Hong-Wu Zhu

MULTI-SOLITON SOLUTIONS AND THEIR INTERACTIONS FOR THE (2+1)-DIMENSIONAL SAWADA-KOTERA MODEL WITH TRUNCATED PAINLEVÉ EXPANSION, HIROTA BILINEAR METHOD AND SYMBOLIC COMPUTATION

In this paper, the (2+1)-dimensional Sawada-Kotera equation is studied by the truncated Painleve expansion and Hirota bilinear method. Firstly, based on the truncation of the Painleve series we obtain two distinct transformations which can transform the (2+1)-dimensional Sawada-Kotera equation into two bilinear equations of different forms (which are shown to be equivalent). Then employing Hirota bilinear method, we derive the analytic one-, two- and three-soliton solutions for the bilinear equations via symbolic computation. A formula which denotes the N-soliton solution is given simultaneously. At last, the evolutions and interactions of the multi-soliton solutions are graphically discussed as well. It is worthy to be noted that the truncated Painleve expansion provides a useful dependent variable transformation which transforms a partial differential equation into its bilinear form and by means of the bilinear form, further study of the original partial differential equation can be conducted.

Symbolic computation on integrable properties of a variable-coefficient Korteweg?de Vries equation from arterial mechanics and Bose?Einstein condensates

Applicable in arterial mechanics, Bose gases of impenetrable bosons and Bose–Einstein condensates, a variable-coefficient Korteweg–de Vries (vcKdV) equation is investigated in this paper with symbolic computation. Based on the Ablowitz–Kaup–Newell–Segur system, the Lax pair and auto-Backlund transformation are constructed. Furthermore, the nonlinear superposition formula and an infinite number of conservation laws for the vcKdV equation are also derived. Special attention is paid to the analytic one- and two-solitonic solutions with their physical properties and possible applications discussed.

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.

Double Wronskian Solution and Conservation Laws for a Generalized Variable-Coefficient Higher-Order Nonlinear Schrödinger Equation in Optical Fibers

Abstract With applications in the higher-power and femtosecond optical transmission regime, a generalized variable-coefficient higher-order nonlinear Schrödinger (VC-HNLS) equation is analytically investigated. The multi-solitonic solutions of the generalized VC-HNLS equation in double Wronskian form is constructed and further verified using the Wronskian technique. Additionally, an infinite number of conservation laws for such an equation are presented. Finally, discussions and conclusions on results are made with figures plotted.

ON THE EXISTENCE OF INFINITE CONSERVATION LAWS OF A VARIABLE-COEFFICIENT KORTEWEG–DE VRIES MODEL WITH SYMBOLIC COMPUTATION

Under investigation in this paper is a generalized variable-coefficient Korteweg–de Vries (vcKdV) model with external-force and perturbed/dissipative terms, which can describe various real dynamical processes of physics from atmosphere blocking and gravity waves, blood vessels, Bose–Einstein condensates, rods and positons and so on. With the aid of symbolic computation, a generalized Miura transformation is proposed to relate the solutions of the vcKdV equation to those of a variable-coefficient modified Korteweg–de Vries equation. Then by using such a Miura transformation and the Galilean invariant transformation, the existence of infinite conservation laws are proved under the Painleve integrable condition. These results may be valuable for the new discoveries in dynamical systems described by integrable vcKdV models and the theoretical study of the relationships among infinite conservation laws, the integrability of the nonlinear evolution equation and inverse scattering transform.