     

NEW TYPE OF NONISOSPECTRAL KP EQUATION WITH SELF-CONSISTENT SOURCES AND ITS BILINEAR BÄCKLUND TRANSFORMATION

A new type of the nonisospectral KP equation with self-consistent sources is constructed by using the source generation procedure. A new feature of the obtained nonisospectral system is that we allow y-dependence of the arbitrary constants in the determinantal solution for the nonisospectral KP equation. In order to further show integrability of the novel nonisospectral KP equation with self-consistent sources, we give a bilinear Bäcklund transformation.     

Soliton Solutions,B/icklund Transformations and Lax Pair for a （3 ＋ 1）-Dimensional Variable-Coefficient Kadomtsev-Petviashvili Equation in Fluids

Under investigation in this paper is a （3 q- 1）-dimensional variable-coefficient Kadomtsev-Petviashvili equation, which describes the propagation of surface and internal water waves. By virtue of the binary Bell polynomials, symbolic computation and auxiliary independent variable, the bilinear forms, soliton solutions, Backlund transformations and Lax pair are obtained. Variable coefficients of the equation can affect the solitonic structure, when they are specially chosen, while curved and linear solitons are illustrated. Elastic collisions between/among two and three solitons are discussed, through which the solitons keep their original shapes invariant except for some phase shifts.     

Binary Bell Polynomials Approach to Generalized Nizhnik-Novikov-Veselov Equation

The elementary and systematic binary Bell polynomials method is applied to the generalized Nizhnik-Novikov-Veselov (GNNV) equation. The bilinear representation, bilinear Bäcklund transformation, Lax pair and infinite conservation laws of the GNNV equation are obtained directly, without too much trick like Hirota's bilinear method.     

ON THE PROLONGATION STRUCTURE OF ERNST EQUATION

An SL(2R) ×R¹(l) prolongation structure of Ernst equation with a real parameter l and the corresponding Riccati equation as well as a pair of linear equations which are in principle equivalent to the inverse scattering problem due to Belinsky and Zakharov are obtained by solving the fundamental equation for the prolongation structure. A necessary condition which should be satisfied by the Bäcklund transformations is pfesented in terms of prolongation structure. And it is indicated that in the, case of Ernst equation the Harrison transformation, Neugebauer transformations and other available Bäcklund transformations as well as Belinsky-Zakharov's Riemann transformation, i.e., the homogeneous Hilbertproblem (HHP), would be covered by this condition.     

A DIRECT PROCEDURE ON THE INTEGRABILITY OF NONISOSPECTRAL AND VARIABLE-COEFFICIENT MKdV EQUATION

An elementary and systematic method based on binary Bell polynomials is applied to nonisospectral and variable-coefficient MKdV (vcMKdV) equation. The bilinear representation, bilinear Bäcklund transformation, Lax pair and infinite local conservation laws are obtained step by step, without too much clever guesswork.     

Bäcklund transformation,infinite number of conservation laws and fission properties of an integro-differential model for ocean internal solitary waves

This paper presents an analytical investigation of the propagation of internal solitary waves in the ocean of finite depth. Using the multi-scale analysis and reduced perturbation methods, the integrodifferential equation is derived, which is called the intermediate long wave(ILW) equation and can describe the amplitude of internal solitary waves. It can reduce to the Benjamin–Ono equation in the deep-water limit, and to the Kd V equation in the shallow-water limit. Little attention has been paid to the features of integro-differential equations, especially for their conservation laws. Here,based on Hirota bilinear method, B?cklund transformations in bilinear form of ILW equation are derived and infinite number of conservation laws are given. Finally, we analyze the fission phenomenon of internal solitary waves theoretically and verify it through numerical simulation. All of these have potential value for the further research on ocean internal solitary waves.     

Residual Symmetry Reduction and Consistent Riccati Expansion of the Generalized Kaup-Kupershmidt Equation

The residual symmetry of the generalized Kaup-Kupershmidt(gKK) equation is obtained from the truncated Painlevé expansion and localized to a Lie point symmetry in a prolonged system. New symmetry reduction solutions of the prolonged system are given by using the standard Lie symmetry method. Furthermore, the g KK equation is proved to integrable in the sense of owning consistent Riccati expansion and some new B¨acklund transformations are given based on this property, from which interaction solutions between soliton and periodic waves are given.     

New Baecklund Transformation and New Exact Solutions to （2＋1）-Dimensional KdV Equation

A new Baecklund transformation for (2 1)-dimensional KdV equation is first obtained by using homogeneous balance method. And making use of the Baecklund transformation and choosing a special seed solution, we get special types of solitary wave solutions. Finally a general variable separation solution containing two arbitrary functions is constructed, from which abundant localized coherent structures of the equation in question can be induced.     

The explicit symmetry breaking solutions of the Kadomtsev-Petviashvili equation

To describe two correlated events, the Alice–Bob (AB) systems were constructed by Lou through the symmetry of the shifted parity, time reversal and charge conjugation. In this paper, the coupled AB system of the Kadomtsev–Petviashvili equation, which is a useful model in natural science, is established. By introducing an extended Bäcklund transformation and its bilinear formation, the symmetry breaking soliton, lump and breather solutions of this system are derived with the aid of some ansatze functions. Figures show these fascinating symmetry breaking structures of the explicit solutions.     

Infinite Sequence of Conservation Laws and Analytic Solutions for a Generalized Variable-Coefficient Fifth-Order Korteweg-de Vries Equation in Fluids

In this paper, an infinite sequence of conservation laws for a generalized variable-coefficient fifth-order Korteweg-de Vries equation in fluids are constructed based on the Bäcklund transformation. Hirota bilinear form and symbolic computation are applied to obtain three kinds of solutions. Variablecoefficients can affect the conserved density, associated flux, andappearance of the characteristic lines. Effects of the wave number on the soliton structures are also discussed and types of soliton structures, e.g., the double-periodic soliton, parallel soliton and soliton complexes, are presented.     

LAX PAIR,BINARY DARBOUX TRANSFORMATION AND NEW GRAMMIAN SOLUTIONS OF NONISOSPECTRAL KADOMTSEV–PETVIASHVILI EQUATION WITH THE TWO-SINGULAR-MANIFOLD METHOD

In this letter, the two-singular-manifold method is applied to the (2+1)-dimensional nonisospectral Kadomtsev–Petviashvili equation with two Painlevé expansion branches to determine auto-Bäcklund transformation, Lax pairs and Darboux transformation. Based on the two obtained Lax pairs, the binary Darboux transformation is constructed and then the Nth iterated transformation formula in the form of Grammian is also presented. By using these Darboux transformations, we obtain some new grammian solutions.     

Bäcklund transformations,kink soliton,breather- and travelling-wave solutions for a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation in fluid dynamics

In this paper, we investigate a (3+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation in fluid dynamics. Based on the Hirota method, we give a bilinear auto-Bäcklund transformation. Via the truncated Painlevé expansion, we get a Painlevé-type auto-Bäcklund transformation. With the aid of the symbolic computation, we derive some one- and two-kink soliton solutions. We present the oblique and parallel elastic interactions between the two-kink solitons. Via the extended homoclinic test technique, we construct some breather-wave solutions. Besides, we derive some lump solutions with the periods of the breather-wave solutions to the infinity. We observe that the shapes of a breather wave and a lump remain unchanged during the propagation. Based on the polynomial-expansion method, travelling-wave solutions are constructed.