Masayuki Oikawa

Interactions of Solitary Waves –A Perturbation Approach to Nonlinear Systems–

An approximate method of studying interactions between two solitary waves which propagate in opposite directions is presented. In the first approximation, the solution is described by s superposition of two solitary waves which are governed by their respective Korteveg-de Vries equation. The second order approximation gives a small correction where the two waves overlap one another. The method is extended to the system, in which there exist n “quasi-simple” waves (the simple waves under the effects of higher derivative terms, such as dispersions of dissipations). The possibility that n “quasi-simple” waves can be superposed to describe nonlinear systems is studied. Applications to ion acoustic waves in collisionless plasmas and shallow water waves are discussed.

Long Internal Waves of Large Amplitude in a Two-Layer Fluid

Long internal waves of large amplitude are studied for a two-layer fluid with a rigid upper boundary. Steady periodic waves are calculated numerically assuming that mean velocities for each layer are equal. In the limit of long wavelength, the waves asymptote to either of a non-dissipative internal bore or a solitary wave of elevation or depression. The conditions for the appearance of each solution are obtained numerically, and one of the conditions is complemented by an analysis based on the conservations of mass, momentum and wave energy. Then it is shown that the interface for the solitary wave of large amplitude is not allowed to cross a critical level similarly to the case of small amplitude. Here the critical level is defined by the distance \(\sqrt{\Delta}h/(1+\sqrt{\Delta})\) from the bottom, and h is the interval between two boundaries, Δ=ρ 2 /ρ 1 and ρ 1 and ρ 2 are densities of upper and lower layers. This result gives a criterion for the largest amplitude of a solitary wave.

Two-Dimensional Resonant Interaction between Long and Short Waves

Two-dimensional resonant interaction between an internal gravity wave and a surface gravity wave packet in a two-layer fluid is investigated. The equations describing this interaction are derived. Modulational instability of a plane wave solution of the quations is discussed. Some interesting solutions which follow from a two-soliton solution to these equations are given.

Soliton solutions of the KP equation with V-shape initial waves

We consider the initial value problems of the Kadomtsev-Petviashvili (KP) equation for symmetric V-shape initial waves consisting of two semi-infinite line solitons with the same amplitude. Numerical simulations show that the solutions of the initial value problem approach asymptotically to certain exact solutions of the KP equation found recently in [Chakravarty and Kodama, JPA, 41 (2008) 275209]. We then use a chord diagram to explain the asymptotic result. We also demonstrate a real experiment of shallow water wave which may represent the solution discussed in this Letter.

A Perturbation Approach to Nonlinear Systems. II. Interaction of Nonlinear Modulated Waves

The reductive perturbation method is extended to apply to a strongly dispersive system, in which mudulated plane waves interact each other through nonlinear interactions. The interaction between two envelope solitons is examined.

Oblique interaction of solitons in an extended Kadomtsev-Petviashvili equation

Oblique interaction of two solitons of the same amplitude in an extended Kadomtsev–Petviashvili (EKP) equation, which is a weakly two-dimensional generalization of an extended Korteweg–de Vries (EKdV) equation, is investigated. This interaction problem is solved numerically under the initial and boundary condition simulating the reflection problem of the obliquely incident soliton due to a rigid wall. The essential parameters are given by Q * ≡ a Q and Ω * ≡Ω/ a 1/2 . Here, Q is the coefficient of the cubic nonlinear term in the EKP quation, a the amplitude of the incident soliton and Ω≡tan θ i , θ i being the angle of incidence. The numerical solutions for various values of these parameters reveal the effect of the cubic nonlinear term on the behavior of the waves generated by the interaction. When Q * is small, the interaction property is very similar to that of the Kadomtsev–Petviashvili equation. Especially, for relatively small Ω * , a new wave of large amplitude and of soliton profile called “stem” ...

Two-component analogue of two-dimensional long wave-short wave resonance interaction equations: A derivation and solutions

The two-component analogue of two-dimensional long wave–short wave resonance interaction equations is derived in a physical setting. Wronskian solutions of the integrable two-component analogue of two-dimensional long wave–short wave resonance interaction equations are presented.

Generalized Casorati determinant and positon-negaton-type solutions of the Toda lattice equation

A set of conditions is presented for Casorati determinants to give solutions to the Toda lattice equation. It is used to establish a relation between the Casorati determinant solutions and the generalized Casorati determinant solutions. Positons, negatons and their interaction solutions of the Toda lattice equation are constructed through the generalized Casorati determinant technique. A careful analysis is also made for general positons and negatons, the resulting positons and negatons of order one being explicitly computed. The generalized Casorati determinant formulation for the two dimensional Toda lattice (2dTL) equation is presented. It is shown that positon, negaton and complexiton type solutions in the 2dTL equation exist and these solutions reduce to positon, negaton and complexiton type solutions in the Toda lattice equation by the standard reduction procedure.

Casorati determinant solution for the discrete-time relativistic Toda lattice equation

Abstract The discrete-time relativistic Toda lattice (dRTL) equation is investigated by using the bilinear formalism. Bilinear equations are systematically constructed with the aid of the singularity confinement method. It is shown that the dRTL equation is decomposed into the Backlund transformations of the discrete-time Toda lattice equation. The N -soliton solution is explicitly constructed in the form of the Casorati determinant.

Two-Dimensional Interaction of Internal Solitary Waves in a Two-Layer Fluid

Two-dimensional interaction of internal solitary waves is studied for a two-layer fluid with a thickness ratio close to a certain critical one depending on the density ratio. It is described by a Modified Kadomtsev-Petviashvili (MKP) equation, provided the propagation directions of the waves are close to each other. The MKP equation is investigated numerically and it is found that the interaction almost satisfying the condition of soliton resonance occurs when the wave amplitudes are small. The interaction is qualitatively similar to the soliton resonance in the Kadomtsev-Petviashvili (KP) equation except that a deformation of the newly generated waves and a radiation can be seen.