Modified scattering for the critical nonlinear Schrödinger equation

We consider the nonlinear Schrödinger equation

$i{u}_t+\mathrm{\Delta }u=\lambda |u{|}^{\frac{2}{N}}u$

in all dimensions $N\ge 1$, where $\lambda \in \mathbb{C}$ and $?\lambda \le 0$. We construct a class of initial values for which the corresponding solution is global and decays as $t\to \infty$, like $t^{?\frac{N}{2}}$ if $?\lambda =0$ and like ${(t\log ?t)}^{?\frac{N}{2}}$ if $?\lambda <0$. Moreover, we give an asymptotic expansion of those solutions as $t\to \infty$. We construct solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at $u=0$. To study the asymptotic behavior, we apply the pseudo-conformal transformation and estimate the solutions by allowing a certain growth of the Sobolev norms which depends on the order of regularity through a cascade of exponents.     

Existence and stability of ground states for fully discrete approximations of the nonlinear Schrödinger equation

In this paper we study the long time behavior of a discrete approximation in time and space of the cubic nonlinear Schrödinger equation on the real line. More precisely, we consider a symplectic time splitting integrator applied to a discrete nonlinear Schrödinger equation with additional Dirichlet boundary conditions on a large interval. We give conditions ensuring the existence of a numerical ground state which is close in energy norm to the continuous ground state. Such result is valid under a CFL condition of the form $\tau h^{-2}\le C$ where $\tau $ and $h$ denote the time and space step size respectively. Furthermore we prove that if the initial datum is symmetric and close to the continuous ground state $\eta $ then the associated numerical solution remains close to the orbit of $\eta ,\Gamma =\cup _\alpha \{e^{i\alpha }\eta \}$ , for very long times.     

Existence and nonexistence of blow-up solutions for a Schrödinger equation involving a nonlinear operator

By introducing a growth condition and using an iterative technique, we establish the results for the nonexistence and existence of positive entire blow-up solutions for a Schrödinger equation involving a nonlinear operator. Our main results improve and extend some existing works. In addition, we also give an example to illustrate our results.     

Existence of ground state solutions for quasilinear Schrödinger equations with super-quadratic condition

In this paper, we study the following quasilinear Schrödinger equation $-\Delta u+u-\Delta \left(u^2\right)u=h\left(u\right),x\in {\mathbb{R}}^N,$where $N\ge 3$, $2^1=\frac{2N}{N-2}$, $h$ is a continuous function. By using a change of variable, we obtain the existence of ground state solutions. Unlike the condition ${lim}_{\left|u\right|\to \infty }\frac{{\int }_0^{u}h\left(s\right)ds}{{\left|u\right|}^4}=\infty$, we only need to assume that ${lim}_{\left|u\right|\to \infty }\frac{{\int }_0^{u}h\left(s\right)ds}{{\left|u\right|}^2}=\infty$.     

Inverse scattering transform for a nonlocal derivative nonlinear Schrödinger equation

Theoretical and Mathematical Physics - We give a detailed discussion of a nonlocal derivative nonlinear Schrödinger (NL-DNLS) equation with zero boundary conditions at infinity in terms of the...     

Blow-up rate for critical nonlinear Schrödinger equation with Stark potential

We consider the blow-up solutions of the Cauchy problem for the critical nonlinear Schrödinger equation with Stark potential and the sharp lower and upper bounds of blow-up rate are established.     

Existence of positive solutions with peaks on a Clifford torus for a fractional nonlinear Schrödinger equation

Science China Mathematics - We consider the fractional nonlinear Schrödinger equation in this paper. Applying the finite reduction method, we prove that the equation has positive solutions...     

Existence of periodic and solitary waves for a nonlinear Schrödinger equation with nonlocal integral term of convolution type

We prove the existence of periodic solutions and solitons in the nonlinear Schrödinger equation with a nonlocal integral term of convolution type. By separating phase and amplitude, the problem is reduced to an integro-differential formulation that can be written as a fixed point problem for a suitable operator on a Banach space. Then a fixed point theorem due to Krasnoselskii can be applied.     

The nonlinear Schrödinger equation with a potential

We consider the cubic nonlinear Schrödinger equation with a potential in one space dimension. Under the assumptions that the potential is generic, sufficiently localized, with no bound states, we obtain the long-time asymptotic behavior of small solutions. In particular, we prove that, as time goes to infinity, solutions exhibit nonlinear phase corrections that depend on the scattering matrix associated to the potential. The proof of our result is based on the use of the distorted Fourier transform – the so-called Weyl–Kodaira–Titchmarsh theory – a precise understanding of the “nonlinear spectral measure” associated to the equation, and nonlinear stationary phase arguments and multilinear estimates in this distorted setting.     

Existence of solutions for a nonperiodic semilinear Schrödinger equation

In this paper we consider a class of semilinear Schrödinger equation which terms are asymptotically periodic at infinity. Under a weaker superquadratic condition on the nonlinearity, the existence of a ground state solution is established. The main tools employed here to overcome the new difficulties are the concentration-compactness principle and the Local Mountain Pass Theorem.     

Positive solutions of a Schrödinger equation with critical nonlinearity

We study the nonlinear Schröodinger equation with critical exponent 2^*= 2 N/( N-2), N 4, where a 0, has a potential well. Using variational methods we establish existence and multiplicity of positive solutions which localize near the potential well for small and large.