Some recent achievements on scalar and vector nonlinear Schrodinger type equations
Session Organizers: Marco Squassina, Barbara Prinari
+11 Speakers (Bonheure, Candela, Musesti, Pisani, Prinari, Salvatore, Sciunzi, Servadei, Sirakov, Squassina,Yomba):
*Session coordinates: >REGENCY HALL 9<, MONDAY, >JULY 7<, 2008.
*All the talks will be of 45 minutes
*Last update of this page: 20 June 2008
Bonheure Denis (Louvain-la-Neuve) [denis.bonheure@uclouvain.be] On Schrodinger equations with potentials vanishing at infinity.
[9:00-9:45 AM]
Abstract: We discuss various recent results concerning positive solutions of the Schrodinger type equation $$-\Delta u + V(x) u = K(x) u^p,\ x\in \mathbb R^N,$$ where $p>1$ and $V,\ K$ are positive potentials that vanish at infinity. We consider the existence of both ground state and bound state solutions, the concentration phenomena in the semi-classical limit and the regularity (integrability) properties at infinity. We also consider potentials with partial symmetry, leading to various type of solutions including positive solutions with infinite action. This talk is based on collaborations with Jean Van Schaftingen and Jonathan Di Cosmo.
Candela Anna Maria (Bari) [candela@dm.uniba.it] Multiplicity results for a p-Laplacian type equation.
[9:45-10:30 AM]
Abstract: Many authors have investigated the existence of solutions for the $p$-Laplacian type equation
\[
- {\rm div} (p A(x,u) |\nabla u|^{p-2}\nabla u) + A_t(x,u) |\nabla u|^p = G_t(x,u)\quad\hbox{in $\Omega$,}\qquad u|_{\partial\Omega} = 0,
\]
with $\Omega$ bounded domain in $\R^N$, $p > 1$ and $A$, $G$ given functions in $\Omega \times \R$. Recently, the research of the {\sl critical points} for the corresponding functional
\[
J(u) = \int_\Omega A(x,u)|\nabla u|^p dx - \int_\Omega G(x,u) dx
\]
has been developed on the Banach space $X = W^{1,p}_0(\Omega) \cap L^\infty(\Omega)$ and, eventually using a generalization of Cerami's Palais-Smale condition, both an existence and some multiplicity results have been obtained for a more general version of this problem.
Musesti Alessandro (Brescia) [a.musesti@dmf.unicatt.it] Non-ergodicity threshold in interacting systems.
[10:30-11:15 AM]
Abstract: We show the existence of a threshold of disconnection in the energy surface for a one-dimensional anisotropic Heisenberg model, a particular model of distance-interacting systems, with an inter-particle potential $d^{-\alpha}$, $0<\alpha < 1$, where $d$ is the distanceamong particles. The disconnection of the energy region implies in particular the non-ergodicity of the system. However, if $\alpha > n$, the dimension of the space, the ratio between the disconnected energy region and the total energy region goes to zero when the number of particles becomes very large.On the contrary, in the case of a long-range action ($\alpha < n$), numerical simulations support the conclusion that such a ratio remains finite for large $N$ values. The disconnection border can thus be thought as a distinctive property of anisotropic long-range interacting systems.
Pisani Lorenzo (Bari) [pisani@dm.uniba.it] Klein-Gordon-Maxwell system in a bounded domain.
[11:15-12:00 AM]
Abstract: We discuss the existence of standing waves $\psi=u(x)e^{-i\omega t}$ in equilibrium with a purely electrostatic field $\mathbf{E}=-\nabla\phi(x)$. We assume homogeneous Dirichlet boundary conditions on $u$ and inhomogeneous Neumann conditions on $\phi$. For small boundary data, we characterize the existence of nontrivial solutions, that is solutions such that $u\neq0$. With a suitable nonlinear perturbation in the Klein-Gordon equation, we get infinitely many solutions.
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Barbara Prinari (Lecce) [prinari@le.infn.it] Inverse scattering transform for the vector NLS equation with non-vanishing boundary conditions.
[2:00-2:45 PM]
Abstract: The inverse scattering transform for the vector defocusing vector nonlinear Schrodinger NLS equation with non-vanishing boundary values at infinity is constructed. The direct scattering problem is formulated on a two-sheeted covering of the complex plane. On the direct side, two out of the six scattering eigenfunctions do not admit an analytic extension on either sheet of the surface. Two additional analytic solutions are constructed by considering *adjoint* eigenfunctions. The discrete spectrum, bound states and symmetries of the direct problem are discussed. In general a discrete eigenvalue corresponds to a quartet of zeros (poles) of certain scattering data. The inverse scattering problem is formulated in terms of a Riemann-Hilbert (RH) problem in the upper/lower half planes of a suitable uniformization variable. Special soliton solutions, which have dark solitonic behavior in both components and ones which have one dark and one bright component are constructed from the poles in the RH problem. The linear limit is obtained from the RH problem and is shown to correspond to the Fourier solution obtained from the linearized vector NLS system.
