85 research outputs found
Multiple nonsymmetric nodal solutions for quasilinear Schrödinger system
In this paper, we consider the quasilinear Schrödinger system in RN (N ≥ 3): −∆u + A(x)u − 1 2 ∆(u 2 )u = 2α |u| α−2u|v| −∆v + Bv − 1 2 ∆(v 2 )v = 2β |u| |v| β−2 v, where α, β > 1, 2 0 is a constant. By using a constrained minimization on Nehari–Pohožaev set, for any given integer s ≥ 2, we construct a nonradially symmetrical nodal solution with its 2s nodal domains
Multiple nonsymmetric nodal solutions for quasilinear Schrödinger system
In this paper, we consider the quasilinear Schrödinger system in RN (N ≥ 3): −∆u + A(x)u − 1 2 ∆(u 2 )u = 2α |u| α−2u|v| −∆v + Bv − 1 2 ∆(v 2 )v = 2β |u| |v| β−2 v, where α, β > 1, 2 0 is a constant. By using a constrained minimization on Nehari–Pohožaev set, for any given integer s ≥ 2, we construct a nonradially symmetrical nodal solution with its 2s nodal domains
Book of Abstracts
USPCAPESFAPESPCNPqINCTMatICMC Summer Meeting on Differentail Equations.\ud
São Carlos, Brasil. 3-7 february 2014
Approximation of small-amplitude weakly coupled oscillators with discrete nonlinear Schrodinger equations
Small-amplitude weakly coupled oscillators of the Klein-Gordon lattices are
approximated by equations of the discrete nonlinear Schrodinger type. We show
how to justify this approximation by two methods, which have been very popular
in the recent literature. The first method relies on a priori energy estimates
and multi-scale decompositions. The second method is based on a resonant normal
form theorem. We show that although the two methods are different in the
implementation, they produce equivalent results as the end product. We also
discuss applications of the discrete nonlinear Schrodinger equation in the
context of existence and stability of breathers of the Klein--Gordon lattice
International Conference on Nonlinear Differential Equations and Applications
Dear Participants, Colleagues and Friends
It is a great honour and a privilege to give you all a warmest welcome to the first Portugal-Italy Conference on Nonlinear Differential Equations and Applications (PICNDEA).
This conference takes place at the Colégio EspÃrito Santo, University of Évora, located in the beautiful city of Évora, Portugal. The host institution, as well the associated scientific research centres, are committed to the event, hoping that it will be a benchmark for scientific collaboration between the two countries in the area of mathematics.
The main scientific topics of the conference are Ordinary and Partial Differential Equations, with particular regard to non-linear problems originating in applications, and its treatment with the methods of Numerical Analysis. The fundamental main purpose is to bring together Italian and Portuguese researchers in the above fields, to create new, and amplify previous collaboration, and to follow and discuss new topics in the area
Stability and instability of expanding solutions to the Lorentzian constant-positive-mean-curvature flow
We study constant mean curvature Lorentzian hypersurfaces of
from the point of view of its Cauchy problem. We
completely classify the spherically symmetric solutions, which include among
them a manifold isometric to the de Sitter space of general relativity. We show
that the spherically symmetric solutions exhibit one of three (future)
asymptotic behaviours: (i) finite time collapse (ii) convergence to a time-like
cylinder isometric to some and (iii) infinite
expansion to the future converging asymptotically to a time translation of the
de Sitter solution. For class (iii) we examine the future stability properties
of the solutions under arbitrary (not necessarily spherically symmetric)
perturbations. We show that the usual notions of asymptotic stability and
modulational stability cannot apply, and connect this to the presence of
cosmological horizons in these class (iii) solutions. We can nevertheless show
the global existence and future stability for small perturbations of class
(iii) solutions under a notion of stability that naturally takes into account
the presence of cosmological horizons. The proof is based on the vector field
method, but requires additional geometric insight. In particular we introduce
two new tools: an inverse-Gauss-map gauge to deal with the problem of
cosmological horizon and a quasilinear generalisation of Brendle's Bel-Robinson
tensor to obtain natural energy quantities.Comment: Version 2: 60 pages, 1 figure. Changes mostly to fix typographical
errors, with the exception of Remark 1.2 and Section 9.1 which are new and
which explain the extrinsic geometry of the embedding in more detail in terms
of the stability result. Version 3: updated reference
Minimal positive solutions for systems of semilinear elliptic equations
The paper is devoted to a system of nonlinear PDEs containing gradient terms. Applying the approach based on Sattinger's iteration procedure we use sub and supersolutions methods to prove the existence of positive solutions with minimal growth. These results can be applied for both sublinear and superlinear problems
Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains
We consider a semilinear elliptic problem with a nonlinear term which is the
product of a power and the Riesz potential of a power. This family of equations
includes the Choquard or nonlinear Schroedinger--Newton equation. We show that
for some values of the parameters the equation does not have nontrivial
nonnegative supersolutions in exterior domains. The same techniques yield
optimal decay rates when supersolutions exists.Comment: 47 pages, 8 figure
LECTURES ON NONLINEAR DISPERSIVE EQUATIONS I
CONTENTS
J. Bona
Derivation and some fundamental properties of nonlinear dispersive waves equations
F. Planchon
Schr\"odinger equations with variable coecients
P. Rapha\"el
On the blow up phenomenon for the L^2 critical non linear Schrodinger Equatio
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