369 research outputs found
Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs
Algorithms are presented for the tanh- and sech-methods, which lead to
closed-form solutions of nonlinear ordinary and partial differential equations
(ODEs and PDEs). New algorithms are given to find exact polynomial solutions of
ODEs and PDEs in terms of Jacobi's elliptic functions.
For systems with parameters, the algorithms determine the conditions on the
parameters so that the differential equations admit polynomial solutions in
tanh, sech, combinations thereof, Jacobi's sn or cn functions. Examples
illustrate key steps of the algorithms.
The new algorithms are implemented in Mathematica. The package
DDESpecialSolutions.m can be used to automatically compute new special
solutions of nonlinear PDEs. Use of the package, implementation issues, scope,
limitations, and future extensions of the software are addressed.
A survey is given of related algorithms and symbolic software to compute
exact solutions of nonlinear differential equations.Comment: 39 pages. Software available from Willy Hereman's home page at
http://www.mines.edu/fs_home/whereman
A KdV-like advection-dispersion equation with some remarkable properties
We discuss a new non-linear PDE, u_t + (2 u_xx/u) u_x = epsilon u_xxx,
invariant under scaling of dependent variable and referred to here as SIdV. It
is one of the simplest such translation and space-time reflection-symmetric
first order advection-dispersion equations. This PDE (with dispersion
coefficient unity) was discovered in a genetic programming search for equations
sharing the KdV solitary wave solution. It provides a bridge between non-linear
advection, diffusion and dispersion. Special cases include the mKdV and linear
dispersive equations. We identify two conservation laws, though initial
investigations indicate that SIdV does not follow from a polynomial Lagrangian
of the KdV sort. Nevertheless, it possesses solitary and periodic travelling
waves. Moreover, numerical simulations reveal recurrence properties usually
associated with integrable systems. KdV and SIdV are the simplest in an
infinite dimensional family of equations sharing the KdV solitary wave. SIdV
and its generalizations may serve as a testing ground for numerical and
analytical techniques and be a rich source for further explorations.Comment: 15 pages, 4 figures, corrected sign typo in KdV Lagrangian above
equation 3
Transverse Instability of Periodic Traveling Waves in the Generalized Kadomtsev-Petviashvili Equation
In this paper, we investigate the spectral instability of periodic traveling
wave solutions of the generalized Korteweg-de Vries equation to long wavelength
transverse perturbations in the generalized Kadomtsev-Petviashvili equation. By
analyzing high and low frequency limits of the appropriate periodic Evans
function, we derive an orientation index which yields sufficient conditions for
such an instability to occur. This index is geometric in nature and applies to
arbitrary periodic traveling waves with minor smoothness and convexity
assumptions on the nonlinearity. Using the integrable structure of the ordinary
differential equation governing the traveling wave profiles, we are then able
to calculate the resulting orientation index for the elliptic function
solutions of the Korteweg-de Vries and modified Korteweg-de Vries equations.Comment: 26 pages. Sign error corrected in Lemma 3. Statement of main theorem
corrected. Exposition updated and references added
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