Cauchy Matrix Approach Research Articles

A novel discretization of the Yajima-Oikawa equation: Cauchy matrix approach

The Cauchy matrix approach, rooted in the Sylvester equation, plays a crucial role in defining the τ functions of nonlinear evolution equations. In this paper, the Cauchy matrix approach is employed to introduce a novel integrable semi-discrete counterpart of the one-dimensional Yajima-Oikawa (YO) system. This new system is linked with the differential-difference Kadomtsev-Petviashvili (KP) equation with self-consistent sources (SCS). Based on the Cauchy matrix approach of the KP system, we systematically construct multiple soliton and multiple pole solutions. Furthermore, we analyze and illustrate various examples of soliton solutions for further understanding.

Cauchy matrix approach for generalized semi-discrete lattice potential Korteweg–de Vries equations

Starting from a new scattering relation, we generate a generalized semi-discrete lattice potential Korteweg–de Vries(gsd-lpKdV) equation from a determining equation set(DES) and its Lax presentation. Further, the gsd-lpKdV equation with self-consistent sources(gsd-lpKdVSCS) is worked out by introducing two discrete variables to the plane wave factors. Specially, explicit formulae for kink solitons, Jordan-block solutions and diagonal-Jordan-block solutions of gsd-lpKdVSCS are obtained based on the Cauchy matrix approach. Moreover, We illustrate the dynamics of one-kink soliton, two-kink soliton, high order Jordan-block solutions and the diagonal-Jordan-block solution.

Nonisospectral equations from the Cauchy matrix approach

Nonisospectral equations from the Cauchy matrix approach

Cauchy matrix approach for H1 a equation in the torqued Adler–Bobenko–Suris lattice list

As a torqued version of the lattice potential Korteweg–de Vries equation, the H1 a is an integrable nonsymmetric lattice equation with only one spacing parameter. In this paper, we present the Cauchy matrix scheme for this equation. Soliton solutions, Jordan-block solutions and soliton-Jordan-block mixed solutions are constructed by solving the determining equation set. All the obtained solutions have jumping property between constant values for fixed n and demonstrate periodic structure.

Cauchy matrix approach to three non-isospectral nonlinear Schrödinger equations

This paper aims to develop a direct approach, namely, the Cauchy matrix approach, to non-isospectral integrable systems. In the Cauchy matrix approach, the Sylvester equation plays a central role, which defines a dressed Cauchy matrix to provide τ functions for the investigated equations. In this paper, using the Cauchy matrix approach, we derive three non-isospectral nonlinear Schrödinger equations and their explicit solutions. These equations are generically related to the time-dependent spectral parameter in the Zakharov–Shabat–Ablowitz–Kaup–Newell–Segur spectral problem. Their solutions are obtained from the solutions of unreduced non-isospectral nonlinear Schrödinger equations through complex reduction. These solutions are analyzed and illustrated to show the non-isospectral effects in dynamics of solitons.

Cauchy matrix structure and solutions of the spin-1 Gross–Pitaevskii equations

In this paper, we investigate the Cauchy matrix structure of the spin-1 Gross–Pitaevskii (GP) equations. By utilizing the Cauchy matrix approach, we derive a 2 × 2 matrix nonlinear Schrödinger (NLS) equation, which serves as an unreduced model for the spin-1 BEC system and allows solutions with explicit formulae. Then we provide suitable constraints which lead to reductions for obtaining the classical and nonlocal spin-1 GP equations and their solutions. Some obtained solutions, including one-soliton solution, two-soliton solution and double-pole solution, are analyzed and illustrated.

Integrability of the vector nonlinear Schrödinger–Maxwell–Bloch equation and the Cauchy matrix approach

Integrability of the vector nonlinear Schrödinger–Maxwell–Bloch equation and the Cauchy matrix approach

Cauchy matrix approach to the noncommutative Kadomtsev–Petviashvili equation with self-consistent sources

Cauchy matrix approach to the noncommutative Kadomtsev–Petviashvili equation with self-consistent sources

Cauchy matrix approach to the SU(2) self‐dual Yang–Mills equation

AbstractThe Cauchy matrix approach is developed to solve the self‐dual Yang–Mills (SDYM) equation. Starting from a Sylvester matrix equation coupled with certain dispersion relations for an infinite number of coordinates, we derive some new relations that give rise to the SDYM equation under Yang's formulation. By imposing further constraints on complex independent variables, a broad class of explicit solutions of the equation under Yang's formulation are obtained.

