Riemann–Hilbert problems and N-soliton solutions of the nonlocal reverse space-time Chen–Lee–Liu equation

In this paper, the N-soliton solutions to the nonlocal reverse space-time Chen–Lee–Liu equation have been derived. Under the nonlocal symmetry reduction to the matrix spectral problem, the nonlocal reverse space-time Chen–Lee–Liu equation can be obtained. Based on the spectral problem, the specific matrix Riemann–Hilbert problem is constructed for this nonlocal equation.Through solving this associated Riemann–Hilbert problem, the N-soliton solutions to this nonlocal equation can be obtained in t...     

Multi-Soliton Solutions for the Coupled Fokas–Lenells System via Riemann–Hilbert Approach

We aim to construct multi-soliton solutions for the coupled Fokas–Lenells system which arises as a model for describing the nonlinear pulse propagation in optical fibers. Starting from the spectral analysis of the Lax pair, a Riemann–Hilbert problem is presented. Then in the framework of the Riemann–Hilbert problem corresponding to the reflectionless case, N-soliton solutions to the coupled Fokas–Lenells system are derived explicitly.     

Initial-Boundary Value Problem for Two-Component Gerdjikov–Ivanov Equation with 3 × 3 Lax Pair on Half-Line

The Fokas unified method is used to analyze the initial-boundary value problem of two-component Gerdjikov–Ivanonv equation on the half-line. It is shown that the solution of the initial-boundary problem can be expressed in terms of the solution of a 3 × 3 Riemann–Hilbert problem. The Dirichlet to Neumann map is obtained through the global relation.     

Multi-soliton solutions for the coupled modified nonlinear Schrdinger equations via Riemann–Hilbert approach

The coupled modified nonlinear Schrdinger equations are under investigation in this work. Starting from analyzing the spectral problem of the Lax pair, a Riemann–Hilbert problem for the coupled modified nonlinear Schrdinger equations is formulated. And then, through solving the obtained Riemann–Hilbert problem under the conditions of irregularity and reflectionless case, N-soliton solutions for the equations are presented. Furthermore, the localized structures and dynamic behaviors of the one-soliton solution are shown graphically.     

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.     

Nonlocal symmetry and exact solutions of the(2+1)-dimensional modified Bogoyavlenskii–Schiff equation

In this paper,the truncated Painlev′e analysis,nonlocal symmetry,Bcklund transformation of the(2+1)-dimensional modified Bogoyavlenskii–Schiff equation are presented.Then the nonlocal symmetry is localized to the corresponding nonlocal group by the prolonged system.In addition,the(2+1)-dimensional modified Bogoyavlenskii–Schiff is proved consistent Riccati expansion(CRE) solvable.As a result,the soliton–cnoidal wave interaction solutions of the equation are explicitly given,which are difficult to find by other traditional methods.Moreover figures are given out to show the properties of the explicit analytic interaction solutions.     

Construction of Multi-soliton Solutions of the N-Coupled Hirota Equations in an Optical Fiber

This work aims to study the N-coupled Hirota equations in an optical fiber under the zero boundary condition at infinity. By analyzing the spectral problem, a matrix Riemann–Hilbert problem on the real axis is strictly established. Then, by solving the presented matrix Riemann–Hilbert problem under the constraint of no reflection,the bright multi-soliton solutions to the N-coupled Hirota equations are explicitly gained.     

A Riemann–Hilbert Approach to Complex Sharma–Tasso–Olver Equation on Half Line

In this paper, the Fokas unified method is used to analyze the initial-boundary value problem of a complex Sharma–Tasso–Olver(c STO) equation on the half line. We show that the solution can be expressed in terms of the solution of a Riemann–Hilbert problem. The relevant jump matrices are explicitly given in terms of the matrix-value spectral functions spectral functions {a(λ), b(λ)} and {A(λ), B(λ)}, which depending on initial data u_0(x) = u(x, 0) and boundary data g_0(y) = u(0, y), g_1(y) = ux(0, y), g_2(y) = u_(xx)(0, y). These spectral functions are not independent, they satisfy a global relation.     

Global Structure Stability of Riemann Solution of Quasilinear Hyperbolic System of Conservation Law in Presence of Boundary： Shocks and Contact Discontinuities

In this paper, we investigate a class of mixed initial-boundary value problems for a kind of n × n quasilinear hyperbolic systems of conservation laws on the quarter plan. We show that the structure of the pieeewise C^1 solution u = u（t, x） of the problem, which can be regarded as a perturbation of the corresponding Riemann problem, is globally similar to that of the solution u = U（x/t） of the corresponding Riemann problem. The piecewise C^1 solution u = u（t, x） to this kind of problems is globally structure-stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities.     

Nonlocal symmetries,consistent Riccati expansion integrability,and their applications of the (2+1)-dimensional Broer–Kaup–Kupershmidt system

For the (2+1)-dimensional Broer–Kaup–Kupershmidt(BKK) system, the nonlocal symmetries related to the Schwarzian variable and the corresponding transformation group are found. Moreover, the integrability of the BKK system in the sense of having a consistent Riccati expansion(CRE) is investigated. The interaction solutions between soliton and cnoidal periodic wave are explicitly studied.     

New interaction solutions of the Kadomtsev–Petviashvili equation

The residual symmetry relating to the truncated Painlev′e expansion of the Kadomtsev–Petviashvili(KP) equation is nonlocal, which is localized in this paper by introducing multiple new dependent variables. By using the standard Lie group approach, new symmetry reduction solutions for the KP equation are obtained based on the general form of Lie point symmetry for the prolonged system. In this way, the interaction solutions between solitons and background waves are obtained, which are hard to find by other traditional methods.     

