On the Riemann–Hilbert problem of a generalized derivative nonlinear Schrödinger equation

In this work,we present a unified transformation method directly by using the inverse scattering method for a generalized derivative nonlinear Schr?dinger(DNLS)equation.By establishing a matrix Riemann-Hilbert problem and reconstructing potential function q(x,t)from eigenfunctions{Gj(x,t,η)}3/1 in the inverse problem,the initial-boundary value problems for the generalized DNLS equation on the half-line are discussed.Moreover,we also obtain that the spectral functions f(η),s(η),F(η),S(η)are not independent of each other,but meet an important global relation.As applications,the generalized DNLS equation can be reduced to the Kaup-Newell equation and Chen-Lee-Liu equation on the half-line.     

Reduced nonlocal integrable mKdV equations of type (−λ, λ) and their exact soliton solutions

We conduct two group reductions of the Ablowitz–Kaup–Newell–Segur matrix spectral problems to present a class of novel reduced nonlocal reverse-spacetime integrable modified Korteweg–de Vries equations. One reduction is local, replacing the spectral parameter with its negative and the other is nonlocal, replacing the spectral parameter with itself. Then by taking advantage of distribution of eigenvalues, we generate soliton solutions from the reflectionless Riemann–Hilbert problems, where eigenvalues could equal adjoint eigenvalues.     

The unified transform method to the high-order nonlinear Schrödinger equation with periodic initial condition

In this paper, we study the high-order nonlinear Schrödinger equation with periodic initial conditions via the unified transform method extended by Fokas and Lenells. For the high-order nonlinear Schrödinger equation, the initial value problem on the circle can be expressed in terms of the solution of a Riemann–Hilbert problem. The related jump matrix can be explicitly expressed based on the initial data alone. Furthermore, we present the explicit solution, which corresponds to a one-gap solution.     

Multi-soliton solutions for the three types of nonlocal Hirota equations via Riemann–Hilbert approach

The purpose of the paper is to formulate multi-soliton solutions for the nonlocal Hirota equations via the Riemann–Hilbert (RH) approach. The RH problems are constructed and the zero structures are studied via performing spectral analysis of the Lax pair. Then we consider three types of nonlocal Hirota equations by discussing different symmetry reductions of the potential matrix. On the basis of the resulting matrix RH problem under the restriction of the reflectionless case, we successfully obtain the multi-soliton solutions of the nonlocal Hirota equations.     

Integrability,multi-soliton and rational solutions,and dynamical analysis for a relativistic Toda lattice system with one perturbation parameter

Under investigation in this paper is a relativistic Toda lattice system with one perturbation parameter α abbreviated as RTL_(α) system by Suris, which may describe the motions of particles in lattices interacting through an exponential interaction force. First of all, an integrable lattice hierarchy associated with an RTL_(α) system is constructed, from which some relevant integrable properties such as Hamiltonian structures, Liouville integrability and conservation laws are investigated. Secondly, the discrete generalized(m, 2 N-m)-fold Darboux transformation is constructed to derive multi-soliton solutions, higher-order rational and semirational solutions, and their mixed solutions of an RTL_(α) system. The soliton elastic interactions and details of rational solutions are analyzed via the graphics and asymptotic analysis. Finally, soliton dynamical evolutions are investigated via numerical simulations,showing that a small noise has very little effect on the soliton propagation. These results may provide new insight into nonlinear lattice dynamics described by RTL_(α) system.     

Quantitative analysis of soliton interactions based on the exact solutions of the nonlinear Schrödinger equation

We make a quantitative study on the soliton interactions in the nonlinear Schrödinger equation (NLSE) and its variable-coefficient (vc) counterpart. For the regular two-soliton and double-pole solutions of the NLSE, we employ the asymptotic analysis method to obtain the expressions of asymptotic solitons, and analyze the interaction properties based on the soliton physical quantities (especially the soliton accelerations and interaction forces); whereas for the bounded two-soliton solution, we numerically calculate the soliton center positions and accelerations, and discuss the soliton interaction scenarios in three typical bounded cases. Via some variable transformations, we also obtain the inhomogeneous regular two-soliton and double-pole solutions for the vcNLSE with an integrable condition. Based on the expressions of asymptotic solitons, we quantitatively study the two-soliton interactions with some inhomogeneous dispersion profiles, particularly discuss the influence of the variable dispersion function f(t) on the soliton interaction dynamics.     

