406 research outputs found
q-Analogue of Shock Soliton Solution
By using Jackson's q-exponential function we introduce the generating
function, the recursive formulas and the second order q-differential equation
for the q-Hermite polynomials. This allows us to solve the q-heat equation in
terms of q-Kampe de Feriet polynomials with arbitrary N moving zeroes, and to
find operator solution for the Initial Value Problem for the q-heat equation.
By the q-analog of the Cole-Hopf transformation we construct the q-Burgers type
nonlinear heat equation with quadratic dispersion and the cubic nonlinearity.
In q -> 1 limit it reduces to the standard Burgers equation. Exact solutions
for the q-Burgers equation in the form of moving poles, singular and regular
q-shock soliton solutions are found.Comment: 13 pages, 5 figure
Analytic model for a frictional shallow-water undular bore
We use the integrable Kaup-Boussinesq shallow water system, modified by a
small viscous term, to model the formation of an undular bore with a steady
profile. The description is made in terms of the corresponding integrable
Whitham system, also appropriately modified by friction. This is derived in
Riemann variables using a modified finite-gap integration technique for the
AKNS scheme. The Whitham system is then reduced to a simple first-order
differential equation which is integrated numerically to obtain an asymptotic
profile of the undular bore, with the local oscillatory structure described by
the periodic solution of the unperturbed Kaup-Boussinesq system. This solution
of the Whitham equations is shown to be consistent with certain jump conditions
following directly from conservation laws for the original system. A comparison
is made with the recently studied dissipationless case for the same system,
where the undular bore is unsteady.Comment: 24 page
Multicomponent Burgers and KP Hierarchies, and Solutions from a Matrix Linear System
Via a Cole-Hopf transformation, the multicomponent linear heat hierarchy
leads to a multicomponent Burgers hierarchy. We show in particular that any
solution of the latter also solves a corresponding multicomponent (potential)
KP hierarchy. A generalization of the Cole-Hopf transformation leads to a more
general relation between the multicomponent linear heat hierarchy and the
multicomponent KP hierarchy. From this results a construction of exact
solutions of the latter via a matrix linear system.Comment: 18 pages, 4 figure
A multiple exp-function method for nonlinear differential equations and its application
A multiple exp-function method to exact multiple wave solutions of nonlinear
partial differential equations is proposed. The method is oriented towards ease
of use and capability of computer algebra systems, and provides a direct and
systematical solution procedure which generalizes Hirota's perturbation scheme.
With help of Maple, an application of the approach to the dimensional
potential-Yu-Toda-Sasa-Fukuyama equation yields exact explicit 1-wave and
2-wave and 3-wave solutions, which include 1-soliton, 2-soliton and 3-soliton
type solutions. Two cases with specific values of the involved parameters are
plotted for each of 2-wave and 3-wave solutions.Comment: 12 pages, 16 figure
Exact Travelling Wave Solutions of Some Nonlinear Nonlocal Evolutionary Equations
Direct algebraic method of obtaining exact solutions to nonlinear PDE's is
applied to certain set of nonlinear nonlocal evolutionary equations, including
nonlinear telegraph equation, hyperbolic generalization of Burgers equation and
some spatially nonlocal hydrodynamic-type model. Special attention is paid to
the construction of the kink-like and soliton-like solutions.Comment: 13 pages, LaTeX2
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