An integrable generalization of the Fokas–Lenells equation:Darboux transformation, reduction and explicit soliton solutions

Under investigation is an integrable generalization of the Fokas–Lenells equation, which can be derived from the negative power flow of a 2 × 2 matrix spectral problem with three potentials. Based on the gauge transformation of the matrix spectral problem, one kind of Darboux transformation with multi-parameters for the three-component coupled Fokas–Lenells system is constructed. As a reduction, the N-fold Darboux transformation for the generalized Fokas–Lenells equation is obtained, from which the N-soliton solution in a compact Vandermonde-like determinant form is given. Particularly,the explicit one-and two-soliton solutions are presented and their dynamical behaviors are shown graphically.

Diverse soliton solutions and dynamical analysis of the discrete coupled mKdV equation with 4×4 Lax pair

Under consideration in this study is the discrete coupled modified Korteweg-de Vries(mKdV)equation with 4×4 Lax pair.Firstly,through using continuous limit technique,this discrete equation can be mapped to the coupled KdV and mKdV equations,which may depict the development of shallow water waves,the optical soliton propagation in cubic nonlinear media and the Alfven wave in a cold collision-free plasma.Secondly,the discrete generalized(r,N-r)-fold Darboux transformation is constructed and extended to solve this discrete coupled equation with the fourth-order linear spectral problem,from which diverse exact solutions including usual multi-soliton and semi-rational soliton solutions on the vanishing background,higher-order rational soliton and mixed hyperbolic-rational soliton solutions on the non-vanishing background are derived,and the limit states of some soliton and rational soliton solutions are analyzed by the asymptotic analysis technique.Finally,the numerical simulations are used to explore the dynamical behaviors of some exact soliton solutions.These results may be helpful for understanding some physical phenomena in fields of shallow water wave,optics,and plasma physics.

THE SASA-SATSUMA EQUATION ON A NON-ZERO BACKGROUND: THE INVERSE SCATTERING TRANSFORM AND MULTI-SOLITON SOLUTIONS

We concentrate on the inverse scattering transformation for the Sasa-Satsuma equation with 3×3 matrix spectrum problem and a nonzero boundary condition. To circumvent the multi-value of eigenvalues, we introduce a suitable two-sheet Riemann surface to map the original spectral parameter k into a single-valued parameter z. The analyticity of the Jost eigenfunctions and scattering coefficients of the Lax pair for the Sasa-Satsuma equation are analyzed in detail. According to the analyticity of the eigenfunctions and the scattering coefficients, the z-complex plane is divided into four analytic regions of D_(j) : j = 1, 2, 3, 4. Since the second column of Jost eigenfunctions is analytic in D_(j), but in the upper-half or lowerhalf plane, we introduce certain auxiliary eigenfunctions which are necessary for deriving the analytic eigenfunctions in Dj. We find that the eigenfunctions, the scattering coefficients and the auxiliary eigenfunctions all possess three kinds of symmetries;these characterize the distribution of the discrete spectrum. The asymptotic behaviors of eigenfunctions, auxiliary eigenfunctions and scattering coefficients are also systematically derived. Then a matrix Riemann-Hilbert problem with four kinds of jump conditions associated with the problem of nonzero asymptotic boundary conditions is established, from this N-soliton solutions are obtained via the corresponding reconstruction formulae. The reflectionless soliton solutions are explicitly given. As an application of the N-soliton formula, we present three kinds of single-soliton solutions according to the distribution of discrete spectrum.

Riemann-Hilbert approach of the complex Sharma-Tasso-Olver equation and its N-soliton solutions

We study the complex Sharma-Tasso-Olver equation using the Riemann-Hilbert approach.The associated Riemann-Hilbert problem for this integrable equation can be naturally constructed by considering the spectral problem of the Lax pair.Subsequently,in the case that the Riemann-Hilbert problem is irregular,the N-soliton solutions of the equation can be deduced.In addition,the three-dimensional graphic of the soliton solutions and wave propagation image are graphically depicted and further discussed.

Soliton solution and interaction property for a coupled modified Korteweg-de Vries (mKdV) system

Hirota＇s bilinear direct method is applied to constructing soliton solutions to a special coupled modified Korteweg- de Vries （mKdV） system. Some physical properties such as the spatiotemporal evolution, waveform structure, interactive phenomena of solitons are discussed, especially in the two-soliton case. It is found that different interactive behaviours of solitary waves take place under different parameter conditions of overtaking collision in this system. It is verified that the elastic interaction phenomena exist in this （1＋1）-dimensional integrable coupled model.

Darboux Transformations and Soliton Solutions for Classical Boussinesq-Burgers Equation

Two basic Darboux transformations of a spectral problem associated with a classical Boussinesq-Burgersequation are presented in this letter.They are used to generate new solutions of the classical Boussinesq-Burgersequation.

Conservation Laws and Soliton Solutions for Generalized Seventh Order KdV Equation

With the assistance of the symbolic computation system Maple,rich higher order polynomial-type conservation laws and a sixth order t/x-dependent conservation law are constructed for a generalized seventh order nonlinear evolution equation by using a direct algebraic method.From the compatibility conditions that guaranteeing the existence of conserved densities,an integrable unnamed seventh order KdV-type equation is found.By introducing some nonlinear transformations,the one-,two-,and three-solition solutions as well as the solitary wave solutions are obtained.

