Description and classification of 2-solitary waves for nonlinear damped Klein-Gordon equations

Abstract

We describe completely 2-solitary waves related to the ground state of the nonlinear damped Klein-Gordon equation \begin{equation*} \partial_{tt}u+2α\partial_{t}u-Δu+u-|u|^{p-1}u=0 \end{equation*} on ${\mathbf{R}}^{\mathbf{N}}$, for $1\le N\le 5$ and energy subcritical exponents $p>2$. The description is twofold. First, we prove that 2-solitary waves with same sign do not exist. Second, we construct and classify the full family of 2-solitary waves in the case of opposite signs. Close to the sum of two remote solitary waves, it turns out that only the components of the initial data in the unstable direction of each ground state are relevant in the large time asymptotic behavior of the solution. In particular, we show that $2$-solitary waves have a universal behavior: the distance between the solitary waves is asymptotic to $\log t$ as $t\to \infty$. This behavior is due to damping of the initial data combined with strong interactions between the solitary waves.

References in corpus (5)

• Two-bubble dynamics for threshold solutions to the wave maps equation
• Symbolic dynamics for non-uniformly hyperbolic surface maps with discontinuities
• Nonexistence of radial two-bubbles with opposite signs for the energy-critical wave equation
• Dynamics of strongly interacting unstable two-solitons for generalized Korteweg-de Vries equations
• Asymptotic decomposition for nonlinear damped Klein-Gordon equations

Cited by in corpus (1)

• Existence of two-solitary waves with logarithmic distance for the nonlinear Klein-Gordon equation