General N‐Dark–Dark Solitons in the Coupled Nonlinear Schrödinger Equations

N‐dark–dark solitons in the integrable coupled NLS equations are derived by the KP‐hierarchy reduction method. These solitons exist when nonlinearities are all defocusing, or both focusing and defocusing nonlinearities are mixed. When these solitons collide with each other, energies in both components of the solitons completely transmit through. This behavior contrasts collisions of bright–bright solitons in similar systems, where polarization rotation and soliton reflection can take place. It is also shown that in the mixed‐nonlinearity case, two dark–dark solitons can form a stationary bound state.     

Discretization of Coupled Nonlinear Schrödinger Equations

The soliton solutions for discrete coupled nonlinear Schrödinger equations are investigated by using bilinear formalism. Pfaffian expressions of the N -soliton solutions of dark–dark and bright–bright types are explicitly given for the defocusing–defocusing and focusing–focusing cases, respectively.     

Dynamics of Strongly Nonlinear Kinks and Solitons in a Two‐Layer Fluid

Perturbation theory is developed for interaction of strongly nonlinear solitary waves close to the limiting, tabletop solitons (Π‐solitons). The method is based on representing each soliton as a compound of two kinks so that the interaction of N solitons is treated as the interaction of 2N kinks. As an example the Miyata–Choi–Camassa equations for a two‐layer fluid is considered. Equations for kink coordinates are obtained and analyzed. Some nontrivial features of two‐soliton interaction characteristic of the strongly nonlinear case are established.     

A simple but efficient approach for studying on nonlinear differential-difference equations

In this paper, we present a new approach for constructing exact solutions to nonlinear differential-difference equations (NLDDEs). By applying the new method, we have studied the saturable discrete nonlinear Schrodinger equation (SDNLSE) and obtained a number of new exact localized solutions, including discrete bright soliton solution, dark soliton solution, bright and dark soliton solution, alternating phase bright soliton solution, alternating phase dark soliton solution and alternating phase bright and dark soliton solution, provided that a special relation is bound on the coefficients of the equation among the solutions obtained.     

Interaction among a lump,periodic waves,and kink solutions to the fractional generalized CBS‐BK equation

The Hirota bilinear method is prepared for searching the diverse soliton solutions for the fractional generalized Calogero‐Bogoyavlenskii‐Schiff‐Bogoyavlensky‐Konopelchenko (CBS‐BK) equation. Also, the Hirota bilinear method is used to finding the lump and interaction with two stripe soliton solutions. Interaction among lumps, periodic waves, and multi‐kink soliton solutions will be investigated. Also, the solitary wave, periodic wave, and cross‐kink wave solutions will be examined for the fractional gCBS‐BK equation. The graphs for various fractional order α are plotted to contain 3D plot, contour plot, density plot, and 2D plot. We construct the exact lump and interaction among other types solutions, by solving the under‐determined nonlinear system of algebraic equations for the associated parameters. Finally, analysis and graphical simulation are presented to show the dynamical characteristics of our solutions and the interaction behaviors are revealed. The existence conditions are employed to discuss the available got solutions.     

Optical soliton solutions for the generalized Kudryashov equation of propagation pulse in optical fiber with power nonlinearities by three integration algorithms

In this paper, we employ three integration algorithms, namely, the well‐known Kudryashov method, the new Kudryashov method, and the unified Riccati equation expansion method to extract optical soliton solutions for the generalized Kudryashov equation with power nonlinearities. Straddled soliton, bright solitons, dark solitons, and singular solitons have been found.     

Soliton Solutions for a Generalized Inhomogeneous Variable-Coefficient Hirota Equation with Symbolic Computation

Under investigation in this paper is a generalized inhomogeneous variable- coefficient Hirota equation. Through the Hirota bilinear method and symbolic computation, the bilinear form and analytic one-, two- and N-soliton solutions for such an equation are obtained, respectively. Properties of those solitons in the inhomogeneous media are discussed analytically. We get the soliton with the property that the larger the amplitude is, the narrower and slower the pulse is. Dynamics of that soliton can be regarded as a repulsion of the soliton by the external potential barrier. During the interaction of two solitons, we observe that the larger the value of the coefficient β in the equation is, the larger the distance of the two solitons is.     

