耦合KdV方程的约化及求解

利用Clarkson和Kruskal(CK)直接方法,对耦合KdV方程进行相似约化,同时从李群出发对该约化方程作了群论解释.进一步地,借助Ablowitz-Ramani-Segur(ARS)算法对耦合方程展开Painlevé测试,找到了3个Painlevé可积模型.最后通过形变映射法,求得耦合KdV方程的准确解析解. 关键词： 耦合KdV方程 CK直接法 Painlevé分析法 准确解析解     

A simple-looking relative of the Novikov,Hirota-Satsuma and Sawada-Kotera equations

We study the simple-looking scalar integrable equation f_xxt 3( f_x f_t 1) = 0, which is related (in different ways) to the Novikov, Hirota-Satsuma and Sawada-Kotera equations. For this equation we present a Lax pair, a Bäcklund transformation, soliton and merging soliton solutions (some exhibiting instabilities), two infinite hierarchies of conservation laws, an infinite hierarchy of continuous symmetries, a Painlevé series, a scaling reduction to a third order ODE and its Painlevé series, and the Hirota form (giving further multisoliton solutions).     

一类非线性色散方程中的新型奇异孤立波

研究一类非线性色散广义DGH方程的新型奇异孤立波及其Painlevé可积性.利用Painlevé分析发现当对流项强度m=2时广义DGH方程是可积的,这是一个新的可积方程.通过构造新的变量代换以及auto-Backlund变换获得该方程丰富的奇异孤立波解,如紧孤立波（compacton）、尖峰孤立波（peakon）、新型带尖点的双孤立波和带爆破点的双孤立波等. 关键词： 非线性色散方程 可积性 奇异孤立波     

Propagation characteristics of ion-acoustic double layer in multicomponent inhomogeneous auroral zone plasma

We examine the effect of dissipation on travelling waves in nonlinear dispersive systems modelled by Benjamin–Bona–Mahony (BBM)-like equations. In the absence of dissipation, the BBM-like equations are found to support soliton and compacton /anticompacton solutions depending on whether the dispersive term is linear or nonlinear. We study the influence of increasing nonlinearity of the medium on the soliton and compacton dynamics. The dissipative effect is found to convert the solitons either to undular bores or to shock-like waves depending on the degree of nonlinearity of the equations. The anticompacton solutions are also transformed to undular bores by the effect of dissipation. But the compactons tend to vanish due to viscous effects. The local oscillatory structures behind the bores and /or shock-like waves in the case of solitons and anticompactons are found to depend sensitively both on the coefficient of viscosity and solution of the unperturbed problem.     

Painlevé integrability and multi-dromion solutions of the 2+1 dimensional AKNS system

The Painlevé integrability of the 2+1 dimensional AKNS system is proved. Using the standard truncated Painlevé expansion which corresponds to a special B?cklund transformation, some special types of the localized excitations like the solitoff solutions, multi-dromion solutions and multi-ring soliton solutions are obtained. Received 31 January 2001 and Received in final form 15 May 2001     

INTEGRABILITY OF CERTAIN DEFORMED NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

A systematic investigation of certain higher order or deformed soliton equations with (1 + 1) dimensions, from the point of complete integrability, is presented. Following the procedure of Ablowitz, Kaup, Newell and Segur (AKNS) we find that the deformed version of Nonlinear Schrodinger equation, Hirota equation and AKNS equation admit Lax pairs. We report that each of the identified deformed equations possesses the Painlevé property for partial differential equations and admits trilinear representation obtained by truncating the associated Painlevé expansions. Hence the above mentioned deformed equations are completely integrable.     

Connection formulae for painlevé V functions

We study those Painlevé V equations which admit a one-parameter family of solutions analytic at the origin and solve the global problem of connecting the two-parameter asymptotic expansion at t → ∞ with the two-parameter expansion about t = 0. This connection is the one needed to study the correlation functions of the transverse Ising chain at the critical field strength. From these connection formulae, we derive a one-parameter connection formula for the general Painlevé II equation and a two-parameter connection formula for a class of Painlevé III equations.     

