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.     

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.     

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.     

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.     

Integrability of extended (2+1)-dimensional shallow water wave equation with Bell polynomials

We investigate the extended (2+1)-dimensional shallow water wave equation. The binary Bell polynomials are used to construct bilinear equation, bilinear Bäcklund transformation, Lax pair, and Darboux covariant Lax pair for this equation. Moreover, the infinite conservation laws of this equation are found by using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas. The N-soliton solutions are also presented by means of the Hirota bilinear method.     

The solitary waves,quasi-periodic waves and integrability of a generalized fifth-order Korteweg-de Vries equation

Under investigation in this paper is a fifth-order Korteweg-de Vries (fKdV) equation, which can be used to describe many nonlinear phenomena in fluid dynamics and plasma physics. Based on the binary Bell polynomials, a lucid and systematic approach is proposed to systematically study its bilinear representation, bilinear Bäcklund transformations and Lax pairs with explicit formulas, respectively. These results can be reduced to the ones of several integrable equations such as Sawada-Kotera equation, Caudrey-Dodd-Gibbon equation, Lax equation, Kaup-Kuperschmidt equation and Ito equation, etc. Furthermore, the N-solitary wave solutions formula and quasi-periodic wave solutions are obtained by using bilinear form of the fKdV equation. Finally, the relation between the periodic wave solution and solitary wave solution is rigorously established.     

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.     

Painlevé Analysis,Soliton Collision and Bäcklund Transformation for the (3+1)-Dimensional Variable-Coefficient Kadomtsev-Petviashvili Equation in Fluids or Plasmas

In this paper, we investigate a (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili equation, which can describe the nonlinear phenomena in fluids or plasmas. Painlevé analysis is performed for us to study the integrability, and we find that the equation is not completely integrable. By virtue of the binary Bell polynomials, bilinear form and soliton solutions are obtained, and Bäcklund transformation in the binary-Bell-polynomial form and bilinear form are derived. Soliton collisions are graphically discussed: the solitons keep their original shapes unchanged after the collision except for the phase shifts. Variable coefficients are seen to affect the motion of solitons: when the variable coefficients are chosen as the constants, solitons keep their directions unchanged during the collision; with the variable coefficients as the functions of the temporal coordinate, the one soliton changes its direction.     

On Integrable Properties for Two Variable-Coefficient Evolution Equations

With the help of the extended binary Bell polynomials, the new bilinear representations, Bcklund trans-formations, Lax pair and infinite conservation laws for two types of variable-coefficient nonlinear integrable equations are obtained, respectively, which are more straightforward than previous corresponding results obtained. Finally, we obtain new multi-soliton wave solutions of a reduced soliton equations with variable coefficients.     

Binary Bell Polynomials,Bilinear Approach to Exact Periodic Wave Solutions of (2+1)-Dimensional Nonlinear Evolution Equations

In the present letter, we get the appropriate bilinear forms of(2+1)-dimensional KdV equation, extended (2+1)-dimensional shallow water wave equation and (2+1)-dimensional Sawada-Kotera equation in a quick and natural manner, namely by appling the binary Bell polynomials. Then the Hirota direct method and Riemann theta function are combined to construct the periodic wave solutions of the three types nonlinear evolution equations. And the corresponding figures of the periodic wave solutions are given. Furthermore, the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.     

Binary Bell polynomial application in generalized （2＋1）-dimensional KdV equation with variable coefficients

In this paper, we apply the binary Bell polynomial approach to high-dimensional variable-coefficient nonlinear evolution equations. Taking the generalized (2+1)-dimensional KdV equation with variable coefficients as an illustrative example, the bilinear formulism, the bilinear Bäcklund transformation and the Lax pair are obtained in a quick and natural manner. Moreover, the infinite conservation laws are also derived.     