Salvatore Dora (Bari) [salvator@dm.uniba.it] Multiple solutions for some variational elliptic systems.
[2:45-3:30 PM]
Abstract: We prove the existence of infinitely many solutions for a special class of symmetric elliptic systems when one of the two nonlinearities has arbitrary growth. Moreover, if the symmetry of the problem is broken by a small enough perturbation term, we find at least three solutions. The proofs use a variational setting given by de Figueiredo and Ruf and the "algebraic" approach based on the Pohozaev's fibering method.
Servadei Raffaella (Cosenza) [servadei@mat.unical.it] p-Laplacian equations with singular weights: existence and regularity results.
[3:30-4:15 PM]
Abstract: In this talk we consider a quasilinear singular elliptic equation with a nonlinear term $f$ satisfying suitable growth conditions. By means of the Mountain Pass Theorem and the constrained minimization method we prove the existence of positive or compactly supported radial ground states for the equation. We also prove a Pohozaev type identity which allow us to deduce many non--existence results for the problem. We study both the case when the nonlinear term $f$ is positive and the so called 'normal case', i.e. the case when $f$ is negative near the origin and positive at infinity. We also discuss some regularity results and the validity of useful qualitative properties of the solutions of the quasilinear singular elliptic equation under consideration. These results are of independent interest and are proved for a general quasilinear equation involving an elliptic operator $A$ and a nonlinear term $B$ which can be possibly singular at some point. As a model for $A$ we can take the $p$--Laplacian operator, but we consider also the inhomogeneous version of the problem. These regularity results are obtained by means of the Moser iteration scheme and the translation method due to Nirenberg and generalize some theorems already appeared in literature. All these results were obtained in collaboration with Patrizia Pucci.
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Sirakov Boyan (Paris) [sirakov@ehess.fr] A priori bounds and solvability for nonlinear uniformly elliptic systems.
[4:30-5:15 PM]
Abstract: In this talk we introduce new notions of sublinearity and superlinearity for elliptic systems and prove related existence results, which extend the known theorems for scalar equations. We obtain a number of new results on higher order equations with power growth nonlinearities. We also discuss Liouville type (nonexistence) results.
Squassina Marco (Verona) [marco.squassina@univr.it] Spatial patterns of ground state solutions for Gross-Pitaevskii systems in the plane.
[5:15-6:00 PM]
Abstract: We consider a system of Gross-Pitaevskii equations in $\R^2$ modelling a mixture of two Bose--Einstein condensates with repulsive interaction. We aim to study the qualitative behaviour of ground and excited state solutions. We allow two different harmonic and off-centered trapping potentials and study the spatial patterns of the solutions within the Thomas-Fermi approximation as well as phase segregation phenomena within the large-interaction regime. Part of this talk is based upon a collaboration with Marco Caliari.
Berardino Sciunzi (Cosenza) [sciunzi@mat.unical.it] Qualitative properties of stable solutions of p-Laplace equations.
[6:00-6:45 PM]
Abstract: We point out some regularity results for positive solutions of $-\Delta_p(u)=f(u)$, when the nonlinearity $f$ is positive. We show that this makes possible to define the linearized operator at $u$ in a suitable weighted Sobolev space. In this setting a solution $u$ is said to be "stable" if the linearized operator at $u$ is positive definte. As an example, local minimizers of the energy functional are stable solutions and also monotone solutions are stable solutions. This class of solutions exibits interesting qualitative properties.
Emmanuel Yomba (Minneapolis) [eyomba@yahoo.com] Modulational instability and exact solutions for a three-component system of nonlinear Schrodinger Equations.
[6:45-7:30 PM]
Abstract: The modulational instability (MI) of the three-component system of non-linear vector Schrodinger equations is investigated. It is found that there are a number of possibilities for the MI regions due to the generalized nonlinear dispersion relation, which relates the frequency and the wave number of modulating perturbations. Some classes of exact traveling wave solutions are obtained. Under some special parameter values, some representative wave structures are graphically displayed. These solutions are obtained by the use of F-expansion method.A
Verona, 20/6/2008
Info, addictions, corrections et al. to be notified to: marco.squassina@univr.it
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