We consider the nonlinear complex d'Alembert equation Square Operator Psi =F( mod Psi mod ) Psi with Psi represented in terms of amplitude and phase, in (1+n)-dimensional Minkowski space. We exploit a compatible d'Alembert-Hamilton system to construct new types of exact solutions for some nonlinearities.
Certain nonlinear evolution PDEs in 1 + 1 variables (time and space) are identified, featuring a positive parameter ω and evolving, for a large class of initial data, periodically with the fixed period T = 2 π / ω (or perhaps over(T, ̃) = p T with p a small integer). They are autonomous (i.e., they do not feature any explicit dependence on the time variable), but they generally (although not quite all of them) depend explicitly on the space variable hence are not translation-invariant. They are integrable, having been obtained by applying an appropriate change of dependent and independent variables to certain nonlinear evolution PDEs whose integrable character has been recently ascertained. Solutions of some of these PDEs are exhibited.
The techniques of an invertible point transformation and the Painleve analysis can be used to construct integrable ordinary differential equations. The authors compare both techniques for anharmonic oscillators.
The anharmonic oscillator d2x/dt2 + f1(t)dx/dt + f2(t)x + f3(t)xn = 0 (n ≠ 0, 1) is investigated applying the invertible point transformation, Lie symmetries and the Painlevé test.
We apply a list of criteria which leads to a class of fifth-order symmetry-integrable evolution equations. The recursion operators for this class are given explicitly. Multipotentialisations are then applied to the equations in this class in order to extend this class of integrable equations.
Dieses Buch ist der erste Teil einer dreiteiligen Serie mit dem Titel Aufgaben, Theorieund L¨osungen zur Linearen Algebra. Dieser erste Teil behandelt Vektoren des euklidischen Raumes sowie Matrizen, Matrixalgebra und Systeme von linearen Gleichungen. Wir l¨osen lineare System mit Hilfe des Gaußschen Eliminationsverfahrens und auf anderem Wege und untersuchen die Eigenschaften dieser Systeme in Bezug auf Vektoren und Matrizen. Dar¨uber hinaus betrachten wir lineare Abbildungen und berechnendie Standardmatrizen dieser Abbildungen.
We classify all 1st-order integrating factors and the correspondingconservation laws for a class of Camassa-Holm type equations.
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'' [ Math. Phys., 46, 073507, 19 pp., 2005]. 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 multipotentialisation of the Kupershmidt equation.
We reduce the nonlinear wave equation □nu = αF[exp(βu)] to ordinary differential equtions and construct exact solutions, by the use of a compatible d'Alembert-Hamilton system. The solutions of these ordinary differential equations, together with the solutions of the corresponding d'Alembert-Hamilton equations, provide a rich class of exact solutions of the multidimensional wave equations. The wave equations are studied in n-dimensional Minkowski space.
We consider symmetry‐integrable evolution equations in 1 + 1 dimensions of order 3 and order 5. We show that there exist only three equations in this class that are invariant under the Möbius transformation, and we name those Schwarzian equations. We report an interesting relation between the recursion operators of the Schwarzian equations and the corresponding adjoint operators that generate hierarchies of Schwarzian systems in terms of the Schwarzian derivative. This indicates a deep relation between the Schwarzian equations and the Schwarzian derivative. A classification of the fully nonlinear third‐order Schwarzian equations is also reported.
This book is the first part of a three-part series titled Problems, Theory and Solutions in Linear Algebra. This first part contains over 100 solved problems and 100 exercises on vectors, matrices, linear systems, as well as linear transformations in Euclidean space. It is intended as a supplement to a textbook in Linear Algebra and the aim of the series it to provide the student with a well-structured and carefully selected set of solved problems as well as a thorough revision of the material taught in a course on this subject for undergraduate engineering and science students.
We report the recursion operators for a class of symmetry integrable evolution equa- tions of third order which admit a fourth-order integrating factor. Under some as- sumptions we obtain the complete list of equations, one of which is a special case of the Schwarzian Korteweg-de Vries equation.