Soliton Solutions of Coupled KdV System from Hirota's Bilinear Direct Method

With Hirota＇s bilinear direct method, we study the special coupled KdV system to obtain its new soliton solutions. Then we further discuss soliton evolution, corresponding structures, and interesting interactive phenomena in detail with plot. As a result, we find that after the interaction, the solitons make elastic collision and there are no exchanges of their physical quantities including energy, velocity and shape except the phase shift.

Coupled Modified Korteweg-de Vries Lattice in （2＋1） Dimensions and Soliton Solutions

The coupled semi-discrete modified Korteweg-de Vries equation in （2＋1）-dimensions is proposed, it is shown that it, can be decomposed into two （1＋1）-dimensional differential-difference equations belonging to mKdV lattice hierarchy by considering a discrete isospeetral problem. A Darboux transformation is set up for the resulting （2＋1）- dimensional lattice soliton equation with the help of gauge transformations of Lax pairs. As an illustration by example, the soliton solutions of the mKdV lartice equation in （2＋1）-dimensions are explicitly given,

Exact Discrete Soliton Solutions and Periodic Solutions to (2+1)-Dimensional Toda Lattice with a New Algebraic Method

In this paper,with the aid of symbolic computation,we present a uniform method for constructing soliton solutions and periodic solutions to (2+1)-dimensional Toda lattice equation.

Soliton Solutions to the Coupled Gerdjikov-Ivanov Equation with Rogue-Wave-Like Phenomena

Bilinear forms of the coupled Gerdjikov–Ivanov equation are derived. The $N$-soliton solutions to the equation are obtained by Hirota＇s method. It is interesting that the two-soliton solutions can generate the rogue-wave-like phenomena by selecting special parameters. The equation can be reduced to the Gerdjikov–Ivanov equation as well as its bilinear forms and its solutions.

Multisoliton Solutions of the （3＋1）—Dimensional Nizhnik—Novikov—Veselov Equation

Using the standard truncated Painlev? analysis, we can obtain a B?cklund transformation of the (3+1)-dimensional Nizhnik?Novikov?Veselov (NNV) equation and get some (3+1)-dimensional single-, two- and three-soliton solutions and some new types of multisoliton solutions of the (3+1)-dimensional NNV system from the B?cklund transformation and the trivial vacuum solution.

EXISTENCE AND NONEXISTENCE FOR THE INITIAL BOUNDARY VALUE PROBLEM OF ONE CLASS OF SYSTEM OF MULTIDIMENSIONAL NONLINEAR SCHRDINGER EQUATIONS WITH OPERATOR AND THEIR SOLITON SOLUTIONS

The nonlinear interactions between the monochromatic wave have been considered by K. Matsunchi, who also proposed one class of the nonlinear Schrdinger equation system with wave operator. We also obtain the same type of equations, which are satisfied by transverse velocity of higher frequency electron, as we study soliton in plasma physics. In this paper, under some condition we study the existence and nonexistence for this equations in the cases possessing different signs in nonlinear term.

RIEMANN-HILBERT PROBLEMS AND SOLITON SOLUTIONS OF NONLOCAL REVERSE-TIME NLS HIERARCHIES

The paper aims at establishing Riemann-Hilbert problems and presenting soliton solutions for nonlocal reverse-time nonlinear Schrodinger(NLS) hierarchies associated with higher-order matrix spectral problems.The Sokhotski-Plemelj formula is used to transform the Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations.A new formulation of solutions to special Riemann-Hilbert problems with the identity jump matrix,corresponding to the reflectionless inverse scattering transforms,is proposed and applied to construction of soliton solutions to each system in the considered nonlocal reversetime NLS hierarchies.

N-soliton solutions of an integrable equation studied by Qiao

In this paper, we studied N-soliton solutions of a new integrable equation studied by Qiao [J. Math. Phys. 48 082701 （2007）]. Firstly, we employed the Darboux matrix method to construct a Darboux transformation for the modified Korteweg-de Vries equation. Then we use the Darboux transformation and a transformation, introduced by Sakovich [J. Math. Phys. 52 023509 （2011）], to derive N-soliton solutions of the new integrable equation from the seed solution. In particular, the multiple soliton solutions are explicitly obtained and shown through some figures.

Soliton Solutions for Nonisospectral AKNS Equation by Hirota＇s Method

Bilinear form of the nonisospectral AKNS equation is given. The N-soliton solutions are obtained through Hirota＇s method.

Relationship Between Soliton-like Solutions and Soliton Solutions to a Class of Nonlinear Partial Differential Equations

By using the generally projective Riccati equation method, a series of doubly periodic solutions (Jacobi elliptic function solution) for a class of nonlinear partial differential equations are obtained in a unified way. When the module m → 1, these solutions exactly degenerate to the soliton solutions of the equations. Then we reveal the relationship between the soliton-like solutions obtained by other authors and these soliton solutions of the equations.

Soliton Solutions for a Negative Order AKNS Equation

In this short paper,bilinear form of a negative order AKNS equation is given.The N-soliton solutions areobtained through Hiorta's direct method.

Soliton Solutions for a Negative Order AKNS Equation Hierarchy

In this paper,bilinear form of a negative order AKNS equation hierarchy is given.The soliton solutionsare obtained through Hiorta's direct method.

Algebraic Method for Constructing Exact Discrete Soliton Solutions of Toda and mKdV Lattices

In this paper, with the aid of symbolic computation, we present a new method for constructing soliton solutions to nonlinear differentiM-difference equations. And we successfully solve Toda and mKdV lattice.