arXiv subjects

Kenjiro Ishizuka

Publications and source records attributed to Kenjiro Ishizuka.

4 recordsLinked to original sources

arXiv2026

Long-time asymptotics of (1,3)-sign solitary waves for the damped nonlinear Klein-Gordon equation

We consider the damped nonlinear Klein-Gordon equation: \begin{align*} \partial_{t}^2u-Δu+2α\partial_{t}u+u-|u|^{p-1}u=0, \ & (t,x) \in \mathbb{R} \times \mathbb{R}^d, \end{align*} where $α>0$, $2\leq d\leq 5$ and energy sub-critical exponents $p>2$. In this paper, we prove that any solution which is asymptotic to a superposition of four solitons with exactly one soliton of opposite sign evolves so that the three like-signed solitons spread out in an equilateral-triangle configuration centered at the oppositely signed soliton.

math.AP↗

arXiv2026

Long-time asymptotics of 3-solitary waves for the damped nonlinear Klein-Gordon equation

We consider the damped nonlinear Klein-Gordon equation: \begin{align*} \partial_{t}^2u-Δu+2α\partial_{t}u+u-|u|^{p-1}u=0, \ & (t,x) \in \mathbb{R} \times \mathbb{R}^d, \end{align*} where $α>0$, $1\leq d\leq 5$ and energy sub-critical exponents $p>2$. In this paper, we prove that 3-solitary waves behave as if the three solitons are on a line. Furthermore, the solitary waves have alternative signs and their distances are of order $\log{t}$.

math.AP↗

arXiv2024

Long-time asymptotics of the damped nonlinear Klein-Gordon equation with a delta potential

We consider the damped nonlinear Klein-Gordon equation with a delta potential \begin{align*} \partial_{t}^2u-\partial_{x}^2u+2α\partial_{t}u+u-γδ_0u-|u|^{p-1}u=0, \ & (t,x) \in \mathbb{R} \times \mathbb{R}, \end{align*} where $p>2$, $α>0,\ γ<2$, and $δ_0=δ_0 (x)$ denotes the Dirac delta with the mass at the origin. When $γ=0$, Côte, Martel and Yuan proved that any global solution either converges to 0 or to the sum of $K\geq 1$ decoupled solitary waves which have alternative signs. In this paper, we first prove that any global solution either converges to 0 or to the sum of $K\geq 1$ decoupled solitary waves. Next we construct a single solitary wave solution that moves away from the origin when $γ<0$ and construct an even 2-solitary wave solution when $γ\leq -2$. Last we give single solitary wave solutions and even 2-solitary wave solutions an upper bound for the distance between the origin and the solitary wave.

math.AP↗

arXiv2021

Global dynamics around 2-solitons for the nonlinear damped Klein-Gordon equations

Global behavior of solutions is studied for the nonlinear Klein-Gordon equation with a focusing power nonlinearity and a damping term in the energy space on the Euclidean space. We give a complete classification of solutions into 5 types of global behavior for all initial data in a small neighborhood of each superposition of two ground states (2-solitons) with the opposite signs and sufficient spatial distance. The neighborhood contains, for each sign of the ground state, the manifold with codimension one in the energy space, consisting of solutions that converge to the ground state at time infinity. The two manifolds are joined at their boundary by the manifold with codimension two of solutions that are asymptotic to 2-solitons moving away from each other. The connected union of these three manifolds separates the rest of the neighborhood into the open set of global decaying solutions and that of blow-up. The main ingredient in the proof is a difference estimate on two solutions starting near 2-solitons and asymptotic to 1-solitons. The main difficulty is in controlling the direction of the two unstable modes attached to 2-solitons, while the soliton interactions are not uniformly integrable in time. It is resolved by showing that the non-scalar part of the interaction between the unstable modes is uniformly integrable due to the symmetry of the equation and the 2-solitons.

math.AP↗