In this paper the nonlinear wave equation $\partial^2 u/\partial x_0^2-\partial^2 u/\partial x_1^2+f(x_0,x_1,u)=0$, where $f$ is an arbitrary smooth function of its arguments, is considered from the symmetry standpoint. The form of the most general Lie point symmetry generator of this equation is obtained. The classes of functions $f$, for which the equation in question admits a one-parameter Lie point symmetry group, are constructed. Then, the authors investigate the possible form of generators of conformal transformations, assuming the usual form of generators of Lorentz and scaling transformations, and study the wave equations invariant under such operators. The symmetry groups of obtained equations are used for the construction of ansätze and reductions of these equations to ordinary differential equations. $Q$-conditional (nonclassical) symmetries of the wave equation are also considered. Namely, the determining equations for the coefficients of a $Q$-conditional symmetry operator are found and their compatibility is investigated.
Ce livre est le premier d'une série de trois ouvrages intitulés Théorie et Problèmes Résolus d'algèbre linéaire. Cette première partie contient plus de 100 problèmes résolus et plus de 100 exercices sur les vecteurs en espaces euclidiens, les matrices, les systèmes linéaires ainsi que les applications linéaires entre espaces euclidiens. Le but de cette série est de fournir aux étudiants de filières scientifiques et techniques un ensemble structuré de problèmes soigneusement choisis ainsi qu'une opportunité d'approfondir leurs connaissances acquises en cours d'algèbre linéaire.
Summary: We study three methods, based on continuous symmetries, to find approximate solutions for the multidimensional nonlinear heat equation $\partial u/\partial x_0+ \Delta u= au^n+ \varepsilon f(u)$, where $a$ and $n$ are arbitrary real constants, $f$ is a smooth function, and $0< \varepsilon\ll 1$.
We present a differential sequence based upon the Calogero-Degasperis-Ibragimov-Shabat Equation and determine first integrals and the general solution. Under suitable transformations eachmember of the differential sequence can be recast as a product of two factors and we report some of the properties of the factored form.
The concept and use of recursion operators is well-established in the study of evolution, in particular nonlinear, equations. We demonstrate the application of the idea of recursion operators to ordinary differential equations. For the purposes of our demonstration we use two equations, one chosen from the class of linearisable hierarchies of evolution equations studied by Euler et al (Stud Appl Math 111 (2003) 315-337) and the other from the class of integrable but nonlinearisible equations studied by Petersson et al (Stud Appl Math 112 (2004) 201-225). We construct the hierarchies for each equation. The symmetry properties of the first hierarchy are considered in some detail. For both hierarchies we apply the singularity analysis. For both we observe intersting behaviour of the resonances for the different possible leading order behaviours. In particular we note the proliferation of subsidiary solutions as one ascends the hierarchy.
The general d'Alembert equation $\square u+ f(x_0, x_1,u)= 0$ is considered, where $\square$ is the two-dimensional d'Alembert operator. We classify the equation for functions $f$ by which it admits several Lie symmetry algebras, which include the Lorentz symmetry generator. The corresponding symmetry reductions are listed.
We discuss the construction of reciprocal Bäcklund transformations for evolution equations using integrating factors of zeroth and higher orders with their corresponding conservation laws. As an example, we consider the Harry Dym equation and the Schwarzian KdV equation.
We introduce a new type of recursion operator to generate a class of nonlocal symmetries for second-order evolution equations in 1+1 dimensions, namely those evolution equations which allow the complete integration of their stationary equations. We show that this class of evolution equations is C-integrable (linearizable by a point transformation). We also discuss some applications.
In our article [1], "A tree of linearisable second-order evolution equations by generalised hodograph transformations" we present a class of linearizable (C-integrable) second-order evolution equations in (1+1) dimensions, using a generalized hodograph transformation. We report here the complete set of recursion operators for this class and present the resulting linearizable (C-integrable) hierarchies in (1+1) dimensions. The autonomous class of linearizable hierarchies are extended further by considering the equations in potential form followed by the pure hodograph transformation.
In this letter we report a new invariant for the Sawada-Kotera equation that is obtained by a systematic potentialization of the Kupershmidt equation. We show that this result can be derived from nonlocal symmetriesand that, conversely, a previously known invariant of the Kaup-Kupershmidt equation can be recovered using potentializations.
We use the generalized Sundman transformation to obtain a relation between a generalized Emden-Fowler equation and the first Painlevé transcendent.
