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Masahito Ohta

Publications and source records attributed to Masahito Ohta.

16 recordsLinked to original sources
arXiv2026

Global existence for a Zakharov type system in a domain

We study an initial-boundary value problem for a Zakharov type system in three space dimensions in a general domain. Under a smallness assumption on the initial data, we construct a unique global strong solution. The solution is obtained by a direct approach based on higher-order energy estimates and the construction of a Cauchy sequence in suitable Banach spaces, without employing compactness methods. Furthermore, we obtain estimates on the growth of higher-order Sobolev norms of solutions.

math.AP↗
arXiv2020

Instability of the solitary waves for the generalized Boussinesq equations

In this work, we consider the following generalized Boussinesq equation \begin{align*} \partial_{t}^2u-\partial_{x}^2u+\partial_{x}^2(\partial_{x}^2u+|u|^{p}u)=0,\qquad (t,x)\in\mathbb R\times \mathbb R, \end{align*} with $0<p<\infty$. This equation has the traveling wave solutions $ϕ_ω(x-ωt)$, with the frequency $ω\in (-1,1)$ and $ϕ_ω$ satisfying \begin{align*} -\partial_{xx}ϕ_ω+(1-{ω^2})ϕ_ω-ϕ_ω^{p+1}=0. \end{align*} Bona and Sachs (1988) proved that the traveling wave $ϕ_ω(x-ωt)$ is orbitally stable when $0<p<4,$ $\frac p4<ω^2<1$. Liu (1993) proved the orbital instability under the conditions $0<p<4,$ $ω^2<\frac p4$ or $p\ge 4,$ $ω^2<1$. In this paper, we prove the orbital instability in the degenerate case $0<p<4,ω^2=\frac p4$ .

math.AP↗
arXiv2020

Blow-up and strong instability of standing waves for the NLS-$δ$ equation on a star graph

We study strong instability (by blow-up) of the standing waves for the nonlinear Schrödinger equation with $δ$-interaction on a star graph $Γ$. The key ingredient is a novel variational technique applied to the standing wave solutions being minimizers of a specific variational problem. We also show well-posedness of the corresponding Cauchy problem in the domain of the self-adjoint operator which defines $δ$-interaction. This permits to prove virial identity for the $H^1$- solutions to the Cauchy problem. We also prove certain strong instability results for the standing waves of the NLS-$δ'$ equation on the line.

math.AP↗
arXiv2019

Stability of Standing Waves for a Nonlinear Klein-Gordon Equation with Delta Potentials

In this paper, we study local well-posedness and orbital stability of standing waves for a singularly perturbed one-dimensional nonlinear Klein-Gordon equation. We first establish local well-posedness of the Cauchy problem by a fixed point argument. Unlike the unperturbed case, a noteworthy difficulty here arises from the possible non-unitarity of the semigroup generating the corresponding linear evolution. We then show that the equation is Hamiltonian and we establish several stability/instability results for its standing waves. Our analysis relies on a detailed study of the spectral properties of the linearization of the equation, and on the well-known 'slope condition' for orbital stability.

math.AP↗
arXiv2018

Strong instability of standing waves with negative energy for double power nonlinear Schrödinger equations

We study the strong instability of ground-state standing waves $e^{iωt}ϕ_ω(x)$ for $N$-dimensional nonlinear Schrödinger equations with double power nonlinearity. One is $L^2$-subcritical, and the other is $L^2$-supercritical. The strong instability of standing waves with positive energy was proven by Ohta and Yamaguchi (2015). In this paper, we improve the previous result, that is, we prove that if $\partial_λ^2S_ω(ϕ_ω^λ)|_{λ=1}\le0$, the standing wave is strongly unstable, where $S_ω$ is the action, and $ϕ_ω^λ(x)\mathrel{\mathop:}=λ^{N/2}ϕ_ω(λx)$ is the $L^2$-invariant scaling.

math.AP↗
arXiv2018

Strong instability of standing waves for nonlinear Schrödinger equations with attractive inverse power potential

We study the strong instability of standing waves $e^{iωt}ϕ_ω(x)$ for nonlinear Schrödinger equations with an $L^2$-supercritical nonlinearity and an attractive inverse power potential, where $ω\in\mathbb{R}$ is a frequency, and $ϕ_ω\in H^1(\mathbb{R}^N)$ is a ground state of the corresponding stationary equation. Recently, for nonlinear Schrödinger equations with a harmonic potential, Ohta (2018) proved that if $\partial_λ^2S_ω(ϕ_ω^λ)|_{λ=1}\le0$, then the standing wave is strongly unstable, where $S_ω$ is the action, and $ϕ_ω^λ(x)\mathrel{\mathop:}=λ^{N/2}ϕ_ω(λx)$ is the scaling, which does not change the $L^2$-norm. In this paper, we prove the strong instability under the same assumption as the above-mentioned in inverse power potential case. Our proof is applicable to nonlinear Schrödinger equations with other potentials such as an attractive Dirac delta potential.

math.AP↗
arXiv2016

Strong instability of standing waves for nonlinear Schrödinger equations with harmonic potential

We study strong instability of standing waves $e^{iωt} ϕ_ω(x)$ for nonlinear Schrödinger equations with $L^2$-supercritical nonlinearity and a harmonic potential, where $ϕ_ω$ is a ground state of the corresponding stationary problem. We prove that $e^{iωt} ϕ_ω(x)$ is strongly unstable if $\partial_λ^2 E(ϕ_ω^λ) |_{λ=1}\le 0$, where $E$ is the energy and $v^λ(x)=λ^{N/2} v(λx)$ is the $L^2$-invariant scaling.

math.AP↗
arXiv2011

Bifurcation from semi-trivial standing waves and ground states for a system of nonlinear Schrödinger equations

We consider a system of nonlinear Schrödinger equations related to the Raman amplification in a plasma. We study the orbital stability and instability of standing waves bifurcating from the semi-trivial standing wave of the system. The stability and instability of the semi-trivial standing wave at the bifurcation point are also studied. Moreover, we determine the set of the ground states completely.

math.AP↗