Junkichi Satsuma

Soliton Solutions of a Coupled Korteweg-de Vries Equation

Abstract A coupled Korteweg-de Vries equation is presented. In general the equation exhibits a soliton solution and three basic conserved quantities. For a special choice of dispersion relations, the equation exhibits at least three-soliton solutions and two more higher-order conserved quantities.

New-Type of Soliton Solutions for a Higher-Order Nonlinear Schrödinger Equation

It is shown that a higher-order nonlinear Schrodinger equation which describes propagation of pulses in optical fiber is solvable by means of the inverse scattering transform. The soliton solution possesses a remarkable property that it can propagate steadily with two peaks of the same height.

Solitons and rational solutions of nonlinear evolution equations

Rational solutions of certain nonlinear evolution equations are obtained by performing an appropriate limiting procedure on the soliton solutions obtained by direct methods. In this note specific attention is directed at the Korteweg–de Vries equation. However, the methods used are quite general and apply to most nonlinear evolution equations with the isospectral property, including certain multidimensional equations. In the latter case, nonsingular, algebraically decaying, soliton solutions can be constructed.

A Coupled KdV Equation is One Case of the Four-Reduction of the KP Hierarchy

It is shown that the coupled KdV equation introduced by the present authors is a special case of the four-reduced KP hierarchy which is included in the general theory of τ functions. From the fact it is also shown that the soliton solutions can be derived from those of the KP equation. Moreover, the existence of infinitely many conserved quantities are proved by means of the linear scheme giving the coupled KdV equation.

Bilinearization of a generalized derivative nonlinear Schrödinger equation

A generalized derivative nonlinear Schrodinger equation, i q t + q x x +2 iγ∣ q ∣ 2 q x +2 i(γ- 1) q 2 q x * +(γ-1)(γ-2)∣ q ∣ 4 q =0, is studied by means of Hirotas bilinear formalism. Soliton solutions are constructed as quotients of Wronski-type determinants. A relationship between the bilinear structure and gauge transformation is also discussed.

An N-Soliton Solution for the Nonlinear Schrödinger Equation Coupled to the Boussinesq Equation

A system comprised of the nonlinear Schrodinger equation coupled to the Boussinesq equation is proposed. This system reduces to the nonlinear Schrodinger equation, the Zakharov equations, or the equations studied by Nishikawa et al ., (as well as Yajima and Satsuma), for an appropriate choice of parameters. It is found that an N-soliton solution exists for a particular choice of parameters.

On an internal wave equation describing a stratified fluid with finite depth

Abstract In this note we give a Backlund transformation, a recursion scheme for the infinite number of conservation laws and an inverse scattering problem for an integro-differential equation governing long internal gravity waves in a stratified fluid of finite total depth. They are shown to reduce to those of the Korteweg—de Vries equation in the shallow water limit and those of the Benjamin—Ono equation in the deep water limit.

Multi-Soliton Solutions of a Coupled System of the Nonlinear Schrödinger Equation and the Maxwell-Bloch Equations

Multi-soliton solutions of a coupled system of the nonlinear Schrodinger equation and the Maxwell-Bloch equations are given. Solutions of the system are explicitly constructed as a quotient of Casorati-type determinants. By using explicit form of the soliton solutions, the influence of the MB-term on the speed of soliton is evaluated.

Q-difference version of the two-dimensional toda lattice equation

A q -difference version of the two-dimensional Toda lattice equation is proposed. Through a suitable reduction, it reduces to the q -difference version of the cylindrical Toda lattice equation. It is shown that the reduced equation admits solutions expressed by the q -Bessel function.

Solutios of the Kadomtsev-Petviashvili Equation and the Two-Dimensional Toda Equations

It is shown that systems of the Kadomtsev-Petviashvili equation and the two-dimensional Toda equations admit solutions with two-directional Wronskian structures. It is found that the boundary conditions on the Toda equations change these structures substantially.