水面波の弱非線形モデルの一つであるDavey-Stewartson(DS)方程式は多様な厳密解を持つことが知られている. 特に, DS2方程式と呼ばれる場合にはダーク型線ソリトン解が存在し, それらが相互作用する多ダーク型線ソリトン解も存在することが知られているが, 多ダーク型線ソリトン相互作用のより詳しい解析はこれまで殆どなされていない. 本稿では, KPソリトン理論を基礎にしたDS2方程式のダーク型線ソリトン相互作用のより詳しい理論解析について報告する.

In this paper, we propose integrable semi-discrete and full-discrete analogues of the short pulse (SP) equation. The key construction is the bilinear form and determinant structure of solutions of the SP equation. We also give the determinant formulas of N-soliton solutions of the semi-discrete and full-discrete analogues of the SP equations, from which the multi-loop and multi-breather solutions can be generated. In the continuous limit, the full-discrete SP equation converges to the semi-discrete SP equation, and then to the continuous SP equation. Based on the semi- discrete SP equation, an integrable numerical scheme, i.e. a self-adaptive moving mesh scheme, is proposed and used for the numerical computation of the short pulse equation.

The line-soliton solutions of the Kadomtsev-Petviashvili (KP) equation are investigated in this article using the -function formalism. In particular, the Wronskian and the Grammian forms of the -function are discussed, and the equivalence of these two forms are established. Furthermore, the interaction properties of two special types of 2-soliton solutions of the KP equation are studied in details.

An alternate form of discrete potential Boussinesq (BSQ) equation is proposed and its multisoliton solutions are constructed. An ultradiscrete potential BSQ equation is also obtained from the discrete potential BSQ equation using the ultradiscretization technique. The detail of the multisoliton solutions is discussed by using the reduction technique. The lattice potential BSQ equation derived by Nijhoff et al. is also investigated by using the singularity confinement test. The relation between the proposed alternate discrete potential BSQ equation and the lattice potential BSQ equation by Nijhoff et al. is clarified.

An integrable two-component analogue of the two-dimensional long wave - short wave resonance interaction (2c-2d-LSRI) system is studied. Wroliskian solutions of 2c-2d-LSRI system are presented. A reduced case, which describes resonant interaction between an interfacial wave and two surface wave packets a two-layer fluid, is also discussed.

The strong coupling limits of the integrable semi-discrete and fully discrete nonlinear Schrodinger systems are studied by using the Hirota bilinear method. The determinant solutions (in both infinite and finite lattice cases) for the strong coupling limits of semi-discrete and fully discrete nonlinear Schrodinger systems are obtained using a determinant technique. The vector generalizations of the strong coupling limits of semi-discrete and fully discrete nonlinear Schrodinger systems are also presented. The Pfaffian solutions for vector systems are obtained using the Pfaffian technique.

An integrable semi-discretization of the Camassa-Holm (CH) equation is presented. The keys of its construction are bilinear forms and determinant structure of solutions of the CH equation. Determinant formulas of N-soliton solutions of the continuous and semi-discrete Camassa-Holm equations are presented. Based on determinant formulas, we can generate multi-soliton, multi-cuspon and multi-soliton-cuspon solutions. Numerical computations using the integrable semi-discrete Camassa-Holm equation are performed. It is shown that the integrable semi-discrete Camassa-Holm equation gives very accurate numerical results even in the cases of cuspon-cuspon and soliton cuspon interactions. The numerical computation for an initial value condition, which is not an exact solution, is also presented.

An integrable (2 + 1)-dimensional nonlocal nonlinear Schrodinger equation is discussed. The N-soliton solution is given by Gram type determinant. It is found that the localized N-soliton solution has interesting interaction behavior which shows change of amplitude of localized pulses after collisions. Published by Elsevier B.V.

In this paper, we propose integrable discretizations of a two-dimensional Hamiltonian system with a quartic potential. Using either the method of separation of variables or the method based on bilinear forms, we construct the corresponding integrable mappings for the first three among four integrable cases.

