394 research outputs found
Notes on Lie symmetry group methods for differential equations
Fundamentals on Lie group methods and applications to differential equations
are surveyed. Many examples are included to elucidate their extensive
applicability for analytically solving both ordinary and partial differential
equations.Comment: 85 Pages. expanded and misprints correcte
A class of invariant diffusion processes in one dimension
This paper relies on a simple test to decide whether or not nontrivial
symmetries of a large class of inhomogeneous diffusion processes on the real
line exist. When these symmetries are confirmed a priori by just picking
coefficients including physically meaningful diffusion and drift terms, the
transformation to canonical forms with four- and six-dimensional symmetry
groups and the full list of their infinitesimal generators are then immediately
at ones's disposable without any cumbersome calculations. The results are
applied to several models previously treated in a less systematic way in the
literature to demonstrate the efficiency of the current approach.Comment: 15 page
A class of multi-parameter eigenvalue problems for perturbed p-Laplacians
AbstractThis paper is devoted to multi-parameter eigenvalue problems for one-dimensional perturbed p-Laplacians, modelling travelling waves for a class of nonlinear evolution PDE. Dispersion relations between the eigen-parameters, the existence of eigenfunctions and positive eigenfunctions, variational principles for eigenvalues and constructing solutions in the analytical and implicit forms are the main subject of this paper. We use both variational and analytical methods
Exact Solutions of a (2+1)-Dimensional Nonlinear Klein-Gordon Equation
The purpose of this paper is to present a class of particular solutions of a
C(2,1) conformally invariant nonlinear Klein-Gordon equation by symmetry
reduction. Using the subgroups of similitude group reduced ordinary
differential equations of second order and their solutions by a singularity
analysis are classified. In particular, it has been shown that whenever they
have the Painlev\'e property, they can be transformed to standard forms by
Moebius transformations of dependent variable and arbitrary smooth
transformations of independent variable whose solutions, depending on the
values of parameters, are expressible in terms of either elementary functions
or Jacobi elliptic functions.Comment: 16 pages, no figures, revised versio
A variable coefficient nonlinear Schr\"{o}dinger equation with a four-dimensional symmetry group and blow-up of its solutions
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 transformation a known solution that can be derived by truncating its
Painlev\'e expansion and study blow-ups of these solutions in the -norm
for , -norm and in the sense of distributions.Comment: 10 page
- …