Recently, Holm and Ivanov, proposed and studied a class of multi-component generalisations of the Camassa-Holm equations [D D Holm and R I Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, [J. Phys A: Math. Theor, vol. 43, 492001 (20pp), 2010]. We consider two of those systems, denoted by Holm and Ivanov by CH(2,1) and CH(2,2), and report a class of integrating factors and its corresponding conservation laws for these two systems. In particular, we obtain the complete sent of first-order integrating factors for the systems in Cauchy-Kovalevskaya form and evaluate the corresponding sets of conservation laws for CH(2,1) and CH(2,2).
We do a Lie symmetry classification for a system of two nonlinear coupled Schrödinger equations. Our system under consideration is a generalization of the equations which follow from the analysis of optical fibres. Reductions of some special equations are given.
The book consists of lecture notes intended for engineering and science students who are reading a first course in ordinary differential equations and who have already read a course on linear algebra, including general vector spaces and integral calculus for functions of one variable. No knowledge of differential equations is required to read and understand this material. Many examples have been included and most statements are proved in full detail. The aim is to provide the student with a thorough understanding of the methods to obtain solutions of certain classes of mainly linear scalar differential equations.
The Journal of Nonlinear Mathematical Physics (JNMP) publishes research papers on fundamental mathematical and computational methods in mathematical physics in the form of Letters, Articles, and Review Articles. The latest ISI Impact factor (2005) is: 0.508. Bibliografisk note: http://www.atlantis-press.com/publications/jnmp/
The book consists of lecture notes intended for engineering and science students who are reading a first course in ordinary differential equations and who have already read a course on linear algebra, including general vector spaces and integral calculus for functions of one variable. No prior knowledge of differential equations is required to read and understand this material. Many examples have been included and most statements are proved in full detail. The aim is to provide the student with a thorough understanding of methods to obtain solutions of certain classes of mainly linear scalar differential equations.
We make use of linear operators to derive the formulae for the generalsolution of elementary linear scalar ordinary differential equations of order n.The key lies in the factorization of the linear operators in terms of first-orderoperators. These first-order operators are then integrated by applying theircorresponding integral operators.This leads to the solution formulae for bothhomogeneous- and nonhomogeneous linear differential equations in a naturalway without the need for any ansatz (or “educated guess”). For second-orderlinear equations with nonconstant coefficients, the condition of the factorizationis given in terms of Riccati equations.
The book aims to provide a comprehensive account of the state of the art on the mathematical description of nonlinear systems. The book consists of 20 invited contributions written by leading experts in different aspects of nonlinear systems that include ordinary and partial differential equations, difference equations and q-difference equations, discrete or lattice equations, non-commutative and matrix equations, as well as supersymmetric equations.
Summary (translated from the Russian): Using Painlevé analysis we construct a conditional auto-Bäcklund transformation for a two-dimensional four-velocity Boltzmann equation with a quadratic nonlinearity. We construct exact multiparameter solutions of this model.
The author studies the integrability of the nonlinear second-order ordinary differential equation described in the title. Using the invertible point transformation technique, Lie point symmetry technique and the Painlevé analysis, he classifies all integrable cases. The Lie algebra structure of the Lie point symmetries is also discussed. Finally, the three different methods are compared.
We present a list of (1 + 1)-dimensional second-order evolution equations all connected via a proposed generalised hodograph transformation, resulting in a tree of equations transformable to the linear second-order autonomous evolution equation. The list includes autonomous and nonautonomous equations
We discuss the Sundman symmetries and point symmetries of linearisable second-order ordinary differential equations, which include discrete and nonlocal symmetries. The method of construction of such symmetries is also introduced for general n-th order equations.
The Madelung representation $\psi = u\exp(iv)$ is considered for the d'Alembert equation $\square_n\psi - F(|\psi|)\psi = 0$ to develop a technique for finding exact solutions. The authors classify the nonlinear function $F$ for which the amplitude and phase of the d'Alembert equation are related to the solutions of the compatible d'Alembert-Hamiltonian system. The equations are studied in the $n$-dimensional Minkowski space.[A.~Ju.~Obolenskij (Kyïv)]
We derive solution-formulae for the Krichever-Novikov equation by a systematic multipotentialisation of the equation. The formulae are achieved due to the connections of the Krichever-Novikov equations to certain symmetry-integrable 3rd-order evolution equations which admit autopotentialisations.