Bäcklund transformation,infinitely-many conservation laws,solitary and periodic waves of an extended (3 + 1)-dimensional Jimbo–Miwa equation with time-dependent coefficients

In this paper, an extended (3+1)-dimensional Jimbo–Miwa equation with time-dependent coefficients is investigated, which comes from the second member of the Kadomtsev–Petviashvili hierarchy and is shown to be conditionally integrable. Bilinear form, Bäcklund transformation, Lax pair and infinitely-many conservation laws are derived via the binary Bell polynomials and symbolic computation. With the help of the bilinear form, one-, two- and three-soliton solutions are obtained via the Hirota method, one-periodic wave solutions are constructed via the Riemann theta function. Additionally, propagation and interaction of the solitons are investigated analytically and graphically, from which we find that the interaction between the solitons is elastic and the time-dependent coefficients can affect the soliton velocities, but the soliton amplitudes remain unchanged. One-periodic waves approach the one-solitary waves with the amplitudes vanishing and can be viewed as a superposition of the overlapping solitary waves, placed one period apart.     

Quasi-periodic wave solutions,soliton solutions,and integrability to a (2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation

In this paper, a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko (gBK) equation is investigated, which can be used to describe the interaction of a Riemann wave propagating along y-axis and a long wave propagating along x-axis. The complete integrability of the gBK equation is systematically presented. By employing Bell’s polynomials, a lucid and systematic approach is proposed to systematically study its bilinear formalism, bilinear Bäcklund transformations, Lax pairs, respectively. Furthermore, based on multidimensional Riemann theta functions, the periodic wave solutions and soliton solutions of the gBK equation are derived. Finally, an asymptotic relation between the periodic wave solutions and soliton solutions are strictly established under a certain limit condition.     

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.     

Bell Polynomials to the Kadomtsev-Petviashivili Equation with Self-Consistent Sources

Bell Polynomials play an important role in the characterization of bilinear equation. Bell Polynomials are extended to construct the bilinear form, bilinear Bäcklund transformation and Lax pairs for the Kadomtsev-Petviashvili equation with self-consistent sources.     

In oceanography,acoustics and hydrodynamics: An extended coupled (2+1)-dimensional Burgers system

In oceanography, acoustics and hydrodynamics, people pay attention to the Burgers-type equations for different wave processes, one of which is an extended coupled (2+1)-dimensional Burgers system hereby under investigation. Based on the scaling transformation, Bell polynomials, Hirota operators and symbolic computation, we structure out two hetero-Bäcklund transformations, each of which to a solvable linear partial differential equation, and construct two sets of the bilinear forms, with the relevant one- and two-soliton solutions. Results rely on the coefficients in the original system.     

Characteristics of the solitary waves and rogue waves with interaction phenomena in a (2 + 1)-dimensional Breaking Soliton equation

Under inquisition in this paper is a $(2+1)$-dimensional Breaking Soliton equation, which can describe various nonlinear scenarios in fluid dynamics. Using the Bell polynomials, some proficient auxiliary functions are offered to apparently construct its bilinear form and corresponding soliton solutions which are different from the previous literatures. Moreover, a direct method is used to construct its rogue wave and solitary wave solutions using particular auxiliary function with the assist of bilinear formalism. Finally, the interactions between solitary waves and rogue waves are offered with a complete derivation. These results enhance the variety of the dynamics of higher dimensional nonlinear wave fields related to mathematical physics and engineering.     

Bäcklund Transformation,Lax Pair and Solitons of the (2+1)-dimensional Davey-Stewartson-like Equations with Variable Coefficients for the Electrostatic Wave Packets

The (2+1)-dimensional Davey-Stewartson-like equations with variable coefficients have the applications in the ultra-relativistic degenerate dense plasmas and Bose-Einstein condensates. Via the Bell polynomials and symbolic computation, the bilinear form, Bäcklund transformation and Lax pair for such equations are obtained. Based on the Hirota method, we construct the soliton solutions, analyze the elastic collisions with the constant and variable coefficients, and observe that solitons no longer keep rectilinear propagation and display different shapes because of the variable coefficients. Besides, localized excitations are derived through the variable separation.