84 research outputs found
Madelung, Gross-Pitaevskii and Korteweg
This paper surveys various aspects of the hydrodynamic formulation of the
nonlinear Schrodinger equation obtained via the Madelung transform in connexion
to models of quantum hydrodynamics and to compressible fluids of the Korteweg
type.Comment: 32 page
Remarks on the mass constraint for KP type equations
For a rather general class of equations of Kadomtsev-Petviashvili (KP) type,
we prove that the zero-mass (in ) constraint is satisfied at any non zero
time even if it is not satisfied at initial time zero. Our results are based on
a precise analysis of the fundamental solution of the linear part and its anti
-derivative
Boussinesq Systems of Bona-Smith Type on Plane Domains: Theory and Numerical Analysis
We consider a class of Boussinesq systems of Bona-Smith type in two space
dimensions approximating surface wave flows modelled by the three-dimensional
Euler equations. We show that various initial-boundary-value problems for these
systems, posed on a bounded plane domain are well posed locally in time. In the
case of reflective boundary conditions, the systems are discretized by a
modified Galerkin method which is proved to converge in at an optimal
rate. Numerical experiments are presented with the aim of simulating
two-dimensional surface waves in complex plane domains with a variety of
initial and boundary conditions, and comparing numerical solutions of
Bona-Smith systems with analogous solutions of the BBM-BBM system
Existence and properties of travelling waves for the Gross-Pitaevskii equation
This paper presents recent results concerning the existence and qualitative
properties of travelling wave solutions to the Gross-Pitaevskii equation posed
on the whole space R^N. Unlike the defocusing nonlinear Schr\"odinger equations
with null condition at infinity, the presence of non-zero conditions at
infinity yields a rather rich and delicate dynamics. We focus on the case N = 2
and N = 3, and also briefly review some classical results on the
one-dimensional case. The works we survey provide rigorous justifications to
the impressive series of results which Jones, Putterman and Roberts established
by formal and numerical arguments
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