Kouichi Toda

N soliton solutions to the Bogoyavlenskii-Schiff equation and a quest for the soliton solution in (3 1) dimensions

We study the integrable systems in higher dimensions which can be written by the trilinear form instead of by the Hirotas bilinear form. We explicitly discuss the Bogoyavlenskii-Schiff equation in (2 + 1) dimensions. Its analytical proof of multisoliton solution and a new feature are given. Being guided by the strong symmetry, we also propose a new equation in (3 + 1) dimensions.

The investigation into the Schwarz–Korteweg–de Vries equation and the Schwarz derivative in (2+1) dimensions

In this note, we shall introduce a new integrable equation and the Schwarz derivative in (2+1) dimensions. First we show the existence of the Lax pair for an equation which has the relation to the Schwarz–Korteweg–de Vries (SKdV) equation. Next we derive a new equation in (2+1) dimensions by using a well-known higher-dimensional manner to the Lax pair for the SKdV equation. The (2+1) dimensional Schwarz derivative is defined here. Finally we briefly discuss various results which we have obtained about the new equation.

N-soliton solutions to a -dimensional integrable equation

We give explicitly N -soliton solutions of a (2C 1)-dimensional equation, xtC xxxz=4CxxzCxxz=2C@ 1 x zzz=4 D 0. This equation is obtained by unifying two directional generalizations of the potential KdV equation: the closed ring with the potential KP equation, and the Calogero-Bogoyavlenskij-Schiff equation. This equation is also a reduction of the KP hierarchy. We also find the Miura transformation which yields the same ring of the corresponding modified equations. The study of higher-dimensional integrable systems is one of the central themes in integrable systems. A typical example of a higher-dimensional integrable system is obtained by modifying the Lax operators of a basic equation, the potential KdV (p-KdV) equation in this paper. The Lax pair of the p-KdV equation have the form L.x;t/D@ 2 xCx.x;t/ (1) T.x;t/D.L.x;t/ 3 2/

Extensions of Soliton equations to non-commutative

We report a strong method to generate various equations which have Lax representations on noncommutative (1 + 1) and (2 + 1)-dimensional spaces. The generated equations contain noncommutative integrable equations obtained by using bicomplex method and by reductions of noncommutative anti-self-dual Yang-Mills equation. This suggests that noncommutative Lax equations would be integrable.

(2 + 1)

Abstract We prove the existence of a Lax pair for the Calogero Korteweg-de Vries (CKdV) equation. Moreover, we modify the T operator in the the Lax pair of the CKdV equation, in the search of a (2+1)-dimensional case and thereby propose a new equation in (2+1) dimensions. We named this the (2+1)-dimensional CKdV equation. We show that the CKdV equation as well as the (2+1)-dimensional CKdV equation are integrable in the sense that they possess the Painlevé property. Some exact solutions are also constructed.

dimensions

Abstract In this paper we discuss two problems. Firstly, integrable equations have their own higher order integrable equations, like the KdV, mKdV and NLS hierarchies etc. We consider whether the integrable equation in (2 + 1) dimensions — the Bogoyavlenskii-Schiff (BS) equation — has also the analogous hierarchy. We derive the Lax pair of the BS hierarchy. We also investigate the integrability of the 5th order BS equation and find that this equation admits the Painleve property. Secondly, the study of higher dimensional integrable systems is one of the central themes in integrability. A typical way of constructing higher dimensional integrable systems is to modify Lax operators of basic equation, in this paper — the potential KdV(p-KdV) equation. We modified the L and T operators of the p-KdV equation for the search of (3 + 1) dimensional integrable equation. However, the Lax equation is eventually reduced to (2 + 1)-dimensional one. In addition, we also propose the modified equation and the Lax pair.

Lax Pairs, Painlevé Properties and Exact Solutions of the Calogero Korteweg-de Vries Equation and a New (2+1)-Dimensional Equation

The general KdV equation (gKdV) derived by T. Chou is one of the famous (1 + 1) dimensional soliton equations with variable coefficients. It is well-known that the gKdV equation is integrable. In this paper a higher-dimensional gKdV equation, which is integrable in the sense of the Painleve test, is presented. A transformation that links this equation to the canonical form of the Calogero-Bogoyavlenskii-Schiff equation is found. Furthermore, the form and similar transformation for the higher-dimensional modified gKdV equation are also obtained.

The Bogoyavlenskii-Schiff hierarchy and integrable equations in (2+1) dimensions

Abstract First of all, we show the existence of the Lax pair for the Calogero Korteweg-de Vries(CKdV) equation. Next we modify T operator that is one of the Lax pair for the CKdV equation for the search of the (2+1)-dimensional case and propose a new equation in (2+1) dimensions. We call it the (2+1)-dimensional CKdV equation. And then we discuss the modification of L operator that is another of the Lax pair of the CKdV equation. Moreover we attempt the modification of L and T operators.

The Painleve Test and Reducibility to the Canonical Forms for Higher-Dimensional Soliton Equations with Variable-Coefficients

Abstract The Calogero—Degasperis—Fokas (CDF) equation is a (1 + 1)-dimensional integrable equation. We extend the CDF equation to (2+1) dimensions by a dimensional extension of the Lax pair. As a result, the higher-dimensional CDF equation and another higher dimensional equation are constructed. The Painleve properties and (2 + 1)-dimensional traveling solitary solutions for the higher-dimensional equations obtained are also briefly reported. Moreover, the (2 + 1)-dimensional CDF equation can be reduced to some integrable equations.

The Investigation into New Equations in (2 + 1) Dimensions

Variable-coefficient generalizations of integrable nonlinear equations are one of exciting subjects in the study of mechanical, physical, mathematical and engineering sciences. In this paper, we present a KdV family (namely, KdV, modified KdV, Calogero-Degasperis-Fokas and Harry-Dym equations) with variable coefficients in (2 + 1) dimensions.