Existence results for an indefinite unbounded perturbation of a resonant Schrödinger equation

We study existence results for a nonlinear Schrödinger equation at resonance. The nonlinearity is assumed to change sign, be unbounded but sublinear with a power like growth at infinity. Under a suitable coercivity assumption on the primitive of the nonlinear term on the kernel of the Schrödinger operator, we prove the existence of at least one solution.     

Dispersive blow-up for nonlinear Schrödinger equations revisited

The possibility of finite-time, dispersive blow-up for nonlinear equations of Schrödinger type is revisited. This mathematical phenomena is one of the conceivable explanations for oceanic and optical rogue waves. In dimension one, the fact that dispersive blow up does occur for nonlinear Schrödinger equations already appears in [9]. In the present work, the existing results are extended in several ways. In one direction, the theory is broadened to include the Davey–Stewartson and Gross–Pitaevskii equations. In another, dispersive blow up is shown to obtain for nonlinear Schrödinger equations in spatial dimensions larger than one and for more general power-law nonlinearities. As a by-product of our analysis, a sharp global smoothing estimate for the integral term appearing in Duhamel's formula is obtained.     

On the role of quadratic oscillations in nonlinear Schrödinger equations

We consider a nonlinear semi-classical Schrödinger equation for which it is known that quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. If the initial data is an energy bounded sequence, we prove that the nonlinear term has an effect at leading order only if the initial data have quadratic oscillations; the proof relies on a linearizability condition (which can be expressed in terms of Wigner measures). When the initial data is a sum of such quadratic oscillations, we prove that the associate solution is the superposition of the nonlinear evolution of each of them, up to a small remainder term. In an appendix, we transpose those results to the case of the nonlinear Schrödinger equation with harmonic potential.     

Boundary-value problem for the nonlinear Schrödinger equation

A mixed boundary-value problem for the nonlinear Schrödinger equation and its generalization is studied by the method used for the inverse scattering problem. A connection is established between conservation laws and boundary conditions in integrable boundary-value problems for higher nonlinear Schrödinger equations. It is shown that the generalized boundary-value problem requires a joint consideration of regular and singular solutions for the nonlinear Schrödinger equation with repulsion.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 169, pp. 151–165, 1988.     

Direct search for exact solutions to the nonlinear Schrödinger equation

A five-dimensional symmetry algebra consisting of Lie point symmetries is firstly computed for the nonlinear Schrödinger equation, which, together with a reflection invariance, generates two five-parameter solution groups. Three ansätze of transformations are secondly analyzed and used to construct exact solutions to the nonlinear Schrödinger equation. Various examples of exact solutions with constant, trigonometric function type, exponential function type and rational function amplitude are given upon careful analysis. A bifurcation phenomenon in the nonlinear Schrödinger equation is clearly exhibited during the solution process.     

Infinitely many small solutions for a modified nonlinear Schrödinger equations

In the present paper, we study the Modified Nonlinear Schrödinger Equations (MNSE). Without any growth condition on the nonlinear term, we obtain the existence of infinitely many small solutions for MNSE by a dual approach.     

Stability of standing waves for NLS with perturbed Lamé potential

We perturb a linear Schrödinger equation with Lamé potential with a small positive or negative potential. The new perturbed operator has one or more eigenvalues, at most one in each spectral gap. We then add a nonlinear term and study the stability of the corresponding nonlinear stationary waves.     

Existence for bounded positive solutions of Schrödinger equations in two-dimensional exterior domains

In this paper, by using the fixed point theory, under quite general conditions on the nonlinear term, we obtain an existence result of bounded positive solutions of Schrödinger equations in two-dimensional exterior domains.     

广义Zakharov方程解的最佳收敛速度

本文考虑一类含自生磁场效应的广义Zakhaorv方程,并对它的非线性Schrdinger极限行为开展研究,主要考虑解的收敛速度问题.通过对非线性项的细致估计,文中给出收敛速度与第一初始层及第二初始层之间的具体依赖关系.该工作改进了Zhang和Guo在2011年得到的收敛性结果.     

Nodal type bound states of Schrödinger equations via invariant set and minimax methods

For nonlinear Schrödinger equations in the entire space we present new results on invariant sets of the gradient flows of the corresponding variational functionals. The structure of the invariant sets will be built into minimax procedures to construct nodal type bound state solutions of nonlinear Schrödinger type equations.     

