Propagating wave patterns for the ‘resonant’ Davey–Stewartson system

The resonant nonlinear Schrödinger (RNLS) equation exhibits the usual cubic nonlinearity present in the classical nonlinear Schrödinger (NLS) equation together with an additional nonlinear term involving the modulus of the wave envelope. It arises in the context of the propagation of long magneto-acoustic waves in cold, collisionless plasma and in capillarity theory. Here, a natural (2 + 1) (2 spatial and 1 temporal)-dimensional version of the RNLS equation is introduced, termed the ‘resonant’ Davey–Stewartson system. The multi-linear variable separation approach is used to generate a class of exact solutions, which will describe propagating, doubly periodic wave patterns.     

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.     

On the Semiclassical Limit of the General Modified NLS Equation

We study the semiclassical limit of the so-called general modified nonlinear Schrödinger equation for initial data with Sobolev regularity, before shocks appear in the limit system. The strict hyperbolicity and genuine nonlinearity are proved for the dispersion limit of the cubic nonlinear case. The limiting transition from the MNLS equation to the NLS equation is also discussed.     

Homoclinic orbits for a perturbed quintic-cubic nonlinear Schrödinger equation

The existence of homoclinic orbits for a perturbed cubic-quintic nonlinear Schrödinger equation with even periodic boundary conditions under the generalized parameters conditions is established. We combined geometric singular perturbation theory, Melnikov analysis, and integrable theory to prove the persistence of homoclinic orbits.     

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.     

Discrete artificial boundary conditions for nonlinear Schrödinger equations

In this work we construct and analyze discrete artificial boundary conditions (ABCs) for different finite difference schemes to solve nonlinear Schrödinger equations. These new discrete boundary conditions are motivated by the continuous ABCs recently obtained by the potential strategy of Szeftel. Since these new nonlinear ABCs are based on the discrete ABCs for the linear problem we first review the well-known results for the linear Schrödinger equation. We present our approach for a couple of finite difference schemes, including the Crank–Nicholson scheme, the Dùran–Sanz-Serna scheme, the DuFort–Frankel method and several split-step (fractional-step) methods such as the Lie splitting, the Strang splitting and the relaxation scheme of Besse. Finally, several numerical tests illustrate the accuracy and stability of our new discrete approach for the considered finite difference schemes.     

Chern–Simons limit of the standing wave solutions for the Schrödinger equations coupled with a neutral scalar field

We show the existence of standing wave solutions to the Schrödinger equation coupled with a neutral scalar field. We also verify the Chern–Simons limit for these solutions. More precisely we prove that solutions to Eqs. (1.3)–(1.4) converge to the unique positive radially symmetric solution of the nonlinear Schrödinger equation (1.6) as the coupling constant q goes to infinity.     

Remarks on global existence for the supercritical nonlinear Schrödinger equation with a harmonic potential

This paper is concerned with the nonlinear Schrödinger equation with a harmonic potential which describes the attractive Bose–Einstein condensate under the magnetic trap. By combining the best constant of Gagliardo–Nirenberg's inequality with the characteristic of this equation, we derive out a global existence condition for the supercritical equation which coincides with the critical case.     

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.     

Derivation of Korteweg–de Vries flow equations from nonlinear Schrödinger equation

We perform a multiple scales analysis on the nonlinear Schrödinger (NLS) equation in the Hamiltonian form together with the Hamiltonian function. We derive, as amplitude equations, Korteweg–de Vries (KdV) flow equations in the bi-Hamiltonian form with the corresponding Hamiltonian functions.     

Exact travelling wave solutions for the generalized nonlinear Schrödinger (GNLS) equation with a source by Extended tanh–coth, sine–cosine and Exp-Function methods

The capability of Extended tanh–coth, sine–cosine and Exp-Function methods as alternative approaches to obtain the analytic solution of different types of applied differential equations in engineering mathematics has been revealed. In this study, the generalized nonlinear Schrödinger (GNLS) equation is solved by three different methods. To obtain the single-soliton solutions for the equation, the Extended tanh–coth and sine–cosine methods are used. Furthermore, for this nonlinear evolution equation the Exp-Function method is applied to derive various travelling wave solution. Results show that while the first two procedures easily provide a concise solution, the Exp-Function method provides a powerful mathematical means for solving nonlinear evolution equations in mathematical physics.     

Asymptotic stability of ground states in 3D nonlinear Schrödinger equation including subcritical cases

We consider a class of nonlinear Schrödinger equation in three space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L²) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time dependent, Hamiltonian, linearized dynamics around a careful chosen one parameter family of bound states that “shadows” the nonlinear evolution of the system. Due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type 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.     

Two-dimensional differential transform method for solving linear and non-linear Schrödinger equations

In this paper, we propose a reliable algorithm to develop exact and approximate solutions for the linear and nonlinear Schrödinger equations. The approach rest mainly on two-dimensional differential transform method which is one of the approximate methods. The method can easily be applied to many linear and nonlinear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. Several illustrative examples are given to demonstrate the effectiveness of the present method.     

New soliton solutions for a Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation

The repeated homogeneous balance is used to construct a new exact traveling wave solution of the Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many new exact traveling wave solutions are successfully obtained, which contain rational and periodic-like solutions. This method is straightforward and concise, and it can be applied to other nonlinear evolution equations.     

Solitary waves and a stability analysis of an equation of short and long dispersive waves

A universal model for the interaction of long nonlinear waves and packets of short waves with long linear carrier waves is given by a system in which an equation of Korteweg–de Vries (KdV) type is coupled to an equation of nonlinear Schrödinger (NLS) type. The system has solutions of steady form in which one component is like a solitary-wave solution of the KdV equation and the other component is like a ground-state solution of the NLS equation. We study the stability of solitary-wave solutions to an equation of short and long waves by using variational methods based on the use of energy–momentum functionals and the techniques of convexity type. We use the concentration compactness method to prove the existence of solitary waves. We prove that the stability of solitary waves is determined by the convexity or concavity of a function of the wave speed.     

On the Dirac delta as initial condition for nonlinear Schrödinger equations

In this article we will study the initial value problem for some Schrödinger equations with Dirac-like initial data and therefore with infinite L² mass, obtaining positive results for subcritical nonlinearities. In the critical case and in one dimension we prove that after some renormalization the corresponding solution has finite energy. This allows us to conclude a stability result in the defocusing setting. These problems are related to the existence of a singular dynamics for Schrödinger maps through the so-called Hasimoto transformation.     

High accuracy representation of the free propagator

A new matrix approximation to the free propagator of quantum mechanics is proposed. It is a variant of the well-known Fourier pseudospectral approximation. Numerical comparisons involving the linear and nonlinear Schrödinger equations are presented.     

On asymptotic stability in energy space of ground states of NLS in 2D

We transpose work by K. Yajima and by T. Mizumachi to prove dispersive and smoothing estimates for dispersive solutions of the linearization at a ground state of a Nonlinear Schrödinger equation (NLS) in 2D. As an application we extend to dimension 2D a result on asymptotic stability of ground states of NLS proved in the literature for all dimensions different from 2.     

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.