Exact solutions of semilinear radial wave equations in n dimensions

Abstract

Exact solutions are derived for an n-dimensional radial wave equation with a general power nonlinearity. The method, which is applicable more generally to other nonlinear PDEs, involves an ansatz technique to solve a first-order PDE system of group-invariant variables given by group foliations of the wave equation, using the one-dimensional admitted point symmetry groups. (These groups comprise scalings and time translations, admitted for any nonlinearity power, in addition to space-time inversions admitted for a particular conformal nonlinearity power.) This is shown to yield not only group-invariant solutions as derived by standard symmetry reduction, but also other exact solutions of a more general form. In particular, solutions with interesting analytical behavior connected with blow-ups as well as static monopoles are obtained.  2004 Elsevier Inc. All rights reserved.

Lie Symmetries and Criticality of Semilinear Differential Systems

“…For a discussion on specific numbers concerning the scalar case see [1,2]. Again this result as well as the corresponding Pokhozhaev's identity [13] is obtained by the same arguments as before and we omit further details.…”

Exact solutions of semilinear radial Schrödinger equations by separation of group foliation variables

“…[11,12,13,14,15,16] when the group G of point symmetries is infinite-dimensional, and later it was developed in Ref. [6,7,8] when the point symmetry group G is finite-dimensional.…”

“…The separation of variables ansatz (4.17)-(4.18) for (H, G) is more general than the twoterm ansatzes used in previous work [6,7,8] where the terms in G were restricted to contain the same powers as the terms in H, e.g.…”

“…In fact, as summarized in recent work [5], the only explicit solutions which are known to-date for n = 1 (m = 0) consist of the obvious constant solution U = (−ω/k) n/4 exp(ıφ) for the ODE (1.4). In this paper we will obtain new explicit exact solutions to the radial gNLS equation (1.1) for n = 1 (m = 0) by applying a symmetry group method which has been used successfully in previous work [6,7,8] to find explicit blow-up and dispersive solutions to semilinear radial wave equations and semilinear radial heat equations with power nonlinearities in multidimensions. The method uses the group foliation equations associated with one-dimensional subgroups of the point symmetry group of a given nonlinear partial differential equation (PDE) [9].…”

Functionally separable solutions to nonlinear wave equations by group foliation method

“…It is associated to the Lie point symmetries of the considered PDEs, and has been used successfully to obtain solutions of nonlinear PDEs [30][31][32]. According to the group foliation method, the first step of the approach is to foliate the solution space of the equation in question into orbits, choosing a symmetry group for the foliation.…”