Asymptotics for the third‐order nonlinear Schrödinger equation in the critical case

We consider the Cauchy problem for the third‐order nonlinear Schrödinger equation where and is the Fourier transform. Our purpose in this paper is to prove the large time asymptoitic behavior of solutions for the defocusing case λ > 0 with a logarithmic correction under the non zero mass condition Copyright © 2016 John Wiley & Sons, Ltd.     

Nonvanishing at spatial extremity solutions of the defocusing nonlinear Schrödinger equation

We show local existence of certain type of solutions for the Cauchy problem of the defocusing nonlinear Schrödinger equation with pure power nonlinearity, in various cases of open sets, unbounded or bounded. These solutions do not vanish at the boundary or at infinity. We also show, in certain cases, that these solutions are unique and global.     

Construction of solutions of the defocusing nonlinear Schrödinger equation with asymptotically time‐periodic boundary values

We study the defocusing nonlinear Schrödinger equation in the quarter plane with asymptotically periodic boundary values. We use the unified transform method, also known as the Fokas method, and the Deift‐Zhou nonlinear steepest descent method to construct solutions in a sector close to the boundary whose leading behavior is described by a single exponential plane wave. Furthermore, we compute the subleading terms in the long‐time asymptotics of the constructed solutions.     

Controllability of the nonlinear Schrödinger equation in the vicinity of the ground state

Local exact controllability of the one‐dimensional NLS (subject to zero‐boundary conditions) with distributed control is shown to hold in a H¹‐neighbourhood of the nonlinear ground state. The Hilbert Uniqueness Method (HUM), due to Lions, is applied to the linear control problem that arises by linearization around the ground state. The application of HUM crucially depends on the spectral properties of the linearized NLS operator which are given in detail. Copyright © 2007 John Wiley & Sons, Ltd.     

N solutions for a derivative nonlinear Schrödinger‐type equation via Riemann‐Hilbert approach

The inverse scattering transform for the derivative nonlinear Schrödinger‐type equation is studied via the Riemann‐Hilbert approach. In the direct scattering process, the spectral analysis of the Lax pair is performed, from which a Riemann‐Hilbert problem is established for the derivative nonlinear Schrödinger‐type equation. In the inverse scattering process, N‐soliton solutions of the derivative nonlinear Schrödinger‐type equation are obtained by solving Riemann‐Hilbert problems corresponding to the reflectionless cases. Moreover, the dynamics of the exact solutions are discussed.     

Derivation of Korteweg‐de Vries flow equations from fourth order nonlinear Schrödinger equation

We perform a multiple scale analysis on the fourth order nonlinear Schrödinger equation in the Hamiltonian form together with the Hamiltonian function. We derive, as amplitude equations, Korteweg‐de Vries flow equations in the bi‐Hamiltonian form with the corresponding Hamiltonian functions. Copyright © 2013 John Wiley & Sons, Ltd.     

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     

Exact soliton solution for the fourth‐order nonlinear Schrödinger equation with generalized cubic‐quintic nonlinearity

In this paper, we investigate the fourth‐order nonlinear Schrödinger equation with parameterized nonlinearity that is generalized from regular cubic‐quintic formulation in optics and ultracold physics scenario. We find the exact solution of the fourth‐order generalized cubic‐quintic nonlinear Schrödinger equation through modified F‐expansion method, identifying the particular bright soliton behavior under certain external experimental setting, with the system's particular nonlinear features demonstrated. Copyright © 2016 John Wiley & Sons, Ltd.     

A Fokas approach to the coupled modified nonlinear Schrödinger equation on the half‐line

In this work, we discuss the coupled modified nonlinear Schrödinger (CMNLS) equation, which describe the pulse propagation in the picosecond or femtosecond regime of the birefringent optical fibers. By use of the Fokas approach, the initial‐boundary value problem for the CMNLS equation related to a 3×3 matrix Lax pair on the half‐line is to be analyzed. Assuming that the solution {u(x,t),v(x,t)} of CMNLS equation exists, we will prove that it can be expressed in terms of the unique solution of a 3×3 matrix Riemann‐Hilbert problem formulated in the plane of the complex spectral parameter λ. Moreover, we also get that some spectral functions s(λ) and S(λ) are not independent of each other but meet a global relationship.     

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.     

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.     

Superconvergence analysis of Crank‐Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation

The purpose of this article is to apply nonconforming finite element(FE) to solve a generalized nonlinear Schrödinger equation. First, a new important property of nonconforming FE (see ( 2.3 ) of Lemma 2 below) is proved by use of BHX lemma and the integral identities techniques. Second, a linearized Crank‐Nicolson fully discrete scheme is constructed and the superclose error estimate of order for original variable u in broken H¹‐norm is also derived by using the properties of element and the splitting argument for nonlinear terms, while previous works always only obtain convergent error estimates with this element. Furthermore, the global superconvergence is arrived at by the interpolated postprocessing technique. Finally, two numerical experiments are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and τ is the time step.