Sharp Threshold of Global Existence for a Nonlocal Nonlinear Schrdinger System in R~3

In this paper, the authors investigate the sharp threshold of a three-dimensional nonlocal nonlinear Schr¨odinger system. It is a coupled system which provides the mathematical modeling of the spontaneous generation of a magnetic field in a cold plasma under the subsonic limit. The main difficulty of the proof lies in exploring the inner structure of the system due to the fact that the nonlocal effect may bring some hinderance for establishing the conservation quantities of the mass and of the energy, constructing the corresponding variational structure, and deriving the key estimates to gain the expected result. To overcome this, the authors must establish local well-posedness theory, and set up suitable variational structure depending crucially on the inner structure of the system under study, which leads to define proper functionals and a constrained variational problem. By building up two invariant manifolds and then making a priori estimates for these nonlocal terms, the authors figure out a sharp threshold of global existence for the system under consideration.     

The Cauchy Problem for Coupled Nonlinear Schrödinger Equations with Linear Damping: Local and Global Existence and Blowup of Solutions

The authors study, by applying and extending the methods developed by Cazenave(2003), Dias and Figueira(2014), Dias et al.(2014), Glassey(1994–1997), Kato(1987), Ohta and Todorova(2009) and Tsutsumi(1984), the Cauchy problem for a damped coupled system of nonlinear Schrdinger equations and they obtain new results on the local and global existence of H~1-strong solutions and on their possible blowup in the supercritical case and in a special situation, in the critical or supercritical cases.     

Existence Results for Nonlinear Schrödinger Equations With Electromagnetic Fields

 We consider the nonlinear Schr?dinger equation

Existence and Multiplicity Results for a Class of Nonlinear Schrödinger Equations with Magnetic Potential Involving Sign-Changing Nonlinearity

In this work we consider the following class of elliptic problems $\begin{cases} −∆_Au + u = a(x)|u|^{q−2}u + b(x)|u|^{p−2}u & {\rm in} & \mathbb{R}^N, \\u ∈ H^1_A (\mathbb{R}^N), \tag{P} \end{cases}$ with $2 < q < p < 2^∗ = \frac{2N}{N−2},$ $a(x)$ and $b(x)$ are functions that can change sign and satisfy some additional conditions; $u \in H^1_A (\mathbb{R}^N)$ and $A : \mathbb{R}^N → \mathbb{R}^N$ is a magnetic potential. Also using the Nehari method in combination with other complementary arguments, we discuss the existence of infinitely many solutions to the problem in question, varying the assumptions about the weight functions.     

Sharp Condition for Global Existence of Supercritical Nonlinear Schr?dinger Equation with a Partial Confinement

We study the L²-supercritical nonlinear Schr?dinger equation(NLS) with a partial confinement,which is the limit case of the cigar-shaped model in Bose-Einstein condensate(BEC). By constructing a cross constrained variational problem and establishing the invariant manifolds of the evolution fow, we show a sharp condition for global existence.     

Multilinear Eigenfunction Estimates for the Harmonic Oscillator and the Nonlinear Schrödinger Equation with the Harmonic Potential

We consider the Cauchy problem for the cubic nonlinear Schr?dinger equation with the harmonic potential. We prove global well-posedness below the energy class in energy subcritical cases. The main ingredients for the proof are a multilinear eigenfunction estimate for the harmonic oscillator and the I-method. Submitted: January 13, 2008. Accepted: February 11, 2009.     

Sign-Changing Multi-Peak Solutions for Nonlinear Schrödinger Equations with Compactly Supported Potential

In this paper, we aim to prove the existence and concentration of sign-changing H ¹(? ^N ) solutions to the following nonlinear Schrödinger equations $$-\varepsilon^2\Delta u_{\varepsilon}+V(x)u_{\varepsilon}=K(x)|u_{\varepsilon}|^{p-1}u_{\varepsilon} $$ with N≥3, $1<p<\frac{N+2}{N-2}$ , and ε>0. When the potential function V(x) has compact support, $V(x)\not\equiv 0$ and V(x)≥0, K(x) is permitted to be unbounded under some necessary restrictions, we will show that one sign-changing H ¹(? ^N ) solution exists and exhibits concentration profile around local minimum points of the ground energy function $G(\xi)\equiv V^{\theta}(\xi) K^{-\frac{2}{p-1}}(\xi)$ with $\theta=\frac{p+1}{p-1}-\frac{N}{2}$ in the semiclassical limit ε→0.     