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.

is shown that the N-dark soliton solutions of the integrable discrete nonlinear Schrodinger (IDNLS) equation are given in terms of the Casorati determinant. The conditions for reduction, complex conjugacy, and regularity for the Casorati determinant solution are also given explicitly. The relationship between the IDNLS and the relativistic Toda lattice is discussed.

We study soliton solutions to the DKP equation which is defined by the Hirota bilinear form, where τ0 ≤ 1. The τ-functions τn are given by the Pfaffians of a certain skew-symmetric matrix. We identify a one-soliton solution as an element of the Weyl group of D-type, and discuss a general structure of the interaction patterns among the solitons. Soliton solutions are characterized by a 4N × 4N skew-symmetric constant matrix which we call the B-matrix. We then find that one can have M-soliton solutions with M being any number from N to 2N - 1 for some of the 4N × 4N B-matrices having only 2N nonzero entries in the upper-triangular part (the number of solitons obtained from those B-matrices was previously expected to be just N). © 2006 IOP Publishing Ltd.

We study, analytically, the discrete complex cubic-quintic Ginzburg-Landau (dCCQGL) equation with a non-local quintic term. We find a set of exact solutions which includes, as particular cases, bright and dark soliton solutions, constant magnitude solutions with phase shifts, periodic solutions in terms of elliptic Jacobi functions in general forms, and various particular periodic solutions. © 2005 Elsevier B.V. All rights reserved.

We present a class of solutions of the two-dimensional Toda lattice equation, its fully discrete analogue and its ultra-discrete limit. These solutions demonstrate the existence of soliton resonance and web-like structure in discrete integrable systems such as differential-difference equations, difference equations and cellular automata (ultra-discrete equations).

A set of coupled conditions consisting of differential-difference equations is presented for Casorati determinants to solve the Toda lattice equation. One class of the resulting conditions leads to an approach for constructing complexiton solutions to the Toda lattice equation through the Casoratian formulation. An analysis is made for solving the resulting system of differential-difference equations, thereby providing the general solution yielding eigenfunctions required for forming complexitons. Moreover, a feasible way is presented to compute the required eigenfunctions, along with examples of real complexitons of lower order. © 2004 Elsevier B.V. All rights reserved.

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. ©2004 The Physical Society of Japan.

We study, analytically, the discrete complex cubic Ginzburg-Landau (dCCGL) equation. We derive the energy balance equation for the dCCGL and consider various limiting cases. We have found a set of exact solutions which includes as particular cases periodic solutions in terms of elliptic Jacobi functions, bright and dark soliton solutions, and constant magnitude solutions with phase shifts. We have also found the range of parameters where each exact solution exists. We discuss the common features of these solutions and solutions of the continuous complex Ginzburg-Landau model and solutions of Hamiltonian discrete systems and also their differences. © 2003 Elsevier Science B.V. All rights reserved.

We study a new quintic discrete nonlinear Schrödinger (QDNLS) equation which reduces naturally to an interesting symmetric difference equation of the form φn+1 + φn-1 = F(φn). Integrability of the symmetric mapping is checked by singularity confinement criteria and growth properties. Some new exact localized solutions for integrable cases are presented for certain sets of parameters. Although these exact localized solutions represent only a small subset of the large variety of possible solutions admitted by the QDNLS equation, those solutions presented here are the first example of exact localized solutions of the QDNLS equation. We also find chaotic behavior for certain parameters of nonintegrable case. © 2003 Elsevier Science B.V. All rights reserved.

Using a direct ansatz approach, we have found a number of periodic zero-velocity analytic solutions of the complex quintic Swift-Hohenberg equation (CSHE). These find application in assorted optical problems. Particular cases of periodic solutions, where the elliptic function modulus equals 1, are various localized solutions of the CSHE. Each of these solutions exists for a certain relation between the parameters of the equation. As a result, they are particular cases of the complete set of periodic and localised solutions which may exist for this equation. In fact, they are multi-parameter families of solutions and they can serve as a seeding set of solutions which could be useful in other optical studies. We have also derived energy and momentum balance equations for the solutions of CSHE and checked that our stationary solutions satisfy the energy balance equation. © 2003 Elsevier Science B.V. All rights reserved.

