Chun-Yi Zhang

On a generalized Kadomtsev–Petviashvili equation with variable coefficients via symbolic computation

Considering the inhomogeneities of media, a generalized Kadomtsev–Petviashvili equation with time-dependent coefficients is hereby investigated with the aid of symbolic computation. The exact analytic one- and two-soliton solutions under certain constraints are obtained by employing the variable-coefficient balancing-act method and Hirota method. Based on its bilinear form, the Lax pair, auto-Backlund transformation (in both the bilinear form and the Lax pair form) and nonlinear superposition formula for such an equation are presented. Moreover, some figures are plotted to analyze the effects of the coefficient functions on the stabilities and propagation characteristics of the solitonic waves.

Integrable properties of a variable-coefficient Korteweg–de Vries model from Bose–Einstein condensates and fluid dynamics

The phenomena of the trapped Bose–Einstein condensates related to matter waves and nonlinear atom optics can be governed by a variable-coefficient Korteweg–de Vries (vc-KdV) model with additional terms contributed from the inhomogeneity in the axial direction and the strong transverse confinement of the condensate, and such a model can also be used to describe the water waves propagating in a channel with an uneven bottom and/or deformed walls. In this paper, with the help of symbolic computation, the bilinear form for the vc-KdV model is obtained and some exact solitonic solutions including the N-solitonic solution in explicit form are derived through the extended Hirota method. We also derive the auto-B¨ acklund transformation, nonlinear superposition formula, Lax pairs and conservation laws of this model. Finally, the integrability of the variable-coefficient model and the characteristic of the nonlinear superposition formula are discussed.

Analytic Multi-Solitonic Solutions of Variable-Coefficient Higher-Order Nonlinear Schrödinger Models by Modified Bilinear Method with Symbolic Computation

In this paper, the physically interesting variable-coefficient higher-order nonlinear Schr¨odinger models in nonlinear optical fibers with varying higher-order effects such as third-order dispersion, self-steepening, delayed nonlinear response and gain or absorption are investigated. The bilinear transformation method is modified for constructing the analytic solutions of these models directly with sets of parametric conditions. With the aid of symbolic computation, the explicit analytic multisolitonic solutions of the variable-coefficient higher-order nonlinear Schr¨odinger models are presented by employing the modified bilinear transformation method. The one- and two-solitonic solutions in explicit form are given in detail. Finally, solutions are illustrated and discussed through adjusting the parameters, so different dispersion management systems can be obtained.

VARIABLE-COEFFICIENT MIURA TRANSFORMATIONS AND INTEGRABLE PROPERTIES FOR A GENERALIZED VARIABLE-COEFFICIENT KORTEWEG–de VRIES EQUATION FROM BOSE–EINSTEIN CONDENSATES WITH SYMBOLIC COMPUTATION

In this paper, a generalized variable-coefficient Korteweg–de Vries (KdV) equation with the dissipative and/or perturbed/external-force terms is investigated, which arises in arterial mechanics, blood vessels, Bose gases of impenetrable bosons and trapped Bose–Einstein condensates. With the computerized symbolic computation, two variable-coefficient Miura transformations are constructed from such a model to the modified KdV equation under the corresponding constraints on the coefficient functions. Meanwhile, through these two transformations, a couple of auto-Backlund transformations, nonlinear superposition formulas and Lax pairs are obtained with the relevant constraints. Furthermore, the one- and two-solitonic solutions of this equation are explicitly presented and the physical properties and possible applications in some fields of these solitonic structures are discussed and pointed out.

Inelastic Interaction and Non-Traveling-Wave Effects for Two Multi-Dimensional Burgers Models from Fluid Dynamics and Astrophysics with Symbolic Computation

Describing the surface perturbations of a shallow viscous fluid, cosmic-ray-modified shock structures and electromagnetic waves in a saturated ferrite, the (2+1)- and (3+1)-dimensional Burgers equations are investigated in this paper. In view of the higher space dimensionality, the transformations from such two models to a (1+1)-dimensional Burgers equation are constructed with symbolic computation. Via the obtained transformations, three families of multi-dimensional N-shock-wavelike solutions are specially presented, which recover some previously published solutions. The inelastically interacting properties and some non-traveling-wave effects of shock waves are discussed through the figures for several sample solutions. Additionally, possible applications for those solutions and effects in some fields are also pointed out

ANALYTICAL INVESTIGATION OF THE CAUDREY–DODD–GIBBON–KOTERA–SAWADA EQUATION USING SYMBOLIC COMPUTATION

In this paper, the Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation is analytically investigated using the Hirota bilinear method. Based on the bilinear form of the CDGKS equation, its N-soliton solution in explicit form is derived with the aid of symbolic computation. Besides the soliton solutions, several integrable properties such as the Backlund transformation, the Lax pair and the nonlinear superposition formula are also derived for the CDGKS equation.

ANALYTIC DARK SOLITON SOLUTIONS FOR A GENERALIZED VARIABLE-COEFFICIENT HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION IN OPTICAL FIBERS USING SYMBOLIC COMPUTATION

In this paper, a generalized variable-coefficient nonlinear Schrodinger equation with higher-order and gain/loss effects which can be used to describe the femtosecond pulse propagation is analytically investigated via symbolic computation. Under sets of coefficient constraints, such an equation is transformed into a completely integrable constant-coefficient higher-order nonlinear Schrodinger equation. Furthermore, through the transformation, the dark one- and two-soliton solutions for the generalized variable-coefficient higher-order nonlinear Schrodinger equation are derived by means of the bilinear method.

N-SOLITON SOLUTIONS, AUTO-BÄCKLUND TRANSFORMATIONS AND LAX PAIR FOR A NONISOSPECTRAL AND VARIABLE-COEFFICIENT KORTEWEG-DE VRIES EQUATION VIA SYMBOLIC COMPUTATION

In this paper, a nonisospectral and variable-coefficient Korteweg-de Vries equation is investigated based on the ideas of the variable-coefficient balancing-act method and Hirota method. Via symbolic computation, we obtain the analytic N-soliton solutions, variable-coefficient bilinear form, auto-Backlund transformations (in both the bilinear form and Lax pair form), Lax pair and nonlinear superposition formula for such an equation in explicit form. Moreover, some figures are plotted to analyze the effects of the variable coefficients on the stabilities and propagation characteristics of the solitonic waves.