Exactly solving some typical Riemann–Liouville fractional models by a general method of separation of variables

Finding exact solutions for Riemann–Liouville(RL) fractional equations is very difficult. We propose a general method of separation of variables to study the problem. We obtain several general results and, as applications, we give nontrivial exact solutions for some typical RL fractional equations such as the fractional Kadomtsev–Petviashvili equation and the fractional Langmuir chain equation. In particular, we obtain non-power functions solutions for a kind of RL time-fractional reaction–diffusion equation. In addition, we find that the separation of variables method is more suited to deal with high-dimensional nonlinear RL fractional equations because we have more freedom to choose undetermined functions.     

Vector NLS solitons interacting with a boundary

We construct multi-soliton solutions of the n-component vector nonlinear Schrödinger equation on the half-line subject to two classes of integrable boundary conditions (BCs): the homogeneous Robin BCs and the mixed Neumann/Dirichlet BCs. The construction is based on the so-called dressing the boundary, which generates soliton solutions by preserving the integrable BCs at each step of the Darboux-dressing process. Under the Robin BCs, examples, including boundary-bound solitons, are explicitly derived; under the mixed Neumann/Dirichlet BCs, the boundary can act as a polarizer that tunes different components of the vector solitons. Connection of our construction to the inverse scattering transform is also provided.     

Non-universal critical behaviour of heteronuclear dimers on a square lattice––A Monte Carlo study

Monte Carlo simulation within the grand canonical ensemble, the histogram reweighting technique, and finite size scaling analysis are used to explore the phase behaviour of heteronuclear dimers, composed of A and B type atoms, on a square lattice. We have found that for the models with attractive BB and AB nearest-neighbour energy, u_BB=u_AB=−1, and for non-repulsive energy between AA nearest-neighbour sites, u_AA<0, the system belongs to the universality class of the two-dimensional Ising model. However, when u_AA>0, the system exhibits a non-universal critical behaviour. We have evaluated the dependences of the critical point characteristics on the value of u_AA.     

A discrete KdV equation hierarchy: continuous limit,diverse exact solutions and their asymptotic state analysis

In this paper, a discrete KdV equation that is related to the famous continuous KdV equation is studied. First, an integrable discrete KdV hierarchy is constructed, from which several new discrete KdV equations are obtained. Second, we correspond the first several discrete equations of this hierarchy to the continuous KdV equation through the continuous limit. Third, the generalized (m, 2N − m)-fold Darboux transformation of the discrete KdV equation is established based on its known Lax pair. Finally, the diverse exact solutions including soliton solutions, rational solutions and mixed solutions on non-zero seed background are obtained by applying the resulting Darboux transformation, and their asymptotic states and physical properties such as amplitude, velocity, phase and energy are analyzed. At the same time, some soliton solutions are numerically simulated to show their dynamic behaviors. The properties and results obtained in this paper may be helpful to understand some physical phenomena described by KdV equations.     

PROLONGATION STRUCTURE APPROACH TO THE GROUP-THEORETICAL TECHNIQUE FOR GENERATING STATIONARY AXISYMMETRIC SPACE-TIMES

The group-theoreti cal technique for generating stationary axisymmetric gravitational fields is approached by means of the prolongation structure theory for soliton systems. An sp(2)xc(t) structure is obtained via solving the fundamental equation for prolongation structures and the F-equation for Kinnersley-Chitre's generating function is naturally introduced as an inverse scattering equation. A homogeneous Hilbert problem(HRP) associated with the Geroch group K and a corresponding linear singular integral equation are derived based upon a general condition satisfied by the auto-Bäcklund transformations in the sense of prolongation structure theory.     

On soliton interaction in cubic nonlinear media

Using the method of the inverse problem, we consider soliton interaction in a nonlinear cubic medium. We present an example of initial conditions under which the nonlinear Schrödinger equation, which describes soliton interaction, has an exact solution.     

On quantum solitons and their classical relatives: Bethe Ansatz states in soliton sectors of the sine-Gordon system

Previously we have found that the semiclassical sine-Gordon/Thirring spectrum can be received in the absence of quantum solitons via the spin $\text{1}\text{2}$ approximation of the quantized sine-Gordon system on a lattice. Later on, we have recovered the Hilbert space of quantum soliton states for the sine-Gordon system. In the present paper we present a derivation of the Bethe Ansatz eigenstates for the generalized ice model in this soliton Hilbert space. We demonstrate that via “Wick rotation” of a fundamental parameter of the ice model one arrives at the Bethe Ansatz eigenstates of the quantum sine-Gordon system. The latter is a “local transition matrix” ancestor of the conventional sine-Gordon /Thirring model, as derived by Faddeev et al. within the quantum inverse-scattering method. Our result is essentially based on the N < ∞, Δ = 1, m ? 1 regime. Consequently, the spectrum received, though resembling the semiclassical one, does not coincide with it at all.     

