arXiv subjects

Kuranosuke Nishimura

Publications and source records attributed to Kuranosuke Nishimura.

3 recordsLinked to original sources

arXiv2019

Scattering for the quadratic nonlinear Schrödinger system in $\mathbb{R}^5$ without mass-resonance condition

We consider the quadratic nonlinear Schrödinger system (NLS system) \begin{align*}\begin{cases} i\partial_t u + Δu = v \overline{u}, \\ i\partial_t v+κΔv = u^2, \end{cases} \text{ on } I \times \mathbb{R}^5, \end{align*} where $κ>0$. The scattering below the standing wave solutions for NLS system was obtained by the first author when $κ= 1/2$. The condition of $κ=1/2$ is called mass-resonance. In this paper, we prove scattering below the standing wave solutions when $κ\neq 1/2$ under the radially symmetric assumption. Our proof is based on the concentration compactness and the rigidity by Kenig--Merle. Moreover, we discuss the concentration compactness and the rigidity for non-radial solutions.

math.AP↗

arXiv2018

Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition

We consider a mass-critical system of nonlinear Schödinger equations \begin{align*} \begin{cases} i\partial_t u +Δu =\bar{u}v,\\ i\partial_t v +κΔv =u^2, \end{cases} (t,x)\in \mathbb{R}\times \mathbb{R}^4, \end{align*} where $(u,v)$ is a $\mathbb{C}^2$-valued unknown function and $κ>0$ is a constant. If $κ=1/2$, we say the equation satisfies mass-resonance condition. We are interested in the scattering problem of this equation under the condition $M(u,v)<M(ϕ,ψ)$, where $M(u,v)$ denotes the mass and $(ϕ,ψ)$ is a ground state. In the mass-resonance case, we prove scattering by the argument of Dodson \cite{MR3406535}. Scattering is also obtained without mass-resonance condition under the restriction that $(u,v)$ is radially symmetric.

math.AP↗

arXiv2018

Blow-up of the radially symmetric solutions for the quadratic nonlinear Schrödinger system without mass-resonance

We consider the quadratic nonlinear Schrödinger system \begin{align*} \begin{cases} i\partial_t u +Δu =v \overline{u},\\ i\partial_t v +κΔv =u^2, \end{cases} \text{ on } I \times \mathbb{R}^d, \end{align*} where $1\leq d \leq 6$ and $κ>0$. In the lower dimensional case $d=1,2,3$, it is known that the $H^1$-solution is global in time. On the other hand, there are finite time blow-up solutions when $d=4,5,6$ and $κ=1/2$. The condition of $κ=1/2$ is called mass-resonance. In this paper, we prove finite time blow-up under radially symmetric assumption when $d=5,6$ and $κ\neq 1/2$ and we show blow-up or grow-up when $d=4$.

math.AP↗