The consistent Riccati expansion and new interaction solution for a Boussinesq-type coupled system

Starting from the Davey–Stewartson equation,a Boussinesq-type coupled equation system is obtained by using a variable separation approach.For the Boussinesq-type coupled equation system,its consistent Riccati expansion(CRE)solvability is studied with the help of a Riccati equation.It is significant that the soliton–cnoidal wave interaction solution,expressed explicitly by Jacobi elliptic functions and the third type of incomplete elliptic integral,of the system is also given.     

New interaction solutions and nonlocal symmetry of an equation combining the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form

The nonlocal symmetry is derived for an equation combining the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form from the truncated Painlevéexpansion method. The nonlocal symmetries are localized to the Lie point symmetry by introducing new auxiliary dependent variables. The finite symmetry transformation and the Lie point symmetry for the prolonged system are solved directly. Many new interaction solutions among soliton and other types of interaction solutions for the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form can be obtained from the consistent condition of the consistent tanh expansion method by selecting the proper arbitrary constants.     

Multiple soliton solutions and symmetry analysis of a nonlocal coupled KP system

A nonlocal coupled Kadomtsev–Petviashivili(ncKP) system with shifted parity(■) and delayed time reversal(■) symmetries is generated from the local coupled Kadomtsev–Petviashivili(cKP) system. By introducing new dependent variables which have determined parities under the action of ■, the ncKP is transformed to a local system. Through this way, multiple even number of soliton solutions of the ncKPI system are generated from N-soliton solutions of the c KP system, which become breathers by choosin...     

Dynamical stability of dipolar condensate in a parametrically modulated one-dimensional optical lattice

We study the stabilization properties of dipolar Bose–Einstein condensate in a deep one-dimensional optical lattice with an additional external parametrically modulated harmonic trap potential. Through both analytical and numerical methods, we solve a dimensionless nonlocal nonlinear discrete Gross–Pitaevskii equation with both the short-range contact interaction and the long-range dipole–dipole interaction. It is shown that, the stability of dipolar condensate in modulated deep optical lattice can be controled by coupled effects of the contact interaction, the dipolar interaction and the external modulation. The system can be stabilized when the dipolar interaction, the contact interaction, the average strength of potential and the ratio of amplitude to frequency of the modulation satisfy a critical condition. In addition, the breather state, the diffused state and the attractive-interaction-induced-trapped state are predicted. The dipolar interaction and the external modulation of the lattice play important roles in stabilizing the condensate.     

Homoclinic Breather-Wave with Convective Effect for the （1＋1）-Dimensional Boussinesq Equation

A new type of two-wave solution, i.e. a homoclinic breather-wave solution with convective effect, for the （1＋1）- dimensional Boussinesq equation is obtained using the extended homoelinic test method. Moreover, the mechanical feature of the wave solution is investigated and the phenomenon of homoelinic convection of the two-wave is exhibited on both sides of the equilibrium. These results enrich the dynamical behavior of （1＋1）-dimensional nonlinear wave fields.     

Localization of nonlocal symmetries and interaction solutions of the Sawada–Kotera equation

The nonlocal symmetry of the Sawada–Kotera(SK) equation is constructed with the known Lax pair. By introducing suitable and simple auxiliary variables, the nonlocal symmetry is localized and the finite transformation and some new solutions are obtained further. On the other hand, the group invariant solutions of the SK equation are constructed with the classic Lie group method.In particular, by a Galileo transformation some analytical soliton-cnoidal interaction solutions of a asymptotically integrable equation are discussed in graphical ways.     

New solutions from nonlocal symmetry of the generalized fifth order KdV equation

The nonlocal symmetry of the generalized fifth order KdV equation(FOKdV) is first obtained by using the related Lax pair and then localizing it in a new enlarged system by introducing some new variables. On this basis, new Ba¨cklund transformation is obtained through Lie's first theorem. Furthermore, the general form of Lie point symmetry for the enlarged FOKdV system is found and new interaction solutions for the generalized FOKdV equation are explored by using a symmetry reduction method.     

Phonon-dependent transport through a serially coupled double quantum dot system

Using Keldysh nonequilibrium Green function formalism and mapping a many-body electron–phonon interaction onto a one body problem, the electron transport through a serially coupled double quantum dot system is analyzed. The influence of the electron–phonon interaction, temperature, detuning, and interdot tunneling on the transmission coefficient and current is studied. Our results show that the electron–phonon interaction results in the appearance of the side peaks in the transmission coefficient, whose height is strongly dependent on the phonon temperature. We have also found that the inequality of the electron–phonon interaction strength in two dots gives rise to an asymmetry in the current–voltage characteristic. In addition, the temperature difference between the phonon and electron subsystems results in the reduction of the saturated current and the destruction of the step-like behavior of the current. It is also observed that the detuning can improve the magnitude of the current by compensating the mismatch of the quantum dots energy levels induced by the electron–phonon interaction.     

A New Solution to the Hirota–Satsuma Coupled KdV Equations by the Dressing Method

The Hirota–Satsuma coupled KdV equations associated 2×2 matrix spectral problem is discussed by the dressing method,which is based on the factorization of integral operator on a line into a product of two Volterra integral operators.A new solution is obtained by choosing special kernel of integral operator.