The coupled modified nonlinear Schr\"{o}dinger equations on the half-line via the Fokas method

Coupled modified nonlinear Schr\"{o}dinger(CMNLS) equations describe the pulse propagation in the picosecond or femtosecond regime of the birefringent optical fibers. In this paper, we use the Fokas method to analyze the initial-boundary value problem for the CMNLS equations on the half-line. Assume that the solution u(x,t) and v(x,t) of CMNLS equations are exists, and we show that it can be expressed in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter {\lambda}.Comment: 17 pages, 2 figure

The coupled Fokas-Lenells equations by a Riemann-Hilbert approach

In this paper, we use the unified transform method to consider the initial-boundary value problem for the coupled Fokas-Lenells equations on the half-line, assuming that the solution $\{q(x,t),r(x,t)\}$ of the coupled Fokas-Lenells equations exists, we show that $\{q_x(x,t),r_x(x,t)\}$ can be expressed in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter $\lambda$. Thus, the solution $\{q(x,t),r(x,t)\}$ can be obtained by integration with respect to $x$.Comment: 18 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1704.0362

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

In this paper, we use the Fokas method to analyze the complex Sharma-Tasso-Olver(cSTO) equation on the half line. We show that it can be represented in terms of the solution of a matrix RHP formulated in the plane of the complex spectral parameter {\lambda}.Comment: arXiv admin note: text overlap with arXiv:1109.4935, arXiv:1205.1559, arXiv:0808.1534 by other author

The Novel Symmetry Constraint and Binary Nonlinearization of the Super Generalized Broer-Kaup Hierarchy with Self-consistent Sources and Conservation Laws

The super generalized Broer-Kaup(gBK) hierarchy and its super Hamiltonian structure are established based on a loop super Lie algebra and super-trace identity. Then the self-consistent sources, the conservation laws, the novel symmetry constraint and the binary nonlinearization of the super gBK hierarchy are generated, respectively. In addition, the integrals of motion required for Liouville integrability are explicitly given.Comment: 19 pages, 0 figure

An integrable generalization of the super Kaup-Newell soliton hierarchy and its bi-Hamiltonian structure

An integrable generalization of the super Kaup-Newell(KN) isospectral problem is introduced and its corresponding generalized super KN soliton hierarchy are established based on a Lie super-algebra B(0,1) and super-trace identity in this paper. And the resulting super soliton hierarchy can be put into a super bi-Hamiltonian form. In addition, a generalized super KN soliton hierarchy with self-consistent sources is also presented.Comment: 12 page

An initial-boundary value problem for the coupled focusing-defocusing complex short pulse equation with a 4×44\times4 Lax pair

In this paper we investigate the coupled focusing-defocusing complex short pulse equation, which describe the propagation of ultra-short optical pulses in cubic nonlinear media. Through the unified transform method, the initial-boundary value problem for the coupled focusing-defocusing complex short pulse equation with $4\times 4$ Lax pair on the half-line are to be analyzed. Assuming that the solution $\{q_1(x,t),q_2(x,t)\}$ of the coupled focusing-defocusing complex short pulse equation exists, we show that $\{q_{1,x}(x,t),q_{2,x}(x,t)\}$ can be expressed in terms of the unique solution of a $4\times 4$ matrix Riemann-Hilbert problem formulated in the complex $\lambda$-plane. Thus, the solution $\{q_1(x,t),q_2(x,t)\}$ can be obtained by integration with respect to $x$. Moreover, we also get that some spectral functions are not independent and satisfy the so-called global relation.Comment: 20 pages, 4 figures. arXiv admin note: text overlap with arXiv:1704.0362

Riemann-Hilbert approach for a mixed coupled nonlinear Schr\"odinger system and its soliton solutions

In this work, we examine the integrable mixed coupled nonlinear Schr\"odinger (mCNLS) system, which describe the propagation of an optical pulse in a birefringent optical fiber. By the Riemann-Hilbert(RH) approach, the N-soliton solutions of the mCNLS system can be expressed explicitly when the jump matrix of a specific RH problem is a $3\times3$ unit matrix. As a special example, the expression of one- and two-soliton are displayed explicitly. More generally, as a promotion, an integrable generalized multi-component NLS system with its linear spectral problem be discussed. It is hoped that our results can help enrich the nonlinear dynamical behaviors of the mCNLS.Comment: 15 pages, 1 figures. arXiv admin note: text overlap with arXiv:1809.0703

The initial-boundary value problems for the coupled derivative nonlinear Schr\"odinger equations on the half-line

The unified transform method is used to analyze the initial-boundary value problem for the coupled derivative nonlinear Schr\"odinger(CDNLS) equations on the half-line. In this paper, we assume that the solution $u(x,t)$ and $v(x,t)$ of CDNLS equations are exists, and we show that it can be expressed in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter $\lambda$.Comment: arXiv admin note: substantial text overlap with arXiv:1704.03623, arXiv:1712.06449, arXiv:1711.0386

On the Riemann-Hilbert problem for the Chen-Lee-Liu derivative nonlinear Schr\"{o}dinger equation

In this work, we investigated a combined Chen-Lee-Liu derivative nonlinear Schr\"{o}dinger equation(called CLL-NLS equation by Kundu) on the half-line by unified transformation approach. We gives spectral analysis of the Lax pair for CLL-NLS equation, and establish a matrix Riemann-Hilbert problem, so as to reconstruct the solution $r(z,t)$ of the CLL-NLS equation by solving. Furthermore, the spectral functions are not independent, but enjoy by a compatibility condition, which is the so-called global relation

Riemann-Hilbert method and N-soliton solutions for the mixed Chen-Lee-Liu derivative nonlinear Schr\"{o}dinger equation

In this paper, we aim to investigate the mixed Chen-Lee-Liu derivative nonlinear Schr\"{o}dinger(CLL-NLS) equation via the Riemann-Hilbert(RH) method. we construct a RH problem base on the Jost solution of the Lax pair. By solving this RH problem corresponding to the non reflection case, the N-soliton solution of CLL-NLS equation is obtained, which expression is the ratio of $(2N+1)\times(2N+1)$ determinant and $2N\times2N$ determinant