The Bäcklund and the Galilei Invariant Transformations Constructed by Similarity Variables for Soliton Equations

Abstract

The Painlevé-test has been applied to checking the integrability of nonlinear PDEs, since similarity solutions of many soliton equations satisfy the Painlevé equation. As is well known, such similarity solutions can be obtained by the infinitesimal transformation, that is, the classical similarity analysis, and also the dimension of the PDEs can be reduced.

In this paper, the KdV, the mKdV, and the nonlinear Schrödinger equations are considered and are transformed into equations with loss and/or nonuniformity by transformations constructed on a basis of the local similarity variables. The transformations include the Bäcklund and the Galilei invariant ones. It should be noticed that the approach is applicable to other PDEs and for nonlocal similarity variables.     

Bogolubov transformations and completeness

It is well-known that two complete, orthonormal sets of solutions of the Klein-Gordon equation are related by invertible Bogolubov transformations and that the Bogolubov coefficients therefore satisfy certain identities. We show that the converse is false, namely, that the fact that the Bogolubov coefficients defined by two sets of solutions satisfy these identities doesnot imply that either set can be expanded in terms of the other. Several simple counterexamples are given.     

A Generalization of the Sine-Gordon Equation to 2 + 1 Dimensions

Abstract

The Singular Manifold Method (SMM) is applied to an equation in 2 + 1 dimensions [13] that can be considered as a generalization of the sine-Gordon equation. SMM is useful to prove that the equation has two Painlevé branches and, therefore, it can be considered as the modified version of an equation with just one branch, that is the AKNS equation in 2 + 1 dimensions. The solutions of the former split as linear superposition of two solutions of the second, related by a B¨acklund-gauge transformation. Solutions of both equations are obtained by means of an algorithmic procedure derived from these transformations.     

Rational solutions of the general nonlinear Schrödinger equation with derivative

The double Wronskian solutions whose entries satisfy matrix equation of the general nonlinear Schrödinger equation with derivative (GDNLSE) are derived through the Wronskian technique. Soliton solutions and rational solutions of GDNLSE are obtained by taking special cases in general solutions.     

非线性演化方程椭圆函数解的一种简单求法及其应用

非线性演化方程的许多行波解可以写成满足投影Riccati方程的两个基本函数的多项式形式.利用这一性质，通过建立一般的椭圆方程与投影Riccati方程解之间的关系，导出了一个构造这些解的新方法.该方法对类型Ⅰ的方程和类型Ⅱ的方程均有效，同时也回答了如何求出非线性演化方程分式形式椭圆函数解的问题. 关键词： 非线性演化方程 椭圆函数解     

关于铁磁链方程的反散射解法的严格证明和验证

本文对于通常证明中引入的造成困难的因子λ^-1,证明不仅应当略去,而且在略去后,同样可以得到与Takhtajan方程等价的结果,只在λ=0处有预料到的修正。同时,给出了反散射解法的解满足铁磁链方程的普遍验证。 关键词：     

On the construction of analytic and crossing symmetric partial waves

The problem of construction of analytic and crossing symmetric partial waves leads to a generalized Riesz problem. In some cases nontrivial solutions can be constructed from the solutions of a homogeneous Fredholm integral equation. It is suggested that the unitary partial waves satisfy a similar equation.Work performed under contract with the Rumanian Nuclear Energy Committee.     

Propagation of gravitational waves in Robertson-Walker backgrounds

The electric and magnetic parts of the linearized Weyl tensor, when the stress-energy tensor is that of a perfect fluid and the background is of Robertson-Walker type, are known to satisfy wave equations that differ by the presence of a source term for the electric part. It is shown here that all of the allowed solutions of the inhomogeneous equation can be obtained by applying a differential operator to the solutions of the homogeneous equation; consequently, electric-type and magnetic-type gravitational waves have the same propagation properties. The results of a complete integration of the appropriately linearized Newman-Penrose equations are given.     

Formal Linearization and Exact Solutions of Some Nonlinear Partial Differential Equations

Abstract

An efficient method for constructing of particular solutions of some nonlinear partial differential equations is introduced. The method can be applied to nonintegrable equations as well as to integrable ones. Examples include multisoliton and periodic solutions of the famous integrable evolution equation (KdV) and the new solutions, describing interaction of solitary waves of nonintegrable equation.     

INVARIANCE OF THE KAUP–KUPERSHMIDT EQUATION AND TRIANGULAR AUTO-BÄCKLUND TRANSFORMATIONS

We report triangular auto-Bäcklund transformations for the solutions of a fifth-order evolution equation, which is a constraint for an invariance condition of the Kaup–Kupershmidt equation derived by E. G. Reyes in his paper titled "Nonlocal symmetries and the Kaup–Kupershmidt equation" [J. Math. Phys. 46 (2005) 073507, 19 pp.]. These auto-Bäcklund transformations can then be applied to generate solutions of the Kaup–Kupershmidt equation. We show that triangular auto-Bäcklund transformations result from a systematic multipotentialization of the Kupershmidt equation.     

