28,896 research outputs found
A simple and direct method for generating travelling wave solutions for nonlinear equations
We propose a simple and direct method for generating travelling wave
solutions for nonlinear integrable equations. We illustrate how nontrivial
solutions for the KdV, the mKdV and the Boussinesq equations can be obtained
from simple solutions of linear equations. We describe how using this method, a
soliton solution of the KdV equation can yield soliton solutions for the mKdV
as well as the Boussinesq equations. Similarly, starting with cnoidal solutions
of the KdV equation, we can obtain the corresponding solutions for the mKdV as
well as the Boussinesq equations. Simple solutions of linear equations can also
lead to cnoidal solutions of nonlinear systems. Finally, we propose and solve
some new families of KdV equations and show how soliton solutions are also
obtained for the higher order equations of the KdV hierarchy using this method.Comment: RevTex, 15 pages, 3 figures; version with new section and references,
to appear in Annals of Physic
Some unexplored features of the nonlinear compressive magnetoacoustic Alfvenic waves
The theory of nonlinear magnetoacoustic wave in the past has strictly been
focused on purely compressive features of the mode. We show that a complete set
of nonlinear equations necessarily includes both compressional and shear
components of the magnetic field. These two turn out to be described by exactly
the same nonlinear equations, which make the use of such a complete full set of
equations far less complicated than expected. Present results should
considerably enrich the theory of these waves by opening up new frontiers of
investigation and providing some completely new types of nonlinear solutions.Comment: Phys. Scripta, to be publishe
Analytic structure of solutions to multiconfiguration equations
We study the regularity at the positions of the (fixed) nuclei of solutions
to (non-relativistic) multiconfiguration equations (including Hartree--Fock) of
Coulomb systems. We prove the following: Let {phi_1,...,phi_M} be any solution
to the rank--M multiconfiguration equations for a molecule with L fixed nuclei
at R_1,...,R_L in R^3. Then, for any j in {1,...,M} and k in {1,...,L}, there
exists a neighbourhood U_{j,k} in R^3 of R_k, and functions phi^{(1)}_{j,k},
phi^{(2)}_{j,k}, real analytic in U_{j,k}, such that phi_j(x) =
phi^{(1)}_{j,k}(x) + |x - R_k| phi^{(2)}_{j,k}(x), x in U_{j,k} A similar
result holds for the corresponding electron density. The proof uses the
Kustaanheimo--Stiefel transformation, as applied earlier by the authors to the
study of the eigenfunctions of the Schr"odinger operator of atoms and molecules
near two-particle coalescence points.Comment: 15 page
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