Multi solitary waves for nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equation in for any d1, with a nonlinearity such that solitary waves exist and are stable. Let R_k(t,x) be K arbitrarily given solitary waves of the equation with different speeds v₁,v₂,…,v_K. In this paper, we prove that there exists a solution u(t) of the equation such that

     

Fractional logarithmic Schrödinger equations

By means of nonsmooth critical point theory, we obtain existence of infinitely many weak solutions of the fractional Schrödinger equation with logarithmic nonlinearity. We also investigate the Hölder regularity of the weak solutions. Copyright © 2015 JohnWiley & Sons, Ltd     

Ground states of nonlinear Schrödinger equations with potentials

In this paper we study the nonlinear Schrödinger equation:

We give general conditions which assure the existence of ground state solutions. Under a Nehari type condition, we show that the standard Ambrosetti–Rabinowitz super-linear condition can be replaced by a more natural super-quadratic condition.     

Optimal regularity for Schrödinger equations

In this paper we study the regularity theory for the Schrödinger equations under proper conditions. Furthermore, it will be verified that these conditions are optimal.     

On singular quasilinear Schrödinger equations with critical exponents

We consider a class of singular quasilinear Schrödinger equations of the form where are given functions, N ?3,λ is a positive constant, . By using variational methods together with concentration–compactness principle, we prove the existence of positive solutions of the aforementioned equations under suitable conditions on V (x ) and K (x ). Copyright © 2017 John Wiley & Sons, Ltd.     

Local well‐posedness for a system of quadratic nonlinear Schrödinger equations in one or two dimensions

In this article, the local well‐posedness of Cauchy's problem is explored for a system of quadratic nonlinear Schrödinger equations in the space L^p( R ⁿ). In a special case of mass resonant 2 × 2 system, it is well known that this problem is well posed in H^s(s≥0) and ill posed in H^s(s < 0) in two‐space dimensions. By translation on a linear semigroup, we show that the general system becomes locally well posed in L^p( R ²) for 1 < p < 2, for which p can arbitrarily be close to the scaling limit p_c=1. In one‐dimensional case, we show that the problem is locally well posed in L¹( R ); moreover, it has a measure valued solution if the initial data are a Dirac function. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiple solutions for generalized quasilinear Schrödinger equations

In this paper, we study the following generalized quasilinear Schrödinger equations: where N≥3, is a C¹ even function, g(0) = 1, and g′(s)≥0 for all s≥0. Under some suitable conditions, we prove that the equation has a positive solution, a negative solution, and a sequence of high‐energy solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

An application of exp‐function method to approximate general and explicit solutions for nonlinear Schrödinger equations

In this work, we implement a relatively new analytical technique, the exp‐function method, for solving nonlinear equations and absolutely a special form of generalized nonlinear Schrödinger equations, which may contain high‐nonlinear terms. This method can be used as an alternative to obtain analytical and approximate solutions of different types of fractional differential equations, which is applied in engineering mathematics. Some numerical examples are presented to illustrate the efficiency and the reliability of exp‐function method. It is predicted that exp‐function method can be found widely applicable in engineering. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1016–1025, 2011     

Extended Lagrangian approach for the defocusing nonlinear Schrödinger equation

We study the defocusing nonlinear Schrödinger (NLS) equation written in hydrodynamic form through the Madelung transform. From the mathematical point of view, the hydrodynamic form can be seen as the Euler–Lagrange equations for a Lagrangian submitted to a differential constraint corresponding to the mass conservation law. The dispersive nature of the NLS equation poses some major numerical challenges. The idea is to introduce a two‐parameter family of extended Lagrangians, depending on a greater number of variables, whose Euler–Lagrange equations are hyperbolic and accurately approximate NLS equation in a certain limit. The corresponding hyperbolic equations are studied and solved numerically using Godunov‐type methods. Comparison of exact and asymptotic solutions to the one‐dimensional cubic NLS equation (“gray” solitons and dispersive shocks) and the corresponding numerical solutions to the extended system was performed. A very good accuracy of such a hyperbolic approximation was observed.     

Two‐grid mixed finite‐element methods for nonlinear Schrödinger equations

Two‐grid mixed finite element schemes are developed for solving both steady state and unsteady state nonlinear Schrödinger equations. The schemes use discretizations based on a mixed finite‐element method. The two‐grid approach yields iterative procedures for solving the nonlinear discrete equations. The idea is to relegate all of the Newton‐like iterations to grids much coarser than the final one, with no loss in order of accuracy. Numerical tests are performed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 63‐73, 2012     

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.     

Minimal time for the approximate bilinear control of Schrödinger equations

We consider a quantum particle in a potential V(x) subject to a time‐dependent (and spatially homogeneous) electric field E(t) (the control). Boscain, Caponigro, Chambrion, and Sigalotti proved that, under generic assumptions on V, this system is approximately controllable on the unit sphere, in sufficiently large time T. In the present article, we show that, for a large class of initial states (dense in unit sphere), approximate controllability does not hold in arbitrarily small time. This generalizes our previous result for Gaussian initial conditions. Furthermore, we prove that the minimal time can in fact be arbitrarily large.     

The attractor of the dissipative coupled fractional Schrödinger equations

In the present paper, we consider the dissipative coupled fractional Schrödinger equations. The global well‐posedness by the contraction mapping principle is obtained. We study the long time behavior of solutions for the Cauchy problem. We prove the existence of global attractor. Copyright © 2013 John Wiley & Sons, Ltd.     

On solitary wave solutions to the nonlinear Schrödinger equations with two parameters

In this paper, we discuss the existence and multiplicity of positive solitary wave solutions for nonlinear Schrödinger equations with two parameters. The proof is based on the method of upper and lower solutions and the fixed point index.     

Multi‐peak standing waves for nonlinear Schrödinger equations involving critical growth

For the following singularly perturbed problem we construct a solution which concentrates at several given isolated positive local minimum components of V as . Here, the nonlinearity f is of critical growth. Moreover, the monotonicity of and the so‐called Ambrosetti–Rabinowitz condition are not required.     

Ground state solutions of the periodic discrete coupled nonlinear Schrödinger equations

In this paper, by using the Nehari manifold approach in combination with periodic approximations, we obtain the sufficient conditions on the existence of the nontrivial ground state solutions of the periodic discrete coupled nonlinear Schrödinger equations. Copyright © 2014 John Wiley & Sons, Ltd.