Solitary Waves For Some Nonlinear Schrödinger Systems

In this paper we study the existence of radially symmetric positive solutions in Hrad 1 (RN) × Hrad 1 (RN) of the elliptic system:- Δ u + u - (α u2 + β v2) u = 0,- Δ v + ω2 v - (β u2 + γ v2) v = 0,N = 1, 2, 3, where α and γ are positive constants (β will be allowed to be negative). This system has trivial solutions of the form (φ{symbol}, 0) and (0, ψ) where φ{symbol} and ψ are nontrivial solutions of scalar equations. The existence of nontrivial solutions for some values of the parameters α, β, γ, ω has been studied recently by several authors [A. Ambrosetti, E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci. Paris, Ser. I 342 (2006) 453-458; T.C. Lin, J. Wei, Ground states of N coupled nonlinear Schrödinger equations in Rn, n ≤ 3, Comm. Math. Phys. 255 (2005) 629-653; T.C. Lin, J. Wei, Ground states of N coupled nonlinear Schrödinger equations in Rn, n ≤ 3, Comm. Math. Phys., Erratum, in press; L. Maia, E. Montefusco, B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, preprint; B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in RN, preprint; J. Yang, Classification of the solitary waves in coupled nonlinear Schrödinger equations, Physica D 108 (1997) 92-112]. For N = 2, 3, perhaps the most general existence result has been proved in [A. Ambrosetti, E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci. Paris, Ser. I 342 (2006) 453-458] under conditions which are equivalent to ours. Motivated by some numerical computations, we return to this problem and, using our approach, we give a more detailed description of the regions of parameters for which existence can be proved. In particular, based also on numerical evidence, we show that the shape of the region of the parameters for which existence of solution can be proved, changes drastically when we pass from dimensions N = 1, 2 to dimension N = 3. Our approach differs from the ones used before. It relies heavily on the spectral theory for linear elliptic operators. Furthermore, we also consider the case N = 1 which has to be treated more extensively due to some lack of compactness for even functions. This case has not been treated before. © 2007 Elsevier Masson SAS. All rights reserved.251149161Ambrosetti, A., Colorado, E., Bound and ground states of coupled nonlinear Schrödinger equations (2006) C. R. Acad. Sci. Paris, Ser. I, 342, pp. 453-458Bonorino, L., Brietzke, E., Lukaszczyk, J.P., Taschetto, C., Properties of the period function for some hamiltonian systems and homogeneous solutions of a semilinear elliptic equation (2005) J. Differential Equations, 214, pp. 156-175Lin, T.C., Wei, J., Ground states of N coupled nonlinear Schrödinger equations in Rn, n ≤ 3 (2005) Comm. Math. Phys., 255, pp. 629-653Reed, M., Simon, B., (1972) Methods of Modern Mathematical Physics, IV, Analysis of Operators, , Academic Press, New YorkReed, M., Simon, B., (1972) Methods of Modern Mathematical Physics, II, Fourier Analysis, , Academic Press, New YorkYang, J., Classification of the solitary waves in coupled nonlinear Schrödinger equations (1997) Physica D, 108, pp. 92-11

Hamiltonian elliptic systems: a guide to variational frameworks

Consider a Hamiltonian system of type $$-\Delta u=H_{v}(u,v),\ -\Delta v=H_{u}(u,v) \ \ \text{ in } \Omega, \qquad u,v=0 \text{ on } \partial \Omega$$ where $H$ is a power-type nonlinearity, for instance $H(u,v)= |u|^p/p+|v|^q/q$, having subcritical growth, and $\Omega$ is a bounded domain of $\mathbb{R}^N$, $N\geq 1$. The aim of this paper is to give an overview of the several variational frameworks that can be used to treat such a system. Within each approach, we address existence of solutions, and in particular of ground state solutions. Some of the available frameworks are more adequate to derive certain qualitative properties; we illustrate this in the second half of this survey, where we also review some of the most recent literature dealing mainly with symmetry, concentration, and multiplicity results. This paper contains some original results as well as new proofs and approaches to known facts.Comment: 78 pages, 7 figures. This corresponds to the second version of this paper. With respect to the original version, this one contains additional references, and some misprints were correcte

Theorems on existence and global dynamics for the Einstein equations

This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local-in-time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symmetry are discussed. A selection of results from Newtonian theory and special relativity that offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure or late-time asymptotics are given. A conjectural picture of the asymptotic behaviour of general cosmological solutions of the Einstein equations is built up. Some miscellaneous topics connected with the main theme are collected in a separate section.Comment: Submitted to Living Reviews in Relativity, major update of Living Rev. Rel. 5 (2002)

Solitary waves in the Nonlinear Dirac Equation

In the present work, we consider the existence, stability, and dynamics of solitary waves in the nonlinear Dirac equation. We start by introducing the Soler model of self-interacting spinors, and discuss its localized waveforms in one, two, and three spatial dimensions and the equations they satisfy. We present the associated explicit solutions in one dimension and numerically obtain their analogues in higher dimensions. The stability is subsequently discussed from a theoretical perspective and then complemented with numerical computations. Finally, the dynamics of the solutions is explored and compared to its non-relativistic analogue, which is the nonlinear Schr{\"o}dinger equation. A few special topics are also explored, including the discrete variant of the nonlinear Dirac equation and its solitary wave properties, as well as the PT-symmetric variant of the model

Geometric Analysis of Bifurcation and Symmetry Breaking in a Gross-Pitaevskii equation

Gross-Pitaevskii and nonlinear Hartree equations are equations of nonlinear Schroedinger type, which play an important role in the theory of Bose-Einstein condensation. Recent results of Aschenbacher et. al. [AFGST] demonstrate, for a class of 3- dimensional models, that for large boson number (squared L^2 norm), N, the ground state does not have the symmetry properties as the ground state at small N. We present a detailed global study of the symmetry breaking bifurcation for a 1-dimensional model Gross-Pitaevskii equation, in which the external potential (boson trap) is an attractive double-well, consisting of two attractive Dirac delta functions concentrated at distinct points. Using dynamical systems methods, we present a geometric analysis of the symmetry breaking bifurcation of an asymmetric ground state and the exchange of dynamical stability from the symmetric branch to the asymmetric branch at the bifurcation point.Comment: 22 pages, 7 figure