Shunji Kawamoto

Similarity Solutions of the Kadomtsev-Petviashvili Equation

It is shown that the Kadomtsev-Petviashvili equation can be reduced first to the Boussinesq or Korteweg-de Vries equation, and secondly to the first or second Painleve equation by means of infinitesimal transformations of Lies method. The soliton solutions moving in non-steady and non-uniform background are also obtained.

Reduction of KdV and Cylindrical KdV Equations to Painlevé Equation

Similarity solutions of the KdV and cylindrical KdV equations are studied by means of Lies method of infinitesimal transformation groups. It is shown that the KdV equation is reduced to the Painleve transcendental equation of the first or second kind. The similarity solutions of cylindrical KdV equation also satisfy the first or second Painleve equation, and a soliton-like solution is obtained.

Cusp Soliton Solutions of the Ito-Type Coupled Nonlinear Wave Equation

The Ito-type coupled nonlinear wave equation is shown to have cusp (singular spiky) soliton solutions.

Solitary Wave Solutions of the Korteweg-de Vries Equation with Higher Order Nonlinearity

The Korteweg-de Vries equation with higher order nonlinearity is described to have the same new solitary wave solution and singular explode one as the Ito-type and the normalized Boussinesq equations.

Derivation of Nonlinear Partial Differential Equations Reducible to the Painleve Equations

New nonlinear partial differential equations reducible to the Painleve equations are derived through special transformations constructed by similarity variables of well-known one-dimensional soliton equations. Also, it is discussed that there exist identity transformations, which transform an equation into the original one.

Construction of Stationary Solitary Wave Solutions

A construction of stationary solitary wave solutions to nonlinear partial differential equations is presented. First, it is shown that one of fifty Painleve-type ordinary differential equations intimately connects with one-soliton solutions which are stationary solitary wave ones of usual soliton equations. Secondly, if these solutions are named the “1st-class”, by extending the Painleve-type equation, we can newly construct three kinds (sech 2 , sech and combined forms) of the “ N th-class” stationary solitary wave solutions through special potential functions expressed by elementary ones.

The Weierstrass function of chaos map with exact solution

It is shown that the Weierstrass function, which gives a fractal curve, can be derived from a nonlinear map with an exact chaos solution.

Cusp and Usual Solitons of the Normalized Boussinesq Equation

A cusp soliton solution, a usual soliton solution and periodic solutions to the normalized Boussinesq equation are obtained as stationary solutions. Also, the similarity equations are discussed as an integrability problem.

The Nonlinear Schrödinger-Type Equations Reducible to the Second Painlevé Equation

It is shown that similarity solutions of the nonlinear Schrodinger equation with inhomogeneous term satisfy the second Pianleve equation, and a new non-linear partial differential equation reducible to the second Painleve equation can be derived through the similarity variables.

Similarity Reduction of Modified Kadomtsev-Petviashvili Equation

Similarity solutions of the modified Kadomtsev-Petviashvili equation are investigated by using infinitesimal transformations of Lies method. It is shown that the modified Kadomtsev-Petviashvili equation can be transformed, by the first reduction, to modified Boussinesq-type, the modified K-dV equations and others.