Solving Non-Isospectral mKdV Equation and Sine-Gordon Equation Hierarchies with Self-Consistent Sources via Inverse Scattering Transform

N-soliton solutions of the hierarchy of non-isospectral mKdV equation with self-consistent sources andthe hierarchy of non-isospectral sine-Gordon equation with self-consistent sources are obtained via the inverse scatteringtransform.

~ Transform Demonstration of Dark Soliton Solutions Found by Inverse Scattering

One of the basic problems about the inverse scattering transform for solving a completely integrable nonlinear evolutions equation is to demonstrate that the Jost solutions obtained from the inverse scattering equations of Cauchy integral satisfy the Lax equations. Such a basic problem still exists in the procedure of deriving the dark soliton solutions of the NLS equation in normal dispersion with non-vanishing boundary conditions through the inverse scattering transform. In this paper, a pair of Jost solutions with same analytic properties are composed to be a 2 × 2 matrix and then another pair are introduced to be its right inverse confirmed by the Liouville theorem. As they are both 2 × 2 matrices, the right inverse should be the left inverse too, based upon which it is not difficult to show that these Jost solutions satisfy both the first and second Lax equations. As a result of compatibility condition, the dark soliton solutions definitely satisfy the NLS equation in normal dispersion with non-vanishing boundary conditions.

Demonstration of Inverse Scattering Transform for DNLS Equation

Since the Jost solutions of the DNLS equation does not tend to the free Jost solutioins as ｜λ｜→∞, the usual inverse scattering transform （IST） must be revised. Beside the Kaup and Newell＇s approach, we propose a simple revision in constructing the equations of IST, where the usual Zakharov-Shabat kern is revised by multiplying λ^-2 or λ^-1. To justify the revision we show that the Jost solutions obtained do satisfy the pair of compatibility equations.

A Newly Revised Inverse Scattering Transform for DNLS^(+) Equation under Nonvanishing Boundary Condition

A newly revised inverse scattering transform（IST） for the derivative nonlinear Schrdinger（DNLS＋） equation with non-vanishing boundary condition（NVBC） and normal group velocity dispersion is proposed by introducing a suitable affine parameter in Zakharov-Shabat integral kern.The explicit breather-type one-soliton solution,which can reproduce one pure soliton at the de-generate case and one bright soliton solution at the limit of van-ishing boundary,has been derived to verify the validity of the revised IST.

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.

Inverse Scattering Transform of the Coupled Sasa–Satsuma Equation by Riemann–Hilbert Approach

The inverse scattering transform of a coupled Sasa–Satsuma equation is studied via Riemann–Hilbert approach. Firstly, the spectral analysis is performed for the coupled Sasa–Satsuma equation, from which a Riemann–Hilbert problem is formulated. Then the Riemann–Hilbert problem corresponding to the reflection-less case is solved.As applications, multi-soliton solutions are obtained for the coupled Sasa–Satsuma equation. Moreover, some figures are given to describe the soliton behaviors, including breather types, single-hump solitons, double-hump solitons, and two-bell solitons.

Lump Solutions of Kadomtsev-Petviashvili I Equation in Non-uniform Media

N-lump solutions of the Kadomtsev-Petviashvili I equation in non-uniform media are derived through the inverse scattering transform. The obtained solutions describe lump waves with time-dependent amplitudes and velocities. Dynamics of l-lump wave and interactions of two lump wave are illustrated.

On the inverse scattering transform in two spatial and one temporal dimensions

Some results and developments on the extension of the inverse scattering transform to solve non-linear evolution equations in one time and two space dimensions are described.

Inverse scattering transforms of the inhomogeneous fifth-order nonlinear Schrodinger equation with zero/nonzero boundary conditions

The present work studies the inverse scattering transforms(IST)of the inhomogeneous fifth-order nonlinear Schrodinger(NLS)equation with zero boundary conditions(ZBCs)and nonzero boundary conditions(NZBCs).Firstly,the bound-state solitons of the inhomogeneous fifth-order NLS equation with ZBCs are derived by the residue theorem and the Laurent's series for the first time.Then,by combining with the robust IST,the Riemann-Hilbert(RH)problem of the inhomogeneous fifth-order NLS equation with NZBCs is revealed.Furthermore,based on the resulting RH problem,some new rogue wave solutions of the inhomogeneous fifth-order NLS equation are found by the Darboux transformation.Finally,some corresponding graphs are given by selecting appropriate parameters to further analyze the unreported dynamic characteristics of the corresponding solutions.

Explicit Breather-Type and Pure N-Soliton Solution of DNLS^+ Equation with Nonvanishing Boundary Condition

Based on a newly revised inverse scattering transform for the derivative nonlinear SchrSdinger （DNLS＋） equation with nonvanishing boundary condition （NVBC）, the explicit breather- type and pure N-soliton solution has been derived by some algebra techniques. The two-breather solution and the pure double-soliton solution have been given as two typical examples in illustration of the general formula of the multi-soliton solution. The asymptotic behaviors of the N-soliton solution are discussed in detail.

Mixed Breather-Type and Pure Soliton Solution of DNLS Equation

A newly revised inverse scattering transform（IST） is used to solve the derivative nonlinear Schr？dinger（DNLS-＋） equation with non-vanishing boundary condition（NVBC） and normal group-velocity dispersion, and a mixed type of soliton solution is found, which is composed of both pure and breather-type solitons. The mixed-soliton solution can degenerate into breathers and pure solitons when taking the limit of some special parameters, proving the validity of the mixed-type solution.

Hamiltonian Theory for the DNLS Equation with a Squared Spectral Parameter

With a special gauge transformation,the Lax pair of the derivative nonlinear Shcrdinger (DNLS) equation turns to depend on the squared parameter λ = k2instead of the usual spec-tral parameter k. By introducing a new direct product of Jost solu-tions,the complete Hamiltonian theory of the DNLS equation is constructed on the basis of the squared spectral parameter,which shows that the integrability completeness is still preserved. This result will be beneficial to the further study of the DNLS equation,such as the direct perturbation method.

A Necessary and Sufficient Condition for the Solvability of the Nonlinear Schr?dinger Equation on a Finite Interval

The admissibility of the initial-boundary data,which characterizes the existence of solution for the initial-boundary value problem,is important.Based on the Fokas method and the inverse scattering transformation,an approach is developed to solve the initial-boundary value problem of the nonlinear Schrodinger equation on a finite interval.A necessary and sufficient condition for the admissibility of the initial-boundary data is given,and the reconstruction of the potential is obtained.

Asymptotic behavior of solutions of defocusing integrable discrete nonlinear Schrodinger equation

We report our recent result about the long-time asymptotics for the defocusing integrable discrete nonlinear Schrodinger equation of Ablowitz- Ladik. The leading term is a sum of two terms that oscillate with decay of order t-1/2.