Takayuki Tsuchida

The Coupled Modified Korteweg-de Vries Equations

Generalization of the modified KdV equation to a multi-component system, that is expressed by \(\frac{\partial u_i}{\partial t} + 6 \bigl( \sum_{j,k=0}^{M-1} C_{jk} u_j u_k \bigr) \frac{\partial u_i}{\partial x} + \frac{\partial^3 u_{i}}{\partial x^3} =0\), i =0, 1, …, M -1, is studied. We apply a new extended version of the inverse scattering method to this system. It is shown that this system has an infinite number of conservation laws and multi-soliton solutions. Further, the initial value problem of the model is solved.

Integrable semi-discretization of the coupled nonlinear Schrödinger equations

A system of semi-discrete coupled nonlinear Schrodinger equations is studied. To show the complete integrability of the model with multiple components, we extend the discrete version of the inverse scattering method for the single-component discrete nonlinear Schrodinger equation proposed by Ablowitz and Ladik. By means of the extension, the initial-value problem of the model is solved. Further, the integrals of motion and the soliton solutions are constructed within the framework of the extension of the inverse scattering method.

New integrable systems of derivative nonlinear Schrödinger equations with multiple components

Abstract The Lax pair for a derivative nonlinear Schrodinger equation proposed by Chen–Lee–Liu is generalized into matrix form. This gives new types of integrable coupled derivative nonlinear Schrodinger equations. By virtue of a gauge transformation, a new multi-component extension of a derivative nonlinear Schrodinger equation proposed by Kaup–Newell is also obtained.

Integrable semi-discretization of the coupled modified KdV equations

The discrete version of the inverse scattering method proposed by Ablowitz and Ladik is extended to solve multi-component systems. The extension enables one to solve the initial value problem, which proves directly the complete integrability of a semi-discrete version of the coupled modified Korteweg–de Vries (KdV) equations and their hierarchy. It also provides a procedure to obtain conservation laws and multi-soliton solutions of the hierarchy.

Integrable discretizations of derivative nonlinear Schrödinger equations

We propose integrable discretizations of derivative nonlinear Schrodinger (DNLS) equations such as the Kaup–Newell equation, the Chen–Lee–Liu equation and the Gerdjikov–Ivanov equation by constructing Lax pairs. The discrete DNLS systems admit the reduction of complex conjugation between two dependent variables and possess bi-Hamiltonian structure. Through transformations of variables and reductions, we obtain novel integrable discretizations of the nonlinear Schrodinger (NLS), modified KdV (mKdV), mixed NLS, matrix NLS, matrix KdV, matrix mKdV, coupled NLS, coupled Hirota, coupled Sasa–Satsuma and Burgers equations. We also discuss integrable discretizations of the sine-Gordon equation, the massive Thirring model and their generalizations.

Multi-Field Integrable Systems Related to WKI-Type Eigenvalue Problems

Higher flows of the Heisenberg ferromagnet equation and the Wadati-Konno-Ichikawa equation are generalized into multi-component systems on the basis of the Lax formulation. It is shown that there is a correspondence between the multi-component systems through a gauge transformation. An integrable semi-discretization of the multi-component higher Heisenberg ferromagnet system is obtained.

Symmetrically coupled higher-order nonlinear Schrödinger equations: singularity analysis and integrability

The integrability of a system of two symmetrically coupled higher-order nonlinear Schrodinger equations with parameter coefficients is tested by means of singularity analysis. It is proven that the system passes the Painleve test for integrability only in ten distinct cases, of which two are new. For one of the new cases, a Lax pair and a multi-field generalization are obtained; for the other one, the equations of the system are uncoupled by a nonlinear transformation.

New reductions of integrable matrix partial differential equations: Sp(m)-invariant systems

We propose a new type of reduction for integrable systems of coupled matrix partial differential equations; this reduction equates one matrix variable with the transposition of another multiplied by an antisymmetric constant matrix. Via this reduction, we obtain a new integrable system of coupled derivative modified KdV equations and a new integrable variant of the massive Thirring model, in addition to the already known systems. We also discuss integrable semidiscretizations of the obtained systems and present new soliton solutions to both continuous and semidiscrete systems. As a by-product, a new integrable semidiscretization of the Manakov model (self-focusing vector nonlinear Schrodinger equation) is obtained.

Systematic method of generating new integrable systems via inverse Miura maps

We provide a new natural interpretation of the Lax representation for an integrable system; that is, the spectral problem is the linearized form of a Miura transformation between the original system and a modified version of it. On the basis of this interpretation, we formulate a systematic method of identifying modified integrable systems that can be mapped to a given integrable system by Miura transformations. Thus, this method can be used to generate new integrable systems from known systems through inverse Miura maps; it can be applied to both continuous and discrete systems in 1+1 dimensions as well as in 2+1 dimensions. The effectiveness of the method is illustrated using examples such as the nonlinear Schroedinger (NLS) system, the Zakharov-Ito system (two-component KdV), the three-wave interaction system, the Yajima-Oikawa system, the Ablowitz-Ladik lattice (integrable space-discrete NLS), and two (2+1)-dimensional NLS systems.

Bi-Hamiltonian structure of modified Volterra model

Abstract Bi-Hamiltonian structure of a periodic modified Volterra model is studied. A new Hamiltonian structure of the model is derived by use of the variable transformation between the model and the original Volterra model. This result shows the bi-Hamiltonian structure of the model.