Necessary conditions and sufficient conditions for global existence for two-components nonlinear Schrödinger equations in the super critical case

In this paper, we consider two-components nonlinear Schrödinger equations in the super critical case. We establish a necessary condition and a sufficient condition of global existence of the solution for two-components nonlinear Schrödinger equations. These conditions are charge criterion of global existence in the super critical case, thereby extending the results in the critical case. Furthermore, we improve a blow-up condition.     

Coupled nonlinear Schrödinger systems with potentials

Coupled nonlinear Schrödinger 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 Schrödinger system, in presence of potentials. We show how the location of the concentration points depends strictly on the potentials.     

带三次项的非线性四阶Schrdinger方程的一个局部能量守恒格式

本文构造了带三次项的非线性四阶Schodinger方程的一个局部能量守恒格式.证明了该格式是线性稳定的,且能保持离散的整体能量守恒律及离散的电荷守恒律.最后通过数值算例验证了理论结果的正确性.     

Initial-Boundary Value Problems for Linear and Soliton PDEs

We consider evolution PDEs for dispersive waves in both linear and nonlinear integrable cases and formulate the associated initial-boundary value problems in the spectral space. We propose a solution method based on eliminating the unknown boundary values by proper restrictions of the functional space and of the spectral variable complex domain. Illustrative examples include the linear Schrödinger equation on compact and semicompact n-dimensional domains and the nonlinear Schrödinger equation on the semiline.     

Anisotropic subdiffractive solitons

We study solitons in the two-dimensional defocusing nonlinear Schrödinger equation with the spatio-temporal modulation of the external potential. The spatial modulation is due to a square lattice; the resulting macroscopic diffraction is rotationally symmetric in the long-wavelength limit but becomes anisotropic for shorter wavelengths. Anisotropic solitons-solitons with the square (x, y)-geometry - are obtained both in the original nonlinear Schrödinger model and in its averaged amplitude equation.     

Exact analytic solutions of Schrödinger linear and nonlinear equations

Exact analytic solutions of Schrödinger linear partial differential equations are obtained. Moreover, the cubic nonlinear Schrödinger equation is treated with the use of a well-known functional analytic method and the existence of convergent power series solutions is proved. From these solutions, under certain initial conditions, similar results as those presented in the literature are obtained.     

Sharp condition of global existence for second-order derivative nonlinear Schrödinger equations in two space dimensions

This paper discusses a class of second-order derivative nonlinear Schrödinger equations which are used to describe the upper-hybrid oscillation propagation. By establishing a variational problem, applying the potential well argument and the concavity method, we prove that there exists a sharp condition for global existence and blow-up of the solutions to the nonlinear Schrödinger equation. In addition, we also answer the question: how small are the initial data, the global solutions exist?     

Explicit dissipative schemes for boundary problems of generalized Schrödinger systems

In this paper, the author constructs a class of explicit schemes, spanning two time levels, for the initial-boundary-value problems of generalized nonlinear Schrödinger systems, and proves the convergence of these schemes with a series of prior estimates. For a single Schrödinger equation, the schemes are identical with those of the article [1].     

Conservation laws, bright matter wave solitons and modulational instability of nonlinear Schrödinger equation with time-dependent nonlinearity

In this paper, we consider a general form of nonlinear Schrödinger equation with time-dependent nonlinearity. Based on the linear eigenvalue problem, the complete integrability of such nonlinear Schrödinger equation is identified by admitting an infinite number of conservation laws. Using the Darboux transformation method, we obtain some explicit bright multi-soliton solutions in a recursive manner. The propagation characteristic of solitons and their interactions under the periodic plane wave background are discussed. Finally, the modulational instability of solutions is analyzed in the presence of small perturbation.     

Kato-type estimates for NLS equation in a scalar field and unique solvability of NKGS system in energy space

We consider Schrödinger equation in R²⁺¹${\mathbb{R}}^{2+1}$ with nonlinear scalar potential. The potentials are time-independent or determined as solutions to inhomogeneous wave equations. By constructing a modified propagator, we derive Kato-type smoothing estimates for the nonlinear Schrödinger (NLS) equation. With the help of these results, we prove the unique solvability of the nonlinear Klein–Gordon–Schrödinger (NKGS) system for all time in the energy space.