Lax pair, Bäcklund transformation and multi-soliton solutions for the Boussinesq-Burgers equations from shallow water waves

Under investigation in this paper is the set of the Boussinesq-Burgers (BB) equations, which can be used to describe the propagation of shallow water waves. Based on the binary Bell polynomials, Hirota method and symbolic computation, the bilinear form and soliton solutions for the BB equations are derived. Bäcklund transformations (BTs) in both the binary-Bell-polynomial and bilinear forms are obtained. Through the BT in the binary-Bell-polynomial form, a type of solutions and Lax pair for the BB equations are presented as well. Propagation characteristics and interaction behaviors of the solitons are discussed through the graphical analysis. Shock wave and bell-shape solitons are respectively obtained for the horizontal velocity field u and height v of the water surface. In both the head-on and overtaking collisions, the shock waves for the u profile change their shapes, which denotes that the collisions for the u profile are inelastic. However, the collisions for the v profile are proved to be elastic through the asymptotic analysis. Our results might have some potential applications for the harbor and coastal design.     

The Derivative Yajima–Oikawa System: Bright,Dark Soliton and Breather Solutions

In this paper, we study the derivative Yajima–Oikawa (YO) system which describes the interaction between long and short waves (SWs). It is shown that the derivative YO system is classified into three types which are similar to the ones of the derivative nonlinear Schrödinger equation. The general N ‐bright and N ‐dark soliton solutions in terms of Gram determinants are derived by the combination of the Hirota's bilinear method and the Kadomtsev–Petviashvili hierarchy reduction method. Particularly, it is found that for the dark soliton solution of the SW component, the magnitude of soliton can be larger than the nonzero background for some parameters, which is usually called anti‐dark soliton. The asymptotic analysis of two‐soliton solutions shows that for both kinds of soliton only elastic collision exists and each soliton results in phase shifts in the long and SWs. In addition, we derive two types of breather solutions from the different reduction, which contain the homoclinic orbit and Kuznetsov–Ma breather solutions as special cases. Moreover, we propose a new (2+1)‐dimensional derivative Yajima–Oikawa system and present its soliton and breather solutions.     

Bell polynomial approach to an extended Korteweg–de Vries equation

Under investigation in this paper is an extended Korteweg–de Vries equation. Via Bell polynomial approach and symbolic computation, this equation is transformed into two kinds of bilinear equations by choosing different coefficients, namely KdV–SK‐type equation and KdV–Lax‐type equation. On the one hand, N‐soliton solutions, bilinear Bäcklund transformation, Lax pair, Darboux covariant Lax pair, and infinite conservation laws of the KdV–Lax‐type equation are constructed. On the other hand, on the basis of Hirota bilinear method and Riemann theta function, quasiperiodic wave solution of the KdV–SK‐type equation is also presented, and the exact relation between the one periodic wave solution and the one soliton solution is established. It is rigorously shown that the one periodic wave solution tend to the one soliton solution under a small amplitude limit. Copyright © 2013 John Wiley & Sons, Ltd.     

Integrable coupled dispersionless equations

A new type of coupled dispersionless equations, which has a real dark type of soliton and a complex bright type of envelope soliton, is proposed and solved by the inverse scattering method. The N-soliton solution is given and one and two solitons are explicitly shown.     

Discrete Analogues of a Negative Order AKNS Equation

The paper investigates the integrable discretization of a negative order AKNS equation. Two semidiscrete and one fully discrete versions of the system are given via Hirota's bilinear method. Soliton solutions for the derived discrete systems are also presented. Dynamics of one‐soliton and two‐soliton solutions of spacial‐discrete system are characterized.     

Dark solitonic excitations and collisions from a fourth-order dispersive nonlinear Schrödinger model for the alpha helical protein

Recent protein observations motivate the dark-soliton study to explain the energy transfer in the proteins. In this paper we will investigate a fourth-order dispersive nonlinear Schrödinger equation, which governs the Davydov solitons in the alpha helical protein with higher-order effects. Painlevé analysis is performed to prove the equation is integrable. Through the introduction of an auxiliary function, bilinear forms and dark N-soliton solutions are constructed with the Hirota method and symbolic computation. Asymptotic analysis on the two-soliton solutions indicates that the soliton collisions are elastic. Decrease of the coefficient of higher-order effects can increase the soliton velocities. Graphical analysis on the two-soliton solutions indicates that the head-on collision between the two solitons, overtaking collision between the two solitons and collision between a moving soliton and a stationary one are all elastic. Collisions among the three solitons are all pairwise elastic.     

Dark–dark and dark–bright soliton interactions in the two-component defocusing nonlinear Schrödinger equation

We use the Inverse Scattering Transform machinery to construct multisoliton solutions to the 2-component defocusing nonlinear Schrödinger equation. Such solutions include dark–dark solitons, which have dark solitonic behaviour in both components, as well as dark–bright soliton solutions, with one dark and one bright component. We then derive the explicit expressions of two soliton solutions for all possible cases: two dark–dark solitons, two dark–bright solitons, and one dark–dark and one dark–bright soliton. Finally, we determine the long-time asymptotic behaviours of these solutions, which allows us to obtain explicit expressions for the shifts in the phases and in the soliton centers due to the interactions.     