Global Attractor for a Generalized Discrete Nonlinear Schrödinger Equation

A generalized discrete nonlinear Schrödinger equation $$i\dot{u}_n(t)+\sum_{m=-\infty}^{+\infty} J(n-m)u_m(t)+g\bigl(u_n(t)\bigr)+i\gamma u_n(t)=f_n,\quad n\in\mathbb{Z}, $$ with long-range interactions in weighted spaces \(\ell_{\mathbf{{q}}}^{2}\) is considered. Under suitable assumptions on the coupling constants J(m), the damping γ and the weight \(\mathbf{{q}}=(q_{n})_{n\in \mathbb{Z}}\) , the existence of a global attractor is proved.     

The Existence of Infinitely Many Solutions for the Nonlinear Schrödinger–Maxwell Equations

In this paper, by using variational methods and critical point theory, we shall mainly be concerned with the study of the existence of infinitely many solutions for the following nonlinear Schrödinger–Maxwell equations $$\left\{\begin{array}{l@{\quad}l}-\triangle u + V(x)u + \phi u = f(x, u), \quad \; \, {\rm in} \, \mathbb{R}^{3},\\ -\triangle \phi = u^{2}, \quad \quad \qquad \quad \quad \quad \quad {\rm in} \, \mathbb{R}^{3},\end{array}\right.$$ where the potential V is allowed to be sign-changing, under some more assumptions on f, we get infinitely many solutions for the system.     

Smale Horseshoes and Symbolic Dynamics in Perturbed Nonlinear Schrödinger Equations

chaos , which offers an interpretation of the numerical observation on the perturbed NLS system: chaotic center-wing jumping, of course under the ``except one point'—type conditions (A1)—(A3). This study is a generalization of the finite-dimensional study [14] to infinite-dimensional perturbed NLS systems. Received September 16, 1996; revised September 23, 1997; final version received May 12, 1998     

Existence of Solutions for a Quasilinear Schrödinger Equation with Potential Vanishing

Acta Mathematicae Applicatae Sinica, English Series - We study the following quasilinear Schrödinger equation $$ - \Delta u + V(x)u - \Delta ({u^2})u = K(x)g(u),\,\,\,\,\,\,\,\,x \in...     

Modified Wave Operator for a System of Nonlinear Schrödinger Equations in 2d

We consider a system of nonlinear Schrödinger equations with quadratic nonlinearities in two space dimensions. We prove the existence of modified wave operators or wave operators under some mass conditions.     

Stochastic Nonlinear Equations of Schrödinger Type

In this article, the solution for a stochastic nonlinear equation of Schrödinger type, which is perturbed by an infinite dimensional Wiener process, is investigated. The existence of the solution is proved by using the Galerkin method. Moment estimates for the solution are also derived. Examples from physics are given in the final part of the article.     

Group Classification of Nonlinear Schrödinger Equations

We propose an approach to problems of group classification. By using this approach, we perform a complete group classification of nonlinear Schrödinger equations of the form i _t + + F(, *) = 0.     

Global Well-Posedness for Higher-Order Schrödinger Equations in Weighted L2 Spaces

We obtain the global well-posedness for Schrödinger equations of higher orders in weighted L² spaces. This is based on weighted L² Strichartz estimates for the corresponding propagator with higher-order dispersion. Our method is also applied to the Airy equation which is the linear component of Korteweg-de Vries type equations.     

Splitting Integrators for Nonlinear Schrödinger Equations Over Long Times

Conservation properties of a full discretization via a spectral semi-discretization in space and a Lie–Trotter splitting in time for cubic Schrödinger equations with small initial data (or small nonlinearity) are studied. The approximate conservation of the actions of the linear Schrödinger equation, energy, and momentum over long times is shown using modulated Fourier expansions. The results are valid in arbitrary spatial dimension.