Using Painlevé analysis, the Hirota multi-linear method and a direct ansatz technique, we study analytic solutions of the (1+1)-dimensional complex cubic and quintic Swift-Hohenberg equations. We consider both standard and generalized versions of these equations. We have found that a number of exact solutions exist to each of these equations, provided that the coefficients are constrained by certain relations. The set of solutions include particular types of solitary wave solutions, hole (dark soliton) solutions and periodic solutions in terms of elliptic Jacobi functions and the Weierstrass ℘ function. Although these solutions represent only a small subset of the large variety of possible solutions admitted by the complex cubic and quintic Swift-Hohenberg equations, those presented here are the first examples of exact analytic solutions found thus far. © 2002 Elsevier Science B.V. All rights reserved.

© 2003 IEEE. We study, analytically, the discrete complex cubic Ginzburg-Landau (dCCGL) equation. We have found a set of exact solutions which includes bright and dark soliton solutions,

Copyright © 2002 by K Maruno and W X Ma. Hirota's bilinear technique is applied to some integrable lattice systems related to the Bäcklund transformations of the 2DToda, Lotka-Volterra and relativistic Lotka- Volterra lattice systems, which include the modified Lotka-Volterra lattice system, the modified relativistic Lotka-Volterra lattice system, and the generalized Blaszak- Marciniak lattice systems. Determinant solutions are constructed through the resulting bilinear forms, especially for the modified relativistic Lotka-Volterra lattice system and a two-dimensional Blaszak-Marciniak lattice system.

The relativistic Lotka-Volterra (RLV) lattice and the discrete-time relativistic Lotka-Volterra (dRLV) lattice are investigated by using the bilinear formalism. The bilinear equations for them are systematically constructed with the aid of the singularity confinement test. It is shown that the RLV lattice and dRLV lattice are decomposed into the Backlund transformations of the Toda lattice system. The N-soliton solutions are explicitly constructed in the form of the Casorati determinant. (C) 2000 Elsevier Science B.V.

Bilinear equations can be obtained for discrete soliton equations using the singularity confinement test. This shows that the singularity confinement test is useful for detecting integrability and in constructing solutions. Thus, the singularity confinement test is truly a powerful tool for studying discrete integrable systems.

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 Bäcklund transformations of the discrete-time Toda lattice equation. The N-soliton solution is explicitly constructed in the form of the Casorati determinant. © 1998 Published by Elsevier Science B.V.

The singularity confinement method is applied to the systematic derivation of the bilinear equations for discrete soliton equations. Using the bilinear forms, the N-soliton and algebraic solutions of the discrete potential mKdV equation are constructed. © 1997 Published by Elsevier Science B.V.

The mixing process in stratified fluid in the closed region of the ocean by the recirculating flow induced by wind is modelled in the laboratory using the lid-driven cavity flow. The deepening of a mixed layer into a region of constant density gradient is examined and the three-dimensional features of flow structures near a density interface is investigated by visualization experiments. The characteristics of the mixed layer and the stratified layer are determined using a conductivity probe. The flow pattern of a primary circulation formed in the top downstream corner, growing gradually, but bounded by a stratified fluid beneath as it were a wall is quite different from that of homogeneous fluid. The density interface formed by erosion of the basic density gradient is distorted by the primary circulation and at the same time it is subject to the three-dimensional instability in the initial stage of the formation. Then the interface is of a wave shape in the spanwise direction, sharper near the crests and flatter in troughs, the wave length of which increases as the interface descends. It is shown that this is caused by the existence of the vortical structure which consists of the pairs of counter-rotating wise stream vortices in the strong shear layer near the interface. The considerable amount of the spanwise distortion of the interface indicates that the three-dimensional structure possibly makes a significant contribution to the mixing across the density interface along with the primary circulation in the upper layer.