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.     

On global Strichartz estimates for non-trapping metrics

We prove global Strichartz estimates (with spectral cutoff on the low frequencies) for non-trapping metric perturbations of the Schrödinger equation, posed on the Euclidean space.     

Finite dimensional global attractor for a parametric nonlinear Schrödinger system with a trapping potential

We prove that a parametric nonlinear Schrödinger equation possesses a finite dimensional smooth global attractor in a suitable energy space.     

Symmetric form of the Hudson-Parthasarathy stochastic equation

We prove that the Hudson-Parthasarathy equation corresponds, up to unitary equivalence, to the strong resolvent limit of Schrödinger Hamiltonians in Fock space and that the symmetric form of this equation corresponds to the weak limit of the Schrödinger Hamiltonians.Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 726–750, November, 1996.     

Generalized Eigenfunctions for Waves in Inhomogeneous Media

Many wave propagation phenomena in classical physics are governed by equations that can be recast in Schrödinger form. In this approach the classical wave equation (e.g., Maxwell's equations, acoustic equation, elastic equation) is rewritten in Schrödinger form, leading to the study of the spectral theory of its classical wave operator, a self-adjoint, partial differential operator on a Hilbert space of vector-valued, square integrable functions. Physically interesting inhomogeneous media give rise to nonsmooth coefficients. We construct a generalized eigenfunction expansion for classical wave operators with nonsmooth coefficients. Our construction yields polynomially bounded generalized eigenfunctions, the set of generalized eigenvalues forming a subset of the operator's spectrum with full spectral measure.     

Exact explicit traveling wave solutions for two nonlinear Schrödinger type equations

The traveling wave solutions of the generalized nonlinear derivative Schrödinger equation and the high-order dispersive nonlinear Schrödinger equation are studied by using the approach of dynamical systems and the theory of bifurcations. With the aid of Maple, all bifurcations and phase portraits in the parametric space are obtained. All possible explicit parametric representations of the bounded traveling wave solutions (solitary wave solutions, kink and anti-kink wave solutions and periodic wave solutions) are given.     

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.     

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.     

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.     

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.     

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.     

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.     

Some new Strichartz estimates for the Schrödinger equation

We deal with fixed-time and Strichartz estimates for the Schrödinger propagator as an operator on Wiener amalgam spaces. We discuss the sharpness of the known estimates and we provide some new estimates which generalize the classical ones. As an application, we present a result on the wellposedness of the linear Schrödinger equation with a rough time-dependent potential.     

Generalized Dicke model as an integrable dynamical system inverse to the nonlinear Schrödinger equation

We prove that a dynamical system obtained by the space-time inversion of the nonlinear Schrödinger equation is equivalent to a generalized Dicke model. We study the complete Liouville integrability of the obtained dynamical system.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 126–128, January, 1995.Thus, we have shown that the generalized Dicke model, inverse to the nonlinear Schrödinger equation, is a completely Liouville integrable Hamiltonian flow of hydrodynamic type.     

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.     

Multiple solutions for some nonlinear Schrödinger equations with indefinite linear part

We consider a class of nonlinear Schrödinger equation with indefinite linear part in RN. We prove that the problem has at least three nontrivial solutions by means of Linking Theorem and (∇)-Theorem.     

Localizing estimates of the support of solutions of some nonlinear Schrödinger equations – The stationary case

The main goal of this paper is to study the nature of the support of the solution of suitable nonlinear Schrödinger equations, mainly the compactness of the support and its spatial localization. This question touches the very foundations underlying the derivation of the Schrödinger equation, since it is well-known a solution of a linear Schrödinger equation perturbed by a regular potential never vanishes on a set of positive measure. A fact, which reflects the impossibility of locating the particle. Here we shall prove that if the perturbation involves suitable singular nonlinear terms then the support of the solution is a compact set, and so any estimate on its spatial localization implies very rich information on places not accessible by the particle. Our results are obtained by the application of certain energy methods which connect the compactness of the support with the local vanishing of a suitable “energy function” which satisfies a nonlinear differential inequality with an exponent less than one. The results improve and extend a previous short presentation by the authors published in 2006.     

On the concentration properties for the nonlinear Schrödinger equation with a Stark potential

In this paper, we study blow-up solutions of the Cauchy problem to the L2 critical nonlinear Schrdinger equation with a Stark potential. Using the variational characterization of the ground state for nonlinear Schrdinger equation without any potential, we obtain some concentration properties of blow-up solutions, including that the origin is the blow-up point of the radial blow-up solutions, the phenomenon of L2-concentration and rate of L2-concentration of blow-up solutions.