Schemes for Generating Different Nonlinear Schrödinger Integrable Equations and Their Some Properties

Acta Mathematicae Applicatae Sinica, English Series - In the paper, we want to derive a few of nonlinear Schrödinger equations with various formats and investigate their properties, such as...     

Finite Time Extinction for Nonlinear Schrödinger Equation in 1D and 2D

We consider a nonlinear Schrödinger equation with power nonlinearity, either on a compact manifold without boundary, or on the whole space in the presence of harmonic confinement, in space dimension one and two. Up to introducing an extra superlinear damping to prevent finite time blow up, we show that the presence of a sublinear damping always leads to finite time extinction of the solution in 1D, and that the same phenomenon is present in the case of small mass initial data in 2D.     

Modified Wave Operator for a System of Nonlinear Schrödinger Equations in 2d

We consider a system of nonlinear Schrödinger equations with quadratic nonlinearities in two space dimensions. We prove the existence of modified wave operators or wave operators under some mass conditions.     

Breather and rogue wave solutions for the (2+1)-dimensional nonlinear Schrödinger–Maxwell–Bloch equation

In this paper, we mainly study the (2+1)-dimensional Schrödinger–Maxwell–Blochequation (SMBE). We have constructed the generalized N-fold Darboux transformations (DT), and based on the plane wave solutions, the breather and rogue wave solutions are systematically generated, the dynamical features of those solutions are graphically represented.     

On the Whitham system for the (2+1)-dimensional nonlinear Schrödinger equation

Whitham modulation equations are derived for the nonlinear Schrödinger equation in the plane ((2+1)-dimensional nonlinear Schrödinger [2d NLS]) with small dispersion. The modulation equations are obtained in terms of both physical and Riemann-type variables; the latter yields equations of hydrodynamic type. The complete 2d NLS Whitham system consists of six dynamical equations in evolutionary form and two constraints. As an application, we determine the linear stability of one-dimensional traveling waves. In both the elliptic and hyperbolic cases, the traveling waves are found to be unstable. This result is consistent with previous investigations of stability by other methods and is supported by direct numerical calculations.     

The complex 2-sphere in ℂ3 and Schrödinger flows

Science China Mathematics - By using holomorphic Riemannian geometry in ?3, the coupled Landau-Lifshitz equation (CLL) is proved to be exactly the equation of Schrödinger flows from...     

Blow-up Theory for the Coupled L 2-Critical Nonlinear Schrödinger System in the Plane

In this paper, we study the blow-up theory for the L ²-critical coupled nonlinear Schrödinger system in the Euclidean plane ${\mathbb{R}^{2}}In this paper, we study the blow-up theory for the L ²-critical coupled nonlinear Schr?dinger system in the Euclidean plane \mathbbR²{\mathbb{R}^{2}} .We show that at the finite time blow-up, similar results of Weinstein, Merle-Tsutsumi for scalar Schrodinger equations are true for the system.     

The Cauchy Problem for the Klein–Gordon–Schrödinger System in Low Dimensions Below the Energy Space

ABSTRACT

We show that the Klein–Gordon–Schrödinger system in one, two, and three dimensions has a global solution below the energy space. The proof uses the I-method recently introduced by Colliander et al. (2001 Colliander , J. , Keel , M. , Staffilani , G. , Takaoka , H. , Tao , T. ( 2001 ). Global well-posedness for Schrödinger equations with derivative . SIAM J. Math. Anal. 33 ( 3 ): 649 – 669 . [CROSSREF]  [Google Scholar]) and mixed type Strichartz estimates for the solutions of Schrödinger and Klein–Gordon equations, respectively.     

Algebraic Method for Group Classification of (1+1)-Dimensional Linear Schrödinger Equations

In the present paper, we study the existence of solutions for some nonlocal problems involving the \(p(x)\)-Laplacian operator. The approach is based on a new sub-supersolution method.     

Dispersive estimates for 1D discrete Schrödinger and Klein–Gordon equations

We derive the long-time asymptotics for solutions of the discrete 1D Schrödinger and Klein–Gordon equations.     

Sharp Threshold for Blowup and Global Existence in Nonlinear Schrödinger Equations Under a Harmonic Potential

ABSTRACT

We present a variational approach to study the nonlinear Schrödinger equations under a harmonic potential. By constructing a type of cross constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we derive a sharp threshold for global existence and blowup of the solution. The stability of the standing waves is also discussed.     

The Cauchy Problem of the Nonlinear Schr■dinger Equations in R~(1 1)

In this paper,we consider the local and global solution for the nonlinear Schrdinger equationwith data in the homogeneous and nonhomogeneous Besov space and the scattering result for small data.Thetechniques to be used are adapted from the Strichartz type estimate,Kato's smoothing effect and the maximalfunction(in time)estimate for the free Schrdinger operator.     

Uniqueness of the Modified Schrödinger Map in H 3/4+ε(ℝ2)

We establish the local well-posedness of the modified Schrödinger map in H ^3/4+ε(?²).