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The main work of this paper is to construct the Wronskian solution and investigate the integrability characteristics of the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation. Firstly, the Wronskian technique is used to acquire a sufficient condition of the Wronskian solution. According to the Wronskian form, the soliton solution is obtained by selecting the elements in the determinant that satisfy the linear partial differential systems. Secondly, the bilinear Bäcklund transformation and Bell-polynomial-typed Bäcklund transformation are derived directly via the Hirota bilinear method and the Bell polynomial theory, respectively. Finally, Painlevé analysis proves that this equation possesses the Painlevé property, and a Painlevé-typed Bäcklund transformation is constructed to solve a family of exact solutions by selecting appropriate seed solution. It shows that the Wronskian technique, Bäcklund transformation, Bell polynomial and Painlevé analysis are applicable to obtain the exact solutions of the nonlinear evolution equations, e.g., soliton solution, single-wave solution and two-wave solution. [ABSTRACT FROM AUTHOR]

Variable-coefficient equations can be used to describe certain phenomena when the inhomogeneous media and nonuniform boundaries are taken into consideration. It is meaningful to solve the exact solution of variable-coefficient equations. In this paper, a (2+1)-dimensional variable-coefficient Boussinesq equation is investigated. The integrability is firstly examined by the Painlevé analysis method. Secondly, the Bäcklund transformations, one- and two-soliton solutions of the (2+1)-dimensional variable-coefficient Boussinesq equation are studied by virtue of the Hirota bilinear method. Propagation characteristics and interaction behaviors of the solitons are discussed: (i) soliton shapes and interaction behaviors are affected by the variable coefficients, and (ii) the two-soliton interaction is elastic, and the shape and velocity does not change after the collision, and only the shift changes. [ABSTRACT FROM AUTHOR]

Abstract: The Korteweg–de Vries (KdV)-type models are of significance in describing many physical situations in fluid flows (particularly for surface and internal waves), plasma physics, and solid state physics. In fluid dynamics, for example, the shallow water wave equation is utilized as a mathematical description of regular and generalized solitary waves in shallow water. Further, higher-order dispersive (e.g., the Lax fifth-order KdV equation) and higher-dimensional [e.g., the (2+1)- and (3+1)-dimensional breaking soliton equations] generalized nonlinear models are useful in analyzing and obtaining modulation theory, existence and stability of solitary waves, bores, and shocks, as well as other integrable properties. With symbolic computation, Bell-polynomial-typed Bäcklund transformations (BTs) are constructed for some single-field bilinearizable nonlinear evolution equations including the shallow water wave equation, Lax fifth-order KdV equation, and (2+1)- and (3+1)-dimensional breaking soliton equations. Bell-polynomial expressions are derived, which can be cast into the bilinear equations with one Tau-function. Key point lies in the introduction of certain auxiliary independent variable in the Bell-polynomial expression. With one auxiliary independent variable, the Bell-polynomial-typed BTs are then constructed according to the coupled two-field conditions between the primary and replica fields with both the fields satisfying the Bell-polynomial-expression equations. Auxiliary-independent-variable-involved Bell-polynomial-typed BTs are changed into their bilinear forms. Aforementioned equations turn out to be integrable in the sense of possessing the Bell-polynomial-typed BTs. [Copyright &y& Elsevier]

Due to their relevance to physics and technology, the Bose-Einstein condensates (BECs) are of current interest. Certain dynamics of the BECs, such as the cigar-shaped condensate confined in a cylindrically symmetric parabolic trap, can be described by the Gross-Pitaevskii (GP) equation with a time-dependent trap. In this paper, by virtue of the Painlevé analysis and symbolic computation, we derive the integrable condition for the GP equation with the time-dependent scattering length in the presence of a confining or expulsive time-dependent trap. Lax pair for this equation is directly obtained via the Ablowitz-Kaup-Newell-Segur scheme under the integrable condition. Bright one-soliton-like solution of the GP equation is presented via the Bäcklund transformation and some analytic solutions with variable amplitudes are obtained by the ansatz method. In addition, an infinite number of conservation laws are also derived. Those results could be of some value for the studies on the lower-dimensional condensates. [ABSTRACT FROM AUTHOR]

In the framework of Bell-polynomial manipulations, under investigation hereby are three single-field bilinearizable equations: the (1+1)-dimensional shallow water wave model, Boiti-Leon-Manna-Pempinelli model, and (2+1)-dimensional Sawada-Kotera model. Based on the concept of scale invariance, a direct and unifying Bell-polynomial scheme is employed to achieve the Bäcklund transformations and Lax pairs associated with those three soliton equations. Note that the Bell-polynomial expressions and Bell-polynomial-typed Bäcklund transformations for those three soliton equations can be, respectively, cast into the bilinear equations and bilinear Bäcklund transformations with symbolic computation. Consequently, it is also shown that the Bell-polynomial-typed Bäcklund transformations can be linearized into the corresponding Lax pairs. [ABSTRACT FROM AUTHOR]

Abstract: 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 - 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. [Copyright &y& Elsevier]