Bao- Feng

Integrable discretizations of the short pulse equation

In this paper, we propose integrable semi-discrete and full-discrete analogues of the short pulse (SP) equation. The key construction is the bilinear form and determinant structure of solutions of the SP equation. We also give the determinant formulas of N-soliton solutions of the semi-discrete and full-discrete analogues of the SP equations, from which the multi-loop and multi-breather solutions can be generated. In the continuous limit, the full-discrete SP equation converges to the semi-discrete SP equation, and then to the continuous SP equation. Based on the semi-discrete SP equation, an integrable numerical scheme, i.e. a self-adaptive moving mesh scheme, is proposed and used for the numerical computation of the short pulse equation.

Multi-soliton, multi-breather and higher order rogue wave solutions to the complex short pulse equation

Abstract In the present paper, we are concerned with the general analytic solutions to the complex short pulse (CSP) equation including soliton, breather and rogue wave solutions. With the aid of a generalized Darboux transformation, we construct the N -bright soliton solution in a compact determinant form, the N -breather solution including the Akhmediev breather and a general higher order rogue wave solution. The first and second order rogue wave solutions are given explicitly and analyzed. The asymptotic analysis is performed rigorously for both the N -soliton and the N -breather solutions. All three forms of the analytical solutions admit either smoothed-, cusped- or looped-type ones for the CSP equation depending on the parameters. It is noted that, due to the reciprocal (hodograph) transformation, the rogue wave solution to the CSP equation can be a smoothed, cusponed or a looped one, which is different from the rogue wave solution found so far.

An integrable semi-discretization of the Camassa-Holm equation and its determinant solution

An integrable semi-discretization of the Camassa–Holm (CH) equation is presented. The keys of its construction are bilinear forms and determinant structure of solutions of the CH equation. Determinant formulas of N-soliton solutions of the continuous and semi-discrete Camassa–Holm equations are presented. Based on determinant formulas, we can generate multi-soliton, multi-cuspon and multi-soliton–cuspon solutions. Numerical computations using the integrable semi-discrete Camassa–Holm equation are performed. It is shown that the integrable semi-discrete Camassa–Holm equation gives very accurate numerical results even in the cases of cuspon–cuspon and soliton–cuspon interactions. The numerical computation for an initial value condition, which is not an exact solution, is also presented.

Complex short pulse and coupled complex short pulse equations

Abstract In the present paper, we propose a complex short pulse equation and a coupled complex short equation to describe ultra-short pulse propagation in optical fibers. They are integrable due to the existence of Lax pairs and infinite number of conservation laws. Furthermore, we find their multi-soliton solutions in terms of pfaffians by virtue of Hirota’s bilinear method. One- and two-soliton solutions are investigated in details, showing favorable properties in modeling ultra-short pulses with a few optical cycles. Especially, same as the coupled nonlinear Schrodinger equation, there is an interesting phenomenon of energy redistribution in soliton interactions. It is expected that, for the ultra-short pulses, the complex and coupled complex short pulses equation will play the same roles as the nonlinear Schrodinger equation and coupled nonlinear Schrodinger equation.

General N-soliton solution to a vector nonlinear Schrödinger equation

We consider a general N-soliton solution to a vector nonlinear Schrodinger (NLS) equation of all possible combinations of nonlinearities including all-focusing, all-defocusing and mixed types. Based on the KP hierarchy reduction method, we firstly construct general two-bright-one-dark and one-bright-two-dark soliton solutions in a three-coupled NLS equation, then we extend our analysis to a vector NLS equation to obtain a general N-soliton solution in Gram determinant form. This formula unifies the bright, dark and bright-dark soliton solutions, which have been widely studied in the literature. The conditions for the existence of all types of soliton solutions with all possible combinations of nonlinearities are elucidated.

An integrable coupled short pulse equation

An integrable coupled short pulse (CSP) equation is proposed for the propagation of ultra-short pulses in optical fibers. Based on two sets of bilinear equations to a two-dimensional Toda lattice linked by a Backlund transformation, and an appropriate hodograph transformation, the proposed CSP equation is derived. Meanwhile, its N-soliton solutions are given by the Casorati determinant in a parametric form. The properties of one- and two-soliton solutions are investigated in detail. Same as the short pulse equation, the two-soliton solution turns out to be a breather type if the wave numbers are complex conjugate. We also illustrate an example of soliton–breather interaction.

Discrete Integrable Systems and Hodograph Transformations Arising from Motions of Discrete Plane Curves

We consider integrable discretizations of some soliton equations associated with the motions of plane curves: the Wadati–Konno–Ichikawa elastic beam equation, the complex Dym equation and the short pulse equation. They are related to the modified KdV or the sine–Gordon equations by the hodograph transformations. Based on the observation that the hodograph transformations are regarded as the Euler–Lagrange transformations of the curve motions, we construct the discrete analogues of the hodograph transformations, which yield integrable discretizations of those soliton equations.

Multi-Dark Soliton Solutions of the Two-Dimensional Multi-Component Yajima–Oikawa Systems

We present a general form of multi-dark soliton solutions of two-dimensional multi-component soliton systems. Multi-dark soliton solutions of the two-dimensional (2D) and one-dimensional (1D) multi-component Yajima-Oikawa (YO) systems, which are often called the 2D and 1D multi-component long wave-short wave resonance interaction systems, are studied in detail. Taking the 2D coupled YO system with two short wave and one long wave components as an example, we derive the general

General mixed multi-soliton solutions to one-dimensional multicomponent Yajima-Oikawa system

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Defocusing complex short-pulse equation and its multi-dark-soliton solution.

-dark-dark soliton solution in both the Gram type and Wronski type determinant forms for the 2D coupled YO system via the KP hierarchy reduction method. By imposing certain constraint conditions, the general