Multiple-kink-soliton solutions of the nonlinear Schrödinger equation (original) (raw)

From single- to multiple-soliton solutions of the perturbed KdV equation

The solution of the perturbed KdV equation (PKDVE), when the zero-order approximation is a multiplesoliton wave, is constructed as a sum of two components: elastic and inelastic. The elastic component preserves the elastic nature of soliton collisions. Its perturbation series is identical in structure to the series-solution of the PKDVE when the zero-order approximation is a single soliton. The inelastic component exists only in the multiple-soliton case, and emerges from the first order and onwards. Depending on initial data or boundary conditions, it may contain, in every order, a plethora of inelastic processes. Examples are given of sign-exchange soliton-anti-soliton scattering, soliton-anti-soliton creation or annihilation, soliton decay or merging, and inelastic soliton deflection. The analysis has been carried out through third order in the expansion parameter, exploiting the freedom in the expansion to its fullest extent. Both elastic and inelastic components do not modify soliton parameters beyond their values in the zero-order approximation. When the PKDVE is not asymptotically integrable, the new expansion scheme transforms it into a system of two equations: The Normal Form for ordinary KdV solitons, and an auxiliary equation describing the contribution of obstacles to asymptotic integrability to the inelastic component. Through the orders studied, the solution of the latter is a conserved quantity, which contains the dispersive wave that has been observed in previous works.

Outset of multiple soliton solutions to the nonlinear Schrödinger equation and the coupled Burgers equation

Journal of Physics Communications, 2019

The nonlinear Schrödinger equation and the coupled Burgers equation illustrate the status of quantum particles, shock waves, acoustic transmission and traffic flow. Therefore these equations are physically significant in their own right. In this article, the new auxiliary equation method has been contrivanced in order to rummage exact wave solutions to previously stated nonlinear evolution equations (NLEEs). We have developed ample soliton solutions and have to do with the physical importance of the acquired solutions by setting the specific values of the embodied parameters through portraying figures and deciphered the physical phenomena. It has been established that the executed method is powerful, skilled to examine NLEEs, compatible to computer algebra and provides further general wave solutions. Thus, the investigation of exact solutions to other NLEES through the new auxiliary method is prospective and deserves further research.

Exact travelling wave solutions for a generalized nonlinear Schrödinger equation

The capability of Extended tanh-coth, sine-cosine and Exp-Function methods as alternative approaches to obtain the analytic solution of different types of applied differential equations in engineering mathematics has been revealed. In this study, the generalized nonlinear Schrödinger (GNLS) equation is solved by three different methods. To obtain the single-soliton solutions for the equation, the Extended tanh-coth and sine-cosine methods are used. Furthermore, for this nonlinear evolution equation the Exp-Function method is applied to derive various travelling wave solution. Results show that while the first two procedures easily provide a concise solution, the Exp-Function method provides a powerful mathematical means for solving nonlinear evolution equations in mathematical physics. j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a m c

Darboux transformation and multi-soliton solutions for some soliton equations

Chaos, Solitons & Fractals, 2009

In this paper, we propose a new approach (different from the approach presented in Proc. R. Soc. Lond. A 460 2617-2627) to calculate multi-soliton solutions of Camassa-Holm equation (CH) and modified Camassa-Holm (MCH) equation with aid of Darboux transformation (DT). We first map the CH and MCH equation to a negative order KdV (NKdV) equation by a reciprocal transformation. Then we proceed to apply the DT to solve the NKdV equation in the usual way. Finally we invert the reciprocal transformation to recover the solutions of the CH equation and MCH equation.

Two-parameter family of exact solutions of the nonlinear Schrödinger equation describing optical-soliton propagation

Physical Review A, 1993

By using a direct method for obtaining exact solutions of the nonlinear Schrodinger equation that describes the evolution of spatial or temporal optical solitons, a two-parameter family of solutions is given. These exact solutions describe the periodic wave patterns that are generated by the spatial or temporal modulational instability, the periodic evolution of the bright solitons superimposed onto a continuouswave background, and the breakup of a single pulse into two dark waves which move apart with equal and opposite transverse components of the velocities.

Finite and infinite soliton and kink-soliton trains of nonlinear Schrödinger equations

We will first review known results on multi-solitons of dispersive partial differential equations, which are special solutions behaving like the sum of many weakly-interacting solitary waves. We will then describe our recent joint work with Dong Li on nonlinear Schrödinger equations: Assuming the composing solitons have sufficiently large relative speeds, we prove the existence and uniqueness of a soliton train which is a multi-soliton composed of infinitely many solitons. In the 1D case, we can add to the infinite train an additional half-kink, which is a solution with a non-zero background at minus infinity.