Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 9 (2013), 016, 19 pages      arXiv:1302.6000      http://dx.doi.org/10.3842/SIGMA.2013.016
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”

A Generalization of the Hopf-Cole Transformation

Paulius Miškinis
Department of Physics, Faculty of Fundamental Sciences, Vilnius Gediminas Technical University, Saulėtekio Ave 11, LT-10223, Vilnius-40, Lithuania

Received June 04, 2012, in final form February 17, 2013; Published online February 25, 2013

Abstract
A generalization of the Hopf-Cole transformation and its relation to the Burgers equation of integer order and the diffusion equation with quadratic nonlinearity are discussed. The explicit form of a particular analytical solution is presented. The existence of the travelling wave solution and the interaction of nonlocal perturbation are considered. The nonlocal generalizations of the one-dimensional diffusion equation with quadratic nonlinearity and of the Burgers equation are analyzed.

Key words: nonlocality; nonlinearity; diffusion equation; Burgers equation.

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