Investigating the exact integrability of the multiwave nonlinear Schrödinger equation

It is shown that the multiwave nonlinear Schrödinger equation describing the evolution of several quasimonochromatic waves having the same group velocities is not exactly integrable (in the sense that no infinite sequence of local conservation laws and symmetries exists). The exact integrability for systems of the form w _t ⁱ =α_iw _xx ⁱ +a _klm ⁱ w^kw^lw^m is investigated, where α_i are different from zero.     

Exponential time integration of solitary waves of cubic Schrödinger equation

The aim of the present paper is to study the suitability of using exponential methods for the time integration of cubic Schrödinger equation till long times. We center on second-order methods, for which we prove a higher order of accuracy on the main invariants when integrating solitary waves. Some geometric implicit exponential methods are considered as well as some explicit suitably projected ones. The comparison in terms of efficiency is performed.     

Existence time for the semilinear Schrödinger equation

Based on the methods introduced by Klainerman and Ponce, and Cohn, a lower hounded estimate of the existence time for a kind of semilinear Schrödinger equation is ohtained in this paper. The implementation of this method depends on the L ^p ? L ^q estimate and the energy estimate.     

Global H1 solvability of the 3D logarithmic Schrödinger equation

In this paper we investigate the existence of a unique global mild solution in $H^1\left({\mathbb{R}}^3\right)$ of the initial-boundary value problem associated with the logarithmic Schrödinger equation $i{?}_t\psi =?D\Delta \psi +\sigma \log \left(\right|\psi {|}^2)\psi$, with $D>0$ and $\sigma \in \mathbb{R}?\left\{0\right\}$.     

Hölder regularity for the time fractional Schrödinger equation

In this paper, we investigate the Hölder regularity of solutions to the time fractional Schrödinger equation of order 1<α<2, which interpolates between the Schrödinger and wave equations. This is inspired by Hirata and Miao's work which studied the fractional diffusion-wave equation. First, we give the asymptotic behavior for the oscillatory distributional kernels and their Bessel potentials by using Fourier analytic techniques. Then, the space regularity is derived by employing some results on singular Fourier multipliers. Using the asymptotic behavior for the above kernels, we prove the time regularity. Finally, we use mismatch estimates to prove the pointwise convergence to the initial data in Hölder spaces. In addition, we also prove Hölder regularity result for the Schrödinger equation.     

Global existence of small solutions for the fourth-order nonlinear Schrödinger equation

We consider the Cauchy problem for the fourth-order nonlinear Schrödinger equation

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where \(n=1,2\). We prove global existence of small solutions under the growth condition of \(f\left( u\right) \) satisfying \(\left| \partial _{u}^{j}f\left( u\right) \right| \le C\left| u\right| ^{p-j},\) where \(p>1+\frac{4}{n},0\le j\le 3\).

     

Integral representations for the solutions of infinite order of the stationary Schrödinger equation in a cone

Each solution of infinite order of the stationary Schrödinger equation defined in a smooth cone and continuous in the closure can be represented in terms of the modified Poisson integral and an infinite series vanishing continuously on the boundary.     

Global solutions of nonlinear Schrödinger equations

We study the nonlinear Schrödinger equation in \(\mathbb {R}^n\) without making any periodicity assumptions on the potential or on the nonlinear term. This prevents us from using concentration compactness methods. Our assumptions are such that the potential does not change the essential spectrum of the linear operator. This results in \([0, \infty )\) being the absolutely continuous part of the spectrum. If there are an infinite number of negative eigenvalues, they will converge to 0. In each case we obtain nontrivial solutions. We also obtain least energy solutions.     

Theory of bifurcations of the Schrödinger equation

We obtain solvability conditions and a representation of solutions for a boundary value problem for a linear nonstationary Schrödinger equation in a Hilbert space as well as sufficient conditions for the bifurcation of solutions of this equation.     

Fractional Schrödinger equation; solvability and connection with classical Schrödinger equation

We consider the Dirichlet boundary problem for semilinear fractional Schrödinger equation with subcritical nonlinear term. Local and global in time solvability and regularity properties of solutions are discussed. But our main task is to describe the connections of the fractional equation with the classical nonlinear Schrödinger equation, including convergence of the linear semigroups and continuity of the nonlinear semigroups when the fractional exponent α approaches 1.     

Fast evaluation of exact transparent boundary condition for one-dimensional cubic nonlinear Schrödinger equation

Fast evaluation of the exact transparent boundary condition for the one-dimensional cubic nonlinear Schrödinger equation is considered in this paper. In [J. Comput. Math., 2007, 25(6): 730–745], the author proposed a fast evaluation method for the half-order time derivative operator. In this paper, we apply this method for the exact transparent boundary condition for the one-dimensional cubic nonlinear Schrödinger equation. Numerical tests demonstrate the effectiveness of the proposed method.     

Breathers in a damped-driven nonlinear Schrödinger equation

We construct time-periodic solitons of the parametrically driven damped nonlinear Schrödinger equation as solutions of a boundary value problem for the space-dependent Fourier coefficients.     

Global Solutions of Modified One-Dimensional Schrödinger Equation

In this paper, we consider the modified one-dimensional Schrödinger equation:$(D_t-F(D))u=λ|u|^2u,$where F(ξ) is a second order constant coefficients classical elliptic symbol, and with smooth initial datum of size $ε≪1$. We prove that the solution is global-in-time, combining the vector fields method and a semiclassical analysis method introduced by Delort. Moreover, we get a one term asymptotic expansion for $u$when $t→+∞$.