Coupled nonlinear Schrodinger systems describe some physical phenomena such as the propagation in birefringent optical fibers, Kerr-like photorefractive media in optics and Bose-Einstein condensates. In this paper, we study the existence of concentrating solutions of a singularly perturbed coupled nonlinear Schrodinger system, in presence of potentials. We show how the location of the concentration points depends strictly on the potentials.
The initial value problem for some defocusing coupled nonlinear Schrodinger equations is investigated. Global well-posedness and scattering are established.
A regularized $α-$system of the Nonlinear Schrödinger Equation (NLS) with $2σ$ nonlinear power in dimension $N$ is studied. We prove existence and uniqueness of local solution in the case $1 \le σ<\frac{4}{N-2}$ and existence and uniqueness of global solution in the case $1 \le σ< \frac{4}{N}$. When $α\to 0^+$, this regularized system will converge to the classical NLS in the appropriate range. In particular, the purpose of this numerical regularization is to shed light on the profile of the blow up solutions of the original Nonlinear Schrödinger Equation in the range $\frac{2}{N}\le σ<\frac{4…
We prove that for the mass critical nonlinear Schrodinger equations, the wave operators and their inverse are related explicitly in terms of the Fourier transform. We discuss some consequences of this property. In the one-dimensional case, we show a precise similarity between the mass critical nonlinear Schrodinger equation and a nonlinear Schrodinger equation of derivative type.
We prove asymptotic stability of trapped solitons in the generalized nonlinear Schrödinger equation with a potential in dimension 1 and for even potential and even initial conditions.
Using a sharp Gagliardo-Nirenberg type inequality, well-posedness issues of the initial value problem for a fractional inhomogeneous Schrodinger equation are investigated.
In this paper we prove the existence of a nontrivial solution to the nonlinear Schrodinger-Maxwell equations in $\R^3,$ assuming on the nonlinearity the general hypotheses introduced by Berestycki & Lions.
We prove the existence of ground state in a multidimensional nonlinear Schrodinger model of paraxial beam propagation in isotropic local media with saturable nonlinearity. Such ground states exist in the form of bright counterpropagating solitons. From the proof, a general threshold condition on the beam coupling constant for the existence of such fundamental solitons follows.
We consider a system of nonlinear Schrodinger equations with three waves interaction studying the existence of ground state solutions. In particular, we find a vector ground state, namely a ground state with the three components all different from zero.
We consider asymptotic stability of a small solitary wave to supercritical 2-dimensional nonlinear Schrödinger equations $$ iu_t+Δu=Vu\pm |u|^{p-1}u \quad\text{for $(x,t)\in\mathbb{R}^2\times\mathbb{R}$,}$$ in the energy class.
In this paper we consider generalized nonlinear Schrödinger equations with external potentials. we compute the forth and the sixed order Fermi Golden Rules (FGR), conjectured in our previous papers, which is used in a study of the asymptotic dynamics of trapped solitons.
We show that the thermodynamics of the focusing cubic discrete nonlinear Schrodinger equation are exactly solvable in dimensions three and higher. A number of explicit formulas are derived.
We consider construction of ansatzes for nonlinear Schrodinger equations in three space dimensions and arbitrary nonlinearity, and conditions of their reduction to ordinary differential equations. Complete description of ansatzes of certain types is presented. We also discuss the relationship between solutions, and both Lie and conditional symmetry of these equations.
In this paper we use a concentration and compactness argument to prove the existence of a nontrivial nonradial solution to the nonlinear Schrodinger-Poisson equations in R3, assuming on the nonlinearity the general hypotheses introduced by Berestycki & Lions
Some focusing coupled Schrodinger equations are investigated. First, existence of ground state is obtained. Second, global and non global existence of solutions are discussed via potential-well method. Finally, strong instability of standing waves is established.
We investigate the initial value problem for some defocusing coupled nonlinear fourth-order Schrodinger equations. Global well-posedness and scattering in the energy space are obtained.
In this paper, a class of coupled systems of nonlinear Schrodinger equations with sign-changing potential, including the linearly coupled case, is considered. The existence of non-trivial bound state solutions via linking methods for cones in Banach spaces is proved.
We prove existence and multiplicity of bound and ground state solutions, under appropriate conditions on the parameters, for a bi-harmonic stationary system of coupled nonlinear Schrödinger--Korteweg-de Vries equations.
We investigate some focusing fourth-order coupled Schrodinger equations. Existence of ground state and global well-posedness are obtained. Moreover, the best constant of some Gagliardo-Nirenberg inequality is studied.
We give short survey on the question of asymptotic stability of ground states of nonlinear Schrödinger equations, focusing primarily on the so called nonlinear Fermi Golden Rule.
Large deviation principle by the weak convergence approach is established for the stochastic nonlinear Schrodinger equation in one-dimension and as an application the exit problem is investigated.
In this paper, we prove the scattering for radial solutions to energy-critical nonlinear Schrödinger equations with regular potentials in defocusing case.
In this paper we consider the transverse instability for a nonlinear Schrödinger equation with a linear potential on $\mathbb{R} \times \mathbb{T}_L$, where $2πL$ is the period of the torus $\mathbb{T}_L$. Rose and Weinstein showed the existence of a stable standing wave for a nonlinear Schrödinger equation with a linear potential. We regard the standing wave of nonlinear Schrödinger equation on $\mathbb{R}$ as a line standing wave of nonlinear Schrödinger equation on $\mathbb{R} \times \mathbb{T}_L$. We show the stability of line standing waves for all $L>0$.