Darboux transformation and soliton solutions for the generalized coupled variable-coefficient nonlinear Schrödinger-Maxwell-Bloch system with symbolic computation

In an inhomogeneous nonlinear light guide doped with two-level resonant atoms, the generalized coupled variable-coefficient nonlinear Schrödinger-Maxwell-Bloch system can be used to describe the propagation of optical solitons. In this paper, the Lax pair and conservation laws of that model are derived via symbolic computation. Furthermore, based on the Lax pair obtained, the Darboux transformation is constructed and soliton solutions are presented. Figures are plotted to reveal the following dynamic features of the solitons: (1) Periodic mutual attractions and repulsions of four types of bound solitons: of two one-peak bright solitons; of two one-peak dark solitons; of two two-peak bright solitons and of two two-peak dark solitons; (2) Two types of elastic interactions of solitons: of two bright solitons and of two dark solitons; (3) Two types of parallel propagations of parabolic solitons: of two bright solitons and of two dark solitons. Those results might be useful in the study of optical solitons in some inhomogeneous nonlinear light guides.     

Darboux transformation and solitons of Yang-Mills-Higgs equations in R2,1

The Darboux transformations for soliton equations are applied to the Yang-Mills-Higgs equations.New solutions can be obtained from a known one via universal and purely algebraic formulas. SU(N) soliton solutions are constructed with explicit formulas. The interaction of solitons is described by the splitting theorem:each p-soliton is splitting into p single solitons asymptotically as t →±∞.     

Exponential Asymptotics for Line Solitons in Two‐Dimensional Periodic Potentials

As a first step toward a fully two‐dimensional asymptotic theory for the bifurcation of solitons from infinitesimal continuous waves, an analytical theory is presented for line solitons, whose envelope varies only along one direction, in general two‐dimensional periodic potentials. For this two‐dimensional problem, it is no longer viable to rely on a certain recurrence relation for going beyond all orders of the usual multiscale perturbation expansion, a key step of the exponential asymptotics procedure previously used for solitons in one‐dimensional problems. Instead, we propose a more direct treatment which not only overcomes the recurrence‐relation limitation, but also simplifies the exponential asymptotics process. Using this modified technique, we show that line solitons with any rational line slopes bifurcate out from every Bloch‐band edge; and for each rational slope, two line‐soliton families exist. Furthermore, line solitons can bifurcate from interior points of Bloch bands as well, but such line solitons exist only for a couple of special line angles due to resonance with the Bloch bands. In addition, we show that a countable set of multiline‐soliton bound states can be constructed analytically. The analytical predictions are compared with numerical results for both symmetric and asymmetric potentials, and good agreement is obtained.     

N-soliton solutions,Bäcklund transformation and conservation laws for the integro-differential nonlinear Schröbinger equation from the isotropic inhomogeneous Heisenberg spin magnetic chain

Under investigation in this paper is an integro-differential nonlinear Schröbinger (IDNLS) equation, which is equivalent to the spin evolution equation of a classical in-homogeneous Heisenberg magnetic chain in the continuum limit. Based on the Hirota method, the bilinear form and N-soliton solution for the IDNLS equation are derived with the help of symbolic computation. Moreover, N-soliton solution for the IDNLS equation is expressed in terms of the double Wronskian and testified through the direct substitution into the bilinear form. Besides, the bilinear Bäcklund transformation and infinitely many conservation laws are also obtained for the IDNLS equation. Propagation characteristics and interaction behaviors of the solitons are discussed by analysis of such physical quantities as the soliton amplitude, width, velocity and initial phase. Interactions of the solitons are proved to be elastic through the asymptotic analysis. Effect of inhomogeneity on the interaction of the solitons is studied graphically.     

Chiral resonant solitons in Chern–Simons theory and Broer–Kaup type new hydrodynamic systems

New Broer–Kaup type systems of hydrodynamic equations are derived from the derivative reaction–diffusion systems arising in SL(2, R) Kaup–Newell hierarchy, represented in the non-Madelung hydrodynamic form. A relation with the problem of chiral solitons in quantum potential as a dimensional reduction of 2 + 1 dimensional Chern–Simons theory for anyons is shown. By the Hirota bilinear method, soliton solutions are constructed and the resonant character of soliton interaction is found.