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 $L_p$-norm for $p>2$, $L_\infty$-norm and in the sense of distributions.Comment: 10 page