Related papers: Group classification of nonlinear wave equations
A modified algorithm for the construction of nonlinear realizations (sigma models) is applied to the general covariance group. Our method features finite dimensionality of the coset manifold by letting the vacuum stability group be…
In this article we present first an algorithm for calculating the determining equations associated with so-called ``nonclassical method'' of symmetry reductions (a la Bluman and Cole) for systems of partial differentail equations. This…
In this paper, we establish a relation between two seemingly unrelated concepts for solving first-order hyperbolic quasilinear systems of partial differential equations in many dimensions. These concepts are based on a variant of the…
In this paper, we prove the first asymptotic completeness result for a scalar quasilinear wave equation satisfying the weak null condition. The main tool we use in the study of this equation is the geometric reduced system introduced in…
We study conditions of reduction of multidimensional wave equations - a system of d'Alembert and Hamilton equations. Necessary conditions for compatibility of such reduction conditions are proved. Possible types of the reduced equations and…
In this article we consider a system of two Klein-Gordon equations, set on the $d$-dimensional box of size $L$, coupled through quadratic semilinear terms of strength $\varepsilon$ and evolving from well-prepared random initial data. We…
A canonical variable coefficient nonlinear Schr\"{o}dinger equation with a four dimensional symmetry group containing $\SL(2,\mathbb{R})$ group as a subgroup is considered. This typical invariance is then used to transform by a symmetry…
The generalized equation for the study of two-component nonlinear waves in different fields of physics is considered. In special cases, this equation is reduced to a set of the various well-known equations describing nonlinear solitary…
We study a class of nonlinear evolution systems of time fractional partial differential equations using Lie symmetry analysis. We obtain not only infinitesimal symmetries but also a complete group classification and a classification of…
Three dimensional nonlinear wave interactions have been analytically described. The procedure under interest can be applied to three dimensional quasilinear systems of first order, whose hydrodynamic reductions are homogeneous…
The objective of this paper is to construct geometrically Riemann $k$-wave solutions of the general form of first-order quasilinear hyperbolic systems of partial differential equations. To this end, we adapt and combine elements of two…
The sine-Gordon equation is considered in the hamiltonian framework provided by the Adler-Kostant-Symes theorem. The phase space, a finite dimensional coadjoint orbit in the dual space $\grg^*$ of a loop algebra $\grg$, is parametrized by a…
We consider the problem of performing the preliminary "symmetry classification'' of a class of quasi-linear PDE's containing one or more arbitrary functions: we provide an easy condition involving these functions in order that nontrivial…
Reductions of N-wave type equations related to simple Lie algebras and the hierarchy of their Hamiltonian structures are studied. The reduction group G_R is realized as a subgroup of the Weyl group of the corresponding algebra. Some of the…
Two families of solutions of a generalized non-Abelian Toda lattice are considered. These solutions are expressed in terms of quasideterminants, constructed by means of Darboux and binary Darboux transformations. As an example of the…
The authors suggest a new powerful tool for solving group classification problems, that is applied to obtaining the complete group classification in the class of nonlinear Schr\"odinger equations of the form…
We study the symmetry groups and winding numbers of planar curves obtained as images of weighted sums of exponentials. More generally, we study the image of the complex unit circle under a finite or infinite Laurent series using a…
Two different versions of cubic sixth-order generalised Boussinesq-type wave equations are considered in this study. A generalised perturbation reduction method is used to solve these equations, which allows the reduction of considered…
This paper presents a study of nonlinear superpositions of Riemann wave solutions admitted by quasilinear hyperbolic first-order systems of partial differential equations. In particular, we focus on the Euler system and non-elastic wave…
We prove the existence of infinitely many solitary waves for the nonlinear Klein-Gordon or Schr\"odinger equation $$ \Delta u-u+ u^3 =0 , $$ in ${\bf R}^2$, which have finite energy and whose maximal group of symmetry reduces to the…