18 research outputs found
Regularity of ground state solutions of dispersion managed nonlinear schrödinger equations
AbstractWe consider the dispersion managed nonlinear Schrödinger equation (DMNLS) in the case of zero residual dispersion. Using dispersive properties of the equation and estimates in Bourgain spaces we show that the ground state solutions of DMNLS are smooth. The existence of smooth solutions in this case matches the well-known smoothness of the solutions in the case of nonzero residual dispersion. In the case xâR2 we prove that the corresponding minimization problem with zero residual dispersion has no solution
Spectral stability for subsonic traveling pulses of the Boussinesq `abc' system
We consider the spectral stability of certain traveling wave solutions of the
Boussinesq `abc' system. More precisely, we consider the explicit
like solutions of the form (\vp(x-w t), \psi(x- w t)=(\vp, const. \vp),
exhibited by M. Chen (1998) and we provide a complete rigorous characterization
of the spectral stability in all cases for which
A Spectral Mapping Theorem and Invariant Manifolds for Nonlinear Schr\"odinger Equations
A spectral mapping theorem is proved that resolves a key problem in applying
invariant manifold theorems to nonlinear Schr\" odinger type equations. The
theorem is applied to the operator that arises as the linearization of the
equation around a standing wave solution. We cast the problem in the context of
space-dependent nonlinearities that arise in optical waveguide problems. The
result is, however, more generally applicable including to equations in higher
dimensions and even systems. The consequence is that stable, unstable, and
center manifolds exist in the neighborhood of a (stable or unstable) standing
wave, such as a waveguide mode, under simple and commonly verifiable spectral
conditions.Comment: LaTeX, 16 page
On non-local variational problems with lack of compactness related to non-linear optics
We give a simple proof of existence of solutions of the dispersion manage-
ment and diffraction management equations for zero average dispersion,
respectively diffraction. These solutions are found as maximizers of non-linear
and non-local vari- ational problems which are invariant under a large
non-compact group. Our proof of existence of maximizer is rather direct and
avoids the use of Lions' concentration compactness argument or Ekeland's
variational principle.Comment: 30 page
Equations of the Camassa-Holm Hierarchy
The squared eigenfunctions of the spectral problem associated with the
Camassa-Holm (CH) equation represent a complete basis of functions, which helps
to describe the inverse scattering transform for the CH hierarchy as a
generalized Fourier transform (GFT). All the fundamental properties of the CH
equation, such as the integrals of motion, the description of the equations of
the whole hierarchy, and their Hamiltonian structures, can be naturally
expressed using the completeness relation and the recursion operator, whose
eigenfunctions are the squared solutions. Using the GFT, we explicitly describe
some members of the CH hierarchy, including integrable deformations for the CH
equation. We also show that solutions of some - dimensional members of
the CH hierarchy can be constructed using results for the inverse scattering
transform for the CH equation. We give an example of the peakon solution of one
such equation.Comment: 10 page