Multiple-kink-soliton solutions of the nonlinear Schrödinger equation (original) (raw)
Soliton Equations and Their Algebro-Geometric Solutions
2008
The discovery of solitary waves of translation goes back to Scott Russell in 1834, and during the remaining part of the 19th century the true nature of these waves remained controversial. It was only with the derivation by Korteweg and de Vries in 1895 of what is now called the Korteweg-de Vries (KdV) equation, that the one-soliton solution and hence the concept of solitary waves was put on a firm basis. 1 An extraordinary series of events took place around 1965 when Kruskal and Zabusky, while analyzing the numerical results of Fermi, Pasta, and Ulam on heat conductivity in solids, discovered that pulselike solitary wave solutions of the KdV equation, for which the name "solitons" was coined, interact elastically. This was followed by the 1967 discovery of Gardner, Greene, Kruskal, and Miura that the inverse scattering method allows one to solve initial value problems for the KdV equation with sufficiently fast-decaying initial data. Soon thereafter, in 1968, Lax found a new explanation of the isospectral nature of KdV solutions using the concept of Lax pairs and introduced a whole hierarchy of KdV equations. Subsequently, in the early 1970s, Zakharov and Shabat (ZS), and Ablowitz, Kaup, Newell, and Segur (AKNS) extended the inverse scattering method to a wide class of nonlinear partial differential equations of relevance in various scientific contexts ranging from nonlinear optics to condensed matter physics and elementary particle physics. In particular, solitons found numerous applications in classical and quantum field theory and in connection with optical communication devices. Another decisive step forward in the development of completely integrable soliton equations was taken around 1974. Prior to that period, inverse spectral 1 With hindsight, though, it is now clear that other researchers, such as Boussinesq, derived the KdV equation and its one-soliton solution prior to 1895, as described in the notes to Section 1.1. 1 To obtain a closed system of differential equations, one has to express F r (µ j ) solely in terms of µ 1 , . . . , µ n and E 0 , . . . , E 2n+1 ; see (1.222) and (1.223).
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.
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
Three Different Methods for New Soliton Solutions of the Generalized NLS Equation
Three different methods are applied to construct new types of solutions of nonlinear evolution equations. First, the Csch method is used to carry out the solutions; then the Extended Tanh-Coth method and the modified simple equation method are used to obtain the soliton solutions. The effectiveness of these methods is demonstrated by applications to the RKL model, the generalized derivative NLS equation. The solitary wave solutions and trigonometric function solutions are obtained. The obtained solutions are very useful in the nonlinear pulse propagation through optical fibers.
Soliton solutions to coupled nonlinear wave equations in (2 + 1)-dimensions
This paper implemented the tanh method to solve a few coupled nonlinear wave equations in (2 ? 1)dimensions. They are the Konopelchenko-Dubrovsky equation, dispersive long wave equation and the Riemann wave equation. Additionally, the traveling wave hypothesis is used to extract a few more solutons to some of these equations. Finally, the numerical simulations supplement these analytical results.
2017
In this paper, by means of similarity transformations we study exact analytical solutions for a generalized nonlinear Schr$\ddot{\mbox{o}}$dinger equation with variable coefficients. This equation appears in literature describing the evolution of coherent light in a nonlinear Kerr medium, Bose-Einstein condensates phenomena and high intensity pulse propagation in optical fibers. By restricting the coefficients to satisfy Ermakov-Riccati systems with multiparameter solutions, we present conditions for existence of explicit solutions with singularities and a family of oscillating periodic soliton-type solutions. Also, we show the existence of bright-, dark- and Peregrine-type soliton solutions, and by means of a computer algebra system we exemplify the nontrivial dynamics of the solitary wave center of these solutions produced by our multiparameter approach.
In this paper, the higher-order generalized nonlinear Schrödinger equation, which describes the propagation of ultrashort optical pulse in optical fibers, is analytically investigated. By virtue of the Darboux transformation constructed in this paper, some exact soliton solutions on the continuous wave (cw) background are generated. The following propagation characteristics of those solitons are mainly discussed: (1) Propagation of two types of breathers which delineate modulation instability and bright pulse propagation on a cw background respectively; (2) Two types propagation characteristics of twosolitons: elastic interactions and mutual attractions and repulsions bound solitons. Those results might be useful in the study of ultrashort optical solitons in optical fibers.