Nonlinear theory of transverse perturbations of quasi-one-dimensional solitons

An averaged variational principle is applied to analyze the nonlinear effect of transverse perturbations (including diffraction) on quasi-one-dimensional soliton propagation governed by various wave equations. It is shown that parameters of the spatiotemporal solitons described by the cubic Schrödinger equation and the Yajima-Oikawa model of interaction between long-and short-wavelength waves satisfy the spatial quintic nonlinear Schrödinger equation for a complex-valued function composed of the amplitude and eikonal of the soliton. Three-dimensional solutions are found for two-component “bullets” having long-and short-wavelength components. Vortex and hole-vortex structures are found for envelope solitons and for two-component solitons in the regime of resonant long/short-wave coupling. Weakly nonlinear behavior of transverse perturbations of one-dimensional soliton solutions in a self-defocusing medium is described by the Kadomtsev-Petviashvili equation. The corresponding rationally localized “lump” solutions can be considered as secondary solitons propagating along the phase fronts of the primary solitons. This conclusion holds for primary solitons described by a broad class of nonlinear wave equations.     

From 2-dimensional surfaces to cosmological solutions

We construct perfect fluid metrics with two symmetries by means of a recently developed geometrical method [1]. The Einstein equations are reduced to a single equation for a conformal factor. Under additional assumptions we obtain new cosmological solutions of Bianchi type II, VI₀ and VII₀. The solutions depend on an arbitrary function of time, which can be specified in order to satisfy an equation of state.     

Free vibration analysis of the completely free rectangular plate by the method of superposition

While the subject of free vibration analysis of the completely free rectangular plate has a history which goes back nearly two centuries it remains a fact that most theoretical solutions to this classical problem are considered to be at best approximate in nature. This is because of the difficulties which have been encountered in trying to obtain solutions which satisfy the free edge conditions as well as the governing differential equation. In a new approach to this problem, by using the method of superposition, it is shown that solutions which satisfy identically the differential equation and which satisfy the boundary conditions with any desired degree of accuracy are obtained. Eigenvalues of four digit accuracy are provided for a wide range of plate aspect ratios and modal shapes. Exact delineation is made between the three families of modes which are characteristic of this plate vibration problem. Accurate modal shapes are provided for the response of completely free square plates.     

Linear superposition solutions to nonlinear wave equations

The solutions to a linear wave equation can satisfy the principle of superposition,i.e.,the linear superposition of two or more known solutions is still a solution of the linear wave equation.We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic,triangle,and exponential functions,and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structural characteristics.The linear superposition solutions to the generalized KdV equation K(2,2,1),the Oliver water wave equation,and the k(n,n) equation are given.The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed,and the reason why the solutions with the forms of hyperbolic,triangle,and exponential functions can form the linear superposition solutions is also discussed.     

2-Soliton-Solution of the Novikov-Veselov Equation

Based on a superposition method recently proposed to obtain 1-solitary wave solutions of the KdV-Burgers equation (Yuanxi and Jiashi, 2005, International Journal of Theoretical Physics 44, 293–301), we show that this method can also be used to find a 2-solitary wave solution of the Novikov-Veselov equation. Thus, it seems that the method of Yuanxi and Jiashi in general is not restricted to constructing 1-solitary wave solutions of nonlinear wave and evolution equations (NLWEEs).     

A boundary integral method to compute Green's functions for the radiative transport equation

The Green's function for the time-independent radiative transport equation in the whole space can be computed as an expansion in plane wave solutions. Plane wave solutions are a general class of solutions for the radiative transport equation. Because plane wave solutions are not known analytically in general, we calculate them numerically using the discrete ordinate method. We use the whole space Green's function to derive boundary integral equations. Through the solution of the boundary integral equations, we compute the Green's function for bounded domains. In particular we compute the Green's function for the half space, the slab, and the two-layered half space. The boundary conditions used here are in their most general form. Hence, this theory can be applied to boundaries with any kind of reflection and transmission law.     

Multi-bump, self-similar, blow-up solutions of the Ginzburg-Landau equation

For the Ginzburg-Landau equation (GL), we establish the existence and local uniqueness of two classes of multi-bump, self-similar, blow-up solutions for all dimensions 2<d<4 (under certain conditions on the coefficients in the equation). In numerical simulation and via asymptotic analysis, one class of solutions was already found; the second class of multi-bump solutions is new.In the analysis, we treat the GL as a small perturbation of the cubic nonlinear Schrödinger equation (NLS). The existence result given here is a major extension of results established previously for the NLS, since for the NLS the construction only holds for d close to the critical dimension d=2.The behaviour of the self-similar solutions is described by a nonlinear, non-autonomous ordinary differential equation (ODE). After linearisation, this ODE exhibits hyperbolic behaviour near the origin and elliptic behaviour asymptotically. We call the region where the type of behaviour changes the mid-range. All of the bumps of the solutions that we construct lie in the mid-range.For the construction, we track a manifold of solutions of the ODE that satisfy the condition at the origin forward, and a manifold of solutions that satisfy the asymptotic conditions backward, to a common point in the mid-range. Then, we show that these manifolds intersect transversely. We study the dynamics in the mid-range by using geometric singular perturbation theory, adiabatic Melnikov theory, and the Exchange Lemma.     

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.