Extended KdV equations generated by squared eigenfunction symmetry

Two types of extended KdV equations generated by squared eigenfunction symmetries are derived with Cauchy matrix approach opening up different ideas that may be of considerable interest of understanding for the integrable system’s homology. The unified-form solution is discussed through choosing different matrices of determining equation set. Furthermore, under a safe range of parameters and the integrability conditions, it is possible to construct the soliton solution into the unified form, which can also verify that Cauchy matrix approach is an effective mathematical tool for investigating the integrable system.

Dark solitons of the Gross–Neveu model

Abstract We present N-soliton solutions for the classical (1+1)-dimensional Gross–Neveu model which satisfy non-zero boundary conditions. These solutions are obtained by a direct method using some properties of the soliton matrices that appear in the framework of the Cauchy matrix approach.

Cauchy matrix structure of the Mel’nikov model of long–short wave interaction

We propose a systematic method to construct the Mel’nikov model of long–short wave interactions, which is a special case of the Kadomtsev–Petviashvili (KP) equation with self-consistent sources (KPSCS). We show details how the Cauchy matrix approach applies to Mel’nikov's model which is derived as a complex reduction of the KPSCS. As a new result we find that in the dispersion relation of a 1-soliton there is an arbitrary time-dependent function that has previously not reported in the literature about the Mel’nikov model. This function brings time variant velocity for the long wave and also governs the short-wave packet. The variety of interactions of waves resulting from the time-freedom in the dispersion relation is illustrated.

Cauchy matrix approach to integrable equations with self-consistent sources and the Yajima–Oikawa system

A new approach by means of the Cauchy matrix is developed to obtain integrable equations with self-consistent sources. We derive a lower order Kadomtsev–Petviashvili equation with self-consistent sources which is then reduced to the multi-component Yajima–Oikawa system. This approach allows explicit multiple-pole solutions. New solutions of the Yajima–Oikawa system are obtained, of which there is more freedom in dispersion relations and amplitudes.

Discrete potential Ablowitz–Kaup–Newell–Segur equation

We establish a discrete model for the potential Ablowitz–Kaup–Newell–Segur equation via a generalized Cauchy matrix approach. Soliton solutions and Jordan block solutions of this equation are presented by solving the determining equation set. By applying appropriate continuum limits, we obtain two semi-discrete potential Ablowitz–Kaup–Newell–Segur equations. The reductions to real and complex discrete and semi-discrete potential modified Korteweg-de Vries equations are also discussed.

Discrete analogues for two nonlinear Schrödinger type equations

Discrete versions for nonlinear Schrödinger equation and modified nonlinear Schrödinger equation are presented, using the method of generalized Cauchy matrix approach based on a Sylvester equation. By solving the determining equation set, some exact solutions including soliton solutions, Jordan block solutions and mixed solutions for the resulting discrete equations are generated. The associated semi-discrete equations as well as continuous equations are also discussed, applying appropriate continuum limits.

Solitons of the (2+2)-dimensional Toda lattice

We use the generalized Cauchy matrix approach to derive the N-soliton solutions for the -dimensional Toda lattice.

Oscillatory Solutions for Lattice Korteweg-de Vries-Type Equations

Abstract By imposing some shift relations on r which satisfies the Sylvester equation KM + MK = r t c , oscillatory solutions are presented for some lattice Korteweg-de Vries-type equations, including the lattice potential Korteweg de-Vires equation, lattice potential modified Korteweg de-Vires equation, and lattice Schwarzian Korteweg-de Vries equation. This is done through the generalised Cauchy matrix approach.

Discrete and Semidiscrete Models for AKNS Equation

Abstract We establish a discrete model for the Ablowitz–Kaup–Newell–Segur (AKNS) equation via generalised Cauchy matrix approach. Two semidiscrete AKNS equations are obtained by, respectively, introducing straight continuum limit and skew continuum limit. Some reductions are also discussed.

A Discrete Negative Order Potential Korteweg–de Vries Equation

Abstract We investigate a discrete negative order potential Korteweg–de Vries (npKdV) equation via the generalised Cauchy matrix approach. Solutions more than multisoliton solutions of this equation are derived by solving the determining equation set. We also show the semidiscrete equation and continuous equation together with their exact solutions by considering the continuum limits.

Solutions and Three-Dimensional Consistency for Nonautonomous Extended Lattice Boussinesq-Type Equations

In this paper, we investigate the nonautonomous extended lattice Boussinesq-type equations in terms of generalized Cauchy matrix approach. Several kinds of solutions more than multi-soliton solutions to these equations are derived by solving determining equation set. Three-dimensional consistency of these equations is also studied.