Two-Soliton Bound States in a Heisenherg Spin Chain

An inhomogeneous Heisenberg spin Hamiltonian with single ion anisotropy is used to investigate the nonlinear excitations in ferromagnetic chain. By means of the Holstein-Primakoff transformation and Glauber's coherent-state representation, the equation of motion for anni-hilation operator a(j) is reduced to a nonlinear Schrödinger-like equation in the semiclassical approximation and the long wave approximation. For a homogeneous system, the exact and explicitly single soliton (localized magnon state) and two-magnon bound state solutions are Given by the inverse scattering transform.     

Riemann-Hilbert approach and N double-pole solutions for a nonlinear Schrödinger-type equation

We investigate the inverse scattering transform for the Schrödinger-type equation under zero boundary conditions with the Riemann-Hilbert (RH) approach. In the direct scattering process, the properties are given, such as Jost solutions, asymptotic behaviors, analyticity, the symmetries of the Jost solutions and the corresponding spectral matrix. In the inverse scattering process, the matrix RH problem is constructed for this integrable equation base on analyzing the spectral problem. Then, the reconstruction formula of potential and trace formula are also derived correspondingly. Thus, N double-pole solutions of the nonlinear Schrödinger-type equation are obtained by solving the RH problems corresponding to the reflectionless cases. Furthermore, we present a single double-pole solution by taking some parameters, and it is analyzed in detail.     

Collapse arrest in the space-fractional Schrödinger equation with an optical lattice

The soliton solution and collapse arrest are investigated in the one-dimensional space-fractional Schr?dinger equation with Kerr nonlinearity and optical lattice. The approximate analytical soliton solutions are obtained based on the variational approach, which provides reasonable accuracy. Linear-stability analysis shows that all the solitons are linearly stable. No collapses are found when the Lévy index 1 α≤ 2. For α = 1, the collapse is arrested by the lattice potential when the amplitude of perturbations is small enough. It is numerically proved that the energy criterion of collapse suppression in the two-dimensional traditional Schr?dinger equation still holds in the one-dimensional fractional Schr?dinger equation. The physical mechanism for collapse prohibition is also given.     

关于Birkhoff逆问题中Santilli方法的研究

研究构造Birkhoff动力学函数的Santilli方法.首先,基于Cauchy-Kovalevskaya型方程解的存在性定理,采用反证法证明自治系统总有自治Birkhoff表示;其次,给出更简洁的方法证明Santilli第二方法可以被简化;找到Santilli第三方法中所隐含的一种等量关系,提出改进的Santilli第三方法,并研究该方法的MATLAB程序化计算;最后,总结全文并对结果进行讨论.     

Soliton solution to the complex modified Korteweg–de Vries equation on both zero and nonzero background via the robust inverse scattering method

In this paper, based on the robust inverse scattering method, we construct two kinds of solutions to the focusing modified Korteweg–de Vries equation. One is the classical soliton solution under the zero background condition and the other one is given through the nonzero background. Especially, for the nonzero background case, we choose a special spectral parameter such that the nonzero background solution is changed into the rational travelling waves. Finally, we also give a simple analysis of the soliton as the time $t$ is large, then we give the comparison between the exact solution and the asymptotic solution.     

Optical solitons in resonant and nonresonant nonlinear media in the presence of perturbations

We studied the optical solitons in nonlinear resonant and nonresonant media in the presence of perturbations, assuming that the transient effects are stimulated by the light scanning beam. We treated a slight deviation from the exact necessary condition for the soliton existence (2betanu=1), as a small perturbation for the integrable system, studying its influence upon the soliton propagation conditions. The approximation is constructed by the help of an algebraic version of the soliton perturbation theory using a Riemann boundary problem in connection with the inverse scattering method. We have obtained the soliton equation and we have solved it in the presence of a small perturbation in the adiabatic approximation. In this case we have demonstrated that for a Lorentz profile line the amplitude of the soliton remains unchanged, the only effect of the perturbation results in a phase shift.     

The dressing method, symmetries, and invariant solutions

The dressing method associates to a given nonlinear equation for q, a Riemann-Hilbert problem or a problem uniquely determined in terms of certain inverse data ƒ. Thus it generates a map from solutions of a linear system of PDEs (that for ƒ) to a nonlinear system of PDEs (that for q). We show that the corresponding tangent map can be expressed in closed form. Hence, symmetries and invariant solutions of ƒ induce symmetries and invariant solutions for q. The procedure can be used to charaterize solutions of Painlevé equations.