Localized structures for (2+1)-dimensional Boiti–Leon–Pempinelli equation

It is shown that Painlevé integrability of (2+1)-dimensional Boiti–Leon–Pempinelli equation is easy to be verified using the standard Weiss–Tabor–Carnevale (WTC) approach after introducing the Kruskal’s simplification. Furthermore, by employing a singular manifold method based on Painlevé truncation, variable separation solutions are obtained explicitly in terms of two arbitrary functions. The two arbitrary functions provide us a way to study some interesting localized structures. The choice of rational functions leads to the rogue wave structure of Boiti–Leon–Pempinelli equation. In addition, for the other choices, it is observed that two solitons may evolve into breather after interaction. Also, the interaction between two kink compactons is investigated.     

On similarity solutions of the Boussinesq equation

It is shown that the similarity solutions of the Boussinesq equation satisfy the first or second Painlevéequation. We also discuss properties of the soliton solution.     

Bäcklund transformations,kink soliton,breather- and travelling-wave solutions for a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation in fluid dynamics

In this paper, we investigate a (3+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation in fluid dynamics. Based on the Hirota method, we give a bilinear auto-Bäcklund transformation. Via the truncated Painlevé expansion, we get a Painlevé-type auto-Bäcklund transformation. With the aid of the symbolic computation, we derive some one- and two-kink soliton solutions. We present the oblique and parallel elastic interactions between the two-kink solitons. Via the extended homoclinic test technique, we construct some breather-wave solutions. Besides, we derive some lump solutions with the periods of the breather-wave solutions to the infinity. We observe that the shapes of a breather wave and a lump remain unchanged during the propagation. Based on the polynomial-expansion method, travelling-wave solutions are constructed.     

A variety of negative-order integrable KdV equations of higher orders

We develop a variety of negative-order integrable KdV equations of higher orders. We use the inverse recursion operator to construct these new equations. The complete integrability of each established equation is investigated via the Painlevé test, where each equation shows distinct branch of resonances. We use the simplified form of the Hirota’s direct method to obtain multiple soliton solutions for the generalized negative-order KdV equation.     

Compacton-like solutions for modified KdV and nonlinear Schrödinger equation with external sources

We present new types of compacton-like solutions for modified KdV and nonlinear Schrödinger equation with external sources, using a recently developed fractional transformation. In particular, we explicate these novel compactons for the trigonometric case, and compare their properties with those of the compactons and solitons in the case of modified KdV equation. Keeping in mind the significance of nonlinear Schrödinger equation with external source, for pulse propagation through asymmetric twin-core fibres, we hope that the newly found compacton may be launched in a long-haul telecommunication network utilizing asymmetric twin-core fibres.     

Double compactons in the Olver–Rosenau equation

It is showed that the fully nonlinear evolution equations of Olver and Rosenau can be reduced to Hamiltonian form by transformation of variables. The resulting Hamiltonian equations are treated by the dynamical systems theory and a phase-space analysis of their singular points is presented. The results of this study demonstrate that the equations can support double compactons. The new Olver–Rosenau compactons are different from the well-known Rosenau–Hyman compacton and Cooper–Shepard–Sodano compacton, because they are induced by a singular elliptic instead of singular straight line on phase-space.     

Two new integrable Kadomtsev–Petviashvili equations with time-dependent coefficients: multiple real and complex soliton solutions

ABSTRACT

In this work, we develop two new integrable Kadomtsev–Petviashvili (KP) equations with time-dependent coefficients. The integrability property of each equation is explicitly demonstrated exhibiting the Painlevé test to confirm its integrability. Moreover, each equation admits multiple real and multiple complex soliton solutions. We introduce complex forms of the simplified Hirota's method to derive multiple complex soliton solutions. These two model equations are likely to be of applicative relevance, because it may be considered an application of a large class of nonlinear KP equations.     