We justify supercritical geometric optics in small time for the defocusing semiclassical Nonlinear Schrodinger Equation for a large class of non-necessarily homogeneous nonlinearities. The case of a half-space with Neumann boundary condition is also studied.
We prove the existence of bound and ground states for a system of coupled nonlinear Schrödinger-Korteweg-de Vries equations, depending on the size of the coupling coefficient.
In this short note, we present some decay estimates for nonlinear solutions of 3d quintic, 3d cubic and 2d quintic NLS (nonlinear Schrödinger equations).
We look for positive solutions to the nonlinear Schrodinger equation with a potential, under the hypothesis of zero mass on the nonlinearity, in a particular situation. Existence and multiplicity results are provided.
This paper is concerned with the Cauchy problem for an inhomogeneous nonlinear Schrodinger equation with exponential growth nonlinearity and harmonic potential in two space dimensions. We prove global well-posedness, existence of the associated ground state and instability of the standing wave.
We prove wellposedness of the Cauchy problem for the nonlinear Schrodinger equation for any defocusing power nonlinearity on a domain of the plane with Dirichlet boundary conditions. The main argument is based on a generalized Strichartz inequality on manifolds with Lipschitz metric.
This paper is concerned with the quantum Zakharov system. We prove that when the ionic speed of sound goes to infinity, the solution to the fourth order Schrodinger part of the quantum Zakharov system converges to the solution to quantum modified nonlinear Schrodinger eqaution.
We consider systems of weakly coupled Schrödinger equations with nonconstant potentials and we investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.
In this paper, we consider the energy critical nonlinear Schrödinger equation with a repulsive inverse square potential. In particular, we deal with radial initial data, whose energy is equal to the energy of static solution to the corresponding nonlinear Schrödinger equation without a potential. We investigate time behavior of the radial solutions with such initial data.
In this work, we explain in what sense the generic level set of the constants of motion for the periodic nonlinear Schrodinger equation is an infinite dimensional torus on which each generalized nonlinear Schrodinger flow is reduced to straight line almost periodic motion, and describe how neighboring generic infinite dimensional tori are connected.
The present paper is devoted to weighted Nonlinear Schrödinger- Poisson systems with potentials possibly unbounded and vanishing at infinity. Using a purely variational approach, we prove the existence of solutions concentrating on a circle.
We consider the nonlinear Schrödinger equation with a repulsive Dirac delta potential in one dimensional Euclidean space. We classify the global dynamics of even solutions with the same action as the high-frequency ground state standing wave solutions.
We consider the nonlinear derivative Schrodinger equation with a quintic nonlinearity, on the one dimensional torus. We exhibit that the nonlinear dynamic properties consist of four frequency modes initially excited, whose frequencies include the resonant clusters and phase matched resonant interactions of nonlinearities. This phenomena arrests energy transfers between low and high modes, which are quantified by a growth in the Sobolev norm.
We derive new results about existence and uniqueness of local and global solutions for nonlinear Schrodinger equation, including self-similar global solutions. Our analysis is performed in the framework of Marcinkiewicz spaces.
We consider the Cauchy problem of a dissipative nonlinear Schrödinger equation with a time dependent harmonic potential. We find a critical situation that the $L^2$-norm of dissipative solutions decays or not and which is decided by a nonlinear power and time decay order of harmonic potential.
We prove the existence of travelling-wave solutions for a system of coupled nonlinear Schrödinger equations arising in nonlinear optics. Such a system describes second-harmonic generation in optical materials with $χ^{(2)}$ nonlinearity. To prove the existence of travelling waves, we employ the method of concentration compactness to prove the relative compactness of minimizing sequences of the associated variational problem.
We prove that the knowledge of the Dirichlet-to-Neumann map, measured on a part of the boundary of a bounded domain in $\mathbb{R}^n, n\geq2$, can uniquely determine, in a nonlinear magnetic Schrödinger equation, the vector-valued magnetic potential and the scalar electric potential, both being nonlinear in the solution.
The long-time asymptotics is analyzed for finite energy solutions of the 1D discrete Schrödinger equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the phase rotation group. For initial states close to a solitary wave, the solution converges to a sum of another solitary wave and dispersive wave which is a solution to the free Schrödinger equation. The proofs use the strategy of Buslaev-Perelman: the linerization of the dynamics on the solitary manifold, the symplectic orthogonal projection, method of majorants, etc.
A refinement of uniform resolvent estimate is given and several smoothing estimates for Schrodinger equations in the critical case are induced from it. The relation between this resolvent estimate and radiation condition is discussed. As an application of critical smoothing estimates, we show a global existence results for derivative nonlinear Schrodinger equations.
We prove a nonlinear Poisson type formula for the Schrodinger group. Such a formula had been derived in a previous paper by the authors, as a consequence of the study of the asymptotic behavior of nonlinear wave operators for small data. In this note, we propose a direct proof, and extend the range allowed for the power of the nonlinearity to the set of all short range nonlinearities. Moreover, energy-critical nonlinearities are allowed.
