155 research outputs found
A Generalization of the Hopf-Cole Transformation
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
On the localized wave patterns supported by convection-reaction-diffusion equation
A set of traveling wave solution to convection-reaction-diffusion equation is
studied by means of methods of local nonlinear analysis and numerical
simulation. It is shown the existence of compactly supported solutions as well
as solitary waves within this family for wide range of parameter values
Microscopic structure of travelling wave solutions in a class of stochastic interacting particle systems
We obtain exact travelling wave solutions for three families of stochastic
one-dimensional nonequilibrium lattice models with open boundaries. These
solutions describe the diffusive motion and microscopic structure of (i) of
shocks in the partially asymmetric exclusion process with open boundaries, (ii)
of a lattice Fisher wave in a reaction-diffusion system, and (iii) of a domain
wall in non-equilibrium Glauber-Kawasaki dynamics with magnetization current.
For each of these systems we define a microscopic shock position and calculate
the exact hopping rates of the travelling wave in terms of the transition rates
of the microscopic model. In the steady state a reversal of the bias of the
travelling wave marks a first-order non-equilibrium phase transition, analogous
to the Zel'dovich theory of kinetics of first-order transitions. The stationary
distributions of the exclusion process with shocks can be described in
terms of -dimensional representations of matrix product states.Comment: 27 page
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
Group analysis and conservation laws of an integrable Kadomtsev–Petviashvili equation
In this paper, an integrable KP equation is studied using symmetry and conservation laws. First, on the basis of various cases of coefficients, we construct the infinitesimal generators. For the special case, we get the corresponding geometry vector fields, and then from known soliton solutions we derive new soliton solutions. In addition, the explicit power series solutions are derived. Lastly, nonlinear self-adjointness and conservation laws are constructed with symmetries
- …