New integrable (2+1)- and (3+1)-dimensional sinh-Gordon equations with constant and time-dependent coefficients

In this work, we mainly address two new integrable (2+1)- and (3+1)-dimensional sinh-Gordon equations, which naturally appear in surface theory and fluid dynamics. The first equation includes constant coefficients, while the other is characterized with time-dependent coefficients. It is of further value to investigate the integrability of each model. This study puts forward a Painlevé test to reveal the Painlevé integrability. We show that the first equation passes the Painlevé test to confirm its integrability. However, the compatibility conditions of the second model with time-dependent coefficients provides the relation between these coefficients to ensure its integrability. We show that the dispersion relations of the two equations are distinct, whereas the phase shifts are identical. We apply the simplified Hirota's method where four sets of multiple soliton are derived for these equations.     

Lie point symmetries and ODEs passing the Painlevé test

The Lie point symmetries of ordinary differential equations (ODEs) that are candidates for having the Painlevé property are explored for ODEs of order n = 2,?. . .?, 5. Among the 6 ODEs identifying the Painlevé transcendents only P_III, P_V and P_VI have nontrivial symmetry algebras and that only for very special values of the parameters. In those cases the transcendents can be expressed in terms of simpler functions, i.e. elementary functions, solutions of linear equations, elliptic functions or Painlevé transcendents occurring at lower order. For higher order or higher degree ODEs that pass the Painlevé test only very partial classifications have been published. We consider many examples that exist in the literature and show how their symmetry groups help to identify those that may define genuinely new transcendents.     

Contiguity relations for discrete and ultradiscrete Painlevé equations

Abstract

We show that the solutions of ultradiscrete Painlevé equations satisfy contiguity relations just as their continuous and discrete counterparts. Our starting point are the relations for q-discrete Painlevé equations which we then proceed to ultradiscretise. In this paper we obtain results for the one-parameter q-P_III, the symmetric q-P_IV and the q-P_IV. These results show that there exists a perfect parallel between the properties of continuous, discrete and ultradiscrete Painlevé equations.     

The integrable Vakhnenko–Parkes (VP) and the modified Vakhnenko–Parkes (MVP) equations: Multiple real and complex soliton solutions

In this work, we show that the integrable Vakhnenko–Parkes (VP) equation passes the Painlevé test and admits multiple real and multiple complex soliton solutions. We also present, for the first time, the modified Vakhnenko-Parkes (MVP) equation, show its complete integrability, and formally derive its multiple real and multiple complex soliton solutions. To achieve the goal set for this work, we introduce two complex forms of the simplified Hirota’s method, the first works effectively for the VP equation, and the other form is nicely applicable for the MVP equation. We believe that establishing the complex forms will shed light on complex solitons of other integrable equations.     

Self-similar radiation from numerical Rosenau–Hyman compactons

The numerical simulation of compactons, solitary waves with compact support, is characterized by the presence of spurious phenomena, as numerically induced radiation, which is illustrated here using four numerical methods applied to the Rosenau–Hyman K(p, p) equation. Both forward and backward radiations are emitted from the compacton presenting a self-similar shape which has been illustrated graphically by the proper scaling. A grid refinement study shows that the amplitude of the radiations decreases as the grid size does, confirming its numerical origin. The front velocity and the amplitude of both radiations have been studied as a function of both the compacton and the numerical parameters. The amplitude of the radiations decreases exponentially in time, being characterized by a nearly constant scaling exponent. An ansatz for both the backward and forward radiations corresponding to a self-similar function characterized by the scaling exponent is suggested by the present numerical results.     

On a revisit to the Painlevé test for integrability and exact solutions for Yang’s self-dual equations forSU (2) gauge fields

Painlevé test (Jimboet al [1]) for integrability for the Yang’s self-dual equations forSU(2) gauge fields has been revisited. Jimboet al analysed the complex form of the equations with a rather restricted form of singularity manifold. They did not discuss exact solutions in that context. Here the analysis has been done starting from the real form of the same equations and keeping the singularity manifold completely general in nature. It has been found that the equations, in real form, pass the Painlevé test for integrability. The truncation procedure of the same analysis leads to non-trivial exact solutions obtained previously and auto-Backlund transformation between two pairs of those solutions