Sharp conditions of global existence and scattering for a focusing energy-critical Schrödinger equation

We consider a focusing energy-critical Schrödinger equation with subcritical perturbations and address question related to the sharp criterion of global existence and scattering. By analyzing the variational characteristics of this equation, we established two types of invariant flows. Then approximating this equation by the energy-critical nonlinear Schrödinger equations with the same initial data and combining the properties of the invariant flows, we obtain the sharp conditions of global existence for this equation. Moreover, when the solution is globally defined, we prove the scattering.     

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.     

Triple Wronskian solutions of the coupled derivative nonlinear Schrödinger equations in optical fibers

A type of the coupled derivative nonlinear Schrödinger (CDNLS) equations are studied by means of symbolic computation, which can describe the wave propagation in birefringent optical fibers. Soliton solutions in the triple Wronskian form of the CDNLS equations are obtained. Elastic and inelastic collisions are both presented under some parametric conditions. In addition, generalized triple Wronskian solutions of a set of the coupled general derivative nonlinear Schrödinger (CGDNLS) equations are derived. Triple Wronskian identities are given to prove such solutions, which may also be used for other coupled nonlinear equations. Rational solutions of the CGDNLS equations are also obtained.     

A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations

In an abstract setting we prove a nonlinear superposition principle for zeros of equivariant vector fields that are asymptotically additive in a well-defined sense. This result is used to obtain multibump solutions for two basic types of periodic stationary Schrödinger equations with superlinear nonlinearity. The nonlinear term may be of convolution type. If the superquadratic term in the energy functional is convex, our results also apply in certain cases if 0 is in a gap of the spectrum of the Schrödinger operator.     

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.     

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.     

A simple direct method to find equivalence transformations of a generalized nonlinear Schrödinger equation and a generalized KdV equation

A simple direct method is presented to find equivalence transformations of nonlinear mathematical physics equations. By using the direct method, we obtain the continuous equivalence transformations of a class of nonlinear Schröequations with variable coefficients and a family of nonlinear KdV equations with variable coefficients. For the nonlinear Schrödinger equations with variable coefficients, the equivalence transformations obtained by the direct method coincide, in nature, with those obtained via the infinitesimal Lie criterion, but our computation is much simpler.     

On Some Discrete Variational Problems

Making use of a discrete version of the P.-L. Lions concentration-compactness principle, we establish some results on the existence of nontrivial solutions for nonlinear stationary discrete equations of the Schrödinger type.     

Strichartz estimates for the magnetic Schrödinger equation

We prove global, scale invariant Strichartz estimates for the linear magnetic Schrödinger equation with small time dependent magnetic field. This is done by constructing an appropriate parametrix. As an application, we show a global regularity type result for Schrödinger maps in dimensions n?6.     

Standing waves for nonlinear Schrödinger equations with singular potentials

We study semiclassical states of nonlinear Schrödinger equations with anisotropic type potentials which may exhibit a combination of vanishing and singularity while allowing decays and unboundedness at infinity. We give existence of spike type standing waves concentrating at the singularities of the potentials.     

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.     

New exact solutions for some coupled nonlinear partial differential equations using extended coupled sub-equations expansion method

An extended coupled sub-equations expansion method is proposed to seek new traveling wave solutions of the coupled nonlinear partial differential equations. The generalized Schrödinger-Boussinesq and coupled nonlinear Klein-Gordon-Schrödinger equations are used to illustrate the validity and the advantages of this method.     

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?     

Local smoothing effect on the Schrödinger equation with harmonic potential

We consider the linear Schrödinger equation with repulsive harmonic potential. We establish the local smoothing effect of this type of equations. Our work extends the related results obtained by L. Vega and N. Visciglia for the free Schrödinger equation.     

On the Blowing up of Solutions for one System of Schrödinger Equations

We consider one system of nonlinear Schrödinger equations and find the conditions when its solution blows up.     

A solution to Schrödinger's problem of non-linear integral equations

A solution of Schrödinger's system of non-linear integral equations determines the rate function of a large deviation principle for kernels with prescribed marginal distributions. This kind of large deviation principle has some meaning in quantum mechanics.Diffusion equations associated with Schrödinger equations have typically transition functions with singular creation and killing. Hence they provide measurable non-negative generally unbounded kernels which may vanish on sets with positive measure and which can possess infinite mass.For Schrödinger systems with such kernels, a solution is proved to exist uniquely in terms of a product measure. It is obtained from a variational principle for the local adjoint of a product measure endomorphism. The generally unbounded factors of the solution are characterized by integrability properties.     

Cauchy problem for nonlinear p-evolution equations

The local solvability of the Cauchy problem in Sobolev spaces is studied for a class of nonlinear partial differential equations incorporating weakly hyperbolic and Schrödinger equations.     

Singularly perturbed elliptic systems

We study the existence and concentration behavior of positive solutions for a class of Hamiltonian systems (two coupled nonlinear stationary Schrödinger equations). Combining the Legendre–Fenchel transformation with mountain pass theorem, we prove the existence of a family of positive solutions concentrating at a point in the limit, where related functionals realize their minimum energy. In some cases, the location of the concentration point is given explicitly in terms of the potential functions of the stationary Schrödinger equations.