Related papers: Global existence for the defocusing nonlinear Schr…

We consider the Cauchy problem for the fractional nonlinear Schr\"{o}dinger equation (FNLS) on the one-dimensional torus with cubic nonlinearity and high dispersion parameter $\alpha > 1$, subject to a Gaussian random initial data of…

In this work I study the well-posedness of the Cauchy problem associated with the coupled Schr\"odinger equations {with quadratic nonlinearities}, which appears modeling problems in nonlinear optics. I obtain the local well-posedness for…

We study the Cauchy problem for the non-linear Schr\"odinger equation with singular potentials. For point-mass potential and nonperiodic case, we prove existence and asymptotic stability of global solutions in weak-L^{p} spaces. Specific…

We consider the question of global existence of small, smooth, and localized solutions of a certain fractional semilinear cubic NLS in one dimension, $$i\partial_t u - \Lambda u = c_0{|u|}^2 u + c_1 u^3 + c_2 u \bar{u}^2 + c_3 \bar{u}^3,…

The initial-boundary value problem in a bounded domain with moving boundaries and nonhomogeneous boundary conditions for a higher order nonlinear Schr\"odinger (HNLS) equation is considered. Existence and uniqueness of global weak solutions…

We consider the global regularity problem for defocusing nonlinear Schr\"odinger systems $$ i \partial_t + \Delta u = (\nabla_{{\bf R}^m} F)(u) + G $$ on Galilean spacetime ${\bf R} \times {\bf R}^d$, where the field $u\colon {\bf R}^{1+d}…

We study the asymptotic behavior of solutions of discrete nonlinear Schr\"odinger-type (DNLS) equations. For a conservative system, we consider the global in time solvability and the question of existence of standing wave solutions.…

In this paper, we first prove global well-posedness for the defocusing cubic nonlinear Schr\"odinger equation (NLS) on 4-dimensional tori - either rational or irrational - and with initial data in $H^1$. Furthermore, we prove that if a…

In this paper, we study the Cauchy problem of the quasilinear Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{lll} iu_t=\Delta u+2uh'(|u|^2)\Delta h(|u|^2)+F(|u|^2)u \quad {\rm for} \ x\in \mathbb{R}^N, \ t>0\\…

In this paper we consider the nonlinear fractional logarithmic Schr\"{o}dinger equation. By using a compactness method, we construct a unique global solution of the associated Cauchy problem in a suitable functional framework. We also prove…

In this paper, we consider the derivative nonlinear Schr\"odinger (DNLS) equation. While the existence theory has been intensely studied, properties like dispersive estimates for the solutions have not yet been investigated. Here we address…

In the present work we explore the potential of models of the discrete nonlinear Schr\"odinger (DNLS) type to support spatially localized and temporally quasiperiodic solutions on top of a finite background. Such solutions are rigorously…

We consider the Cauchy problem for the nonlinear Schr\"odinger equation $iu_t+ \Delta u+ \lambda |u|^\alpha u=0$ in $\R^N $, in the $H^s$-subcritical and critical cases $0<\alpha \le 4/(N-2s)$, where $0<s<N/2$. Local existence of solutions…

We consider the stochastic NLS with nonlinear Stratonovic noise for initial values in $L^2(R^d)$ and prove local existence and uniqueness of a mild solution for subcritical and critical nonlinearities. The proof is based on deterministic…

In this paper we investigate the global well-posedness and long-term behavior of solutions to the kinetic derivative nonlinear Schr\"odinger equation (KDNLS) on the real line. The equation incorporates both local cubic nonlinearities with…

We prove uniqueness of solutions to the Cauchy problem for the derivative nonlinear Schr\"odinger equation in $L^\infty_tH^{1/2}_x$. Our proof is based on the method of normal form reduction (NFR), which has been employed to obtain the…

We consider the Cauchy problem for the nonlinear Schr\"odinger equation on $\mathbb{R}^d$, where the initial data is in $\dot{H}^1(\mathbb{R}^d)\cap L^p(\mathbb{R}^d)$. We prove local well-posedness for large ranges of $p$ and discuss some…

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schr\"odinger equation with initial data $u_{0}\in X$, where $X\in\{M_{2,q}^{s}(\mathbb R), H^{\sigma}(\mathbb T),…

The existence of nonzero periodic travelling wave solutions for a general discrete nonlinear Schr\"odinger equation (DNLS) on finite one-dimensional lattices is proved. The DNLS features a general nonlinear term and variable range of…

We consider the Cauchy problem for the one-dimensional periodic cubic nonlinear fractional Schr{\"o}dinger equation (FNLS) with initial data distributed via its associated Gibbs measure. We construct global strong solutions with the flow…