The term multipotentialisation of evolution equations in 1+1 dimensions refers to the process of potentialising a given evolution equation, followed by at least one further potentialisation of the resulting potential equation. For certain equations this process can be applied several times to result in a finite chain of potential equations, where each equation in the chain is a potential equation of the previous equation. By a potentialisation of an equation with dependent variable u to an equation with dependent variable v, we mean a differential substitution v_x=\Phi^t, where \Phi^t is a conserved current of the equation in u. The process of multipotentialisation may lead to interesting nonlocal transformations between the equations. Remarkably, this can, in some cases, result in nonlocal invariance transformations for the equations, which then serve as iteration formulas by which solutions can be generated for all the equations in the chain.
In the current paper we give a comprehensive introduction to this subject and report new nonlocal invariance transformations that result from the multipotentialisation of the Kupershmidt equation and its higher-order hierarchies. The recursion operators that define the hierarchies are given explicitly.
We discuss nonlocal symmetries and nonlocal conservation laws that follow from the systematic potentialisation of evolution equations. Those are the Lie point symmetries of the auxiliary systems, also known as potential symmetries.We define higher-degree potential symmetries which then lead to nonlocal conservation laws and nonlocal transformations for the equations. We demonstrate our approach and derive second degree potential symmetries for the Burgers' hierarchy and the Calogero-Degasperis-Ibragimov-Shabat hierarchy.
Sundman symmetries arise from more general transformations than do point or contact symmetries. This paper first shows how to systematically calculate Sundman symmetries of second- and third-order nonlinear ordinary differential equations. Secondly, the authors illustrate the application of these symmetries by computing first integrals of the corresponding equations.
The subject of the paper are symmetries of the nonlinear hyperbolic equation, $$\frac{\partial^2u}{\partial t^2} - \sum_{n=1}^N \frac{\partial^2u}{\partial^2x_n} +m^2u - \varepsilon f(u)\left(\lambda_0 \frac{\partial u}{\partial t} + \sum_{n=1}^N\lambda_n \frac{\partial u}{\partial x_n}\right) =0. $$ The case $f(u)=1-u^2$ corresponds to the generalized van der Pol equation. The equation is expanded in powers of the parameter $\varepsilon$, which stands in front of the nonlinear term, and then symmetries of the resulting chain of approximate equations are studied by means of the Lie-group technique. Emphasis is made on a special type of the function $f(u)$ which admits conformal invariance of the equation.
We propose a method to identify and classify evolution equations and systems that can be multipotentialised in given target equations or target systems. We refer to this as the converse problem. Although we mainly study a method for (1 + 1)-dimensional equations/system, we do also propose an extension of the methodology to higher-dimensional evolution equations. An important point is that the proposed converse method allows one to identify certain types of auto-Bäcklund transformations for the equations/systems. In this respect we define the triangular-auto-Bäcklund transformation and derive its connections to the converse problem. Several explicit examples are given. In particular, we investigate a class of linearisable third-order evolution equations, a fifth-order symmetry-integrable evolution equation as well as linearisable systems.
Non-Lie ansätze and exact solutions are constructed by means of conditional symmetries for a class of nonlinear heat equations in one space dimension. We study approximate symmetries by which one can obtain approximate solutions for multidimensional partial differential equations. The method of approximate symmetries is applied to an $n$-dimensional generalized van der Pol equation. Approximate conditions symmetries are introduced to construct approximate solutions for the latter equation.
We introduce nonlocal auto-hodograph transformations for a hierarchy of nonlinear evolution equations. This is accomplished by composing nonlocal transformations (one of which is a hodograph transformation) which linearize the given equations. This enables one to construct sequences of exact solutions for any equation belonging to the hierarchy.
Summary: We investigate conditional invariance by considering $Q$-symmetry generators of the nonlinear heat equation $\partial u/\partial x\sb 0-\lambda \partial\sp 2u/\partial x\sp 2\sb 1=f(u)$, where $\lambda$ is a real constant and $f$ an arbitrary differentiable function. With the obtained $Q$-generators we construct exact solutions by the use of similarity ansatze and reductions to ordinary differential equations. A generalization to $m$ space dimensions is performed.
We propose a method for constructing first integrals of higher order ordinary differential equations. In particular third, fourth and fifth order equations of the form are considered. The relation of the proposed method to local and nonlocal symmetries are discussed.