On asymptotic stability of stable Good Boussinesq solitary waves

TL;DR

This work proves the asymptotic stability of stable Good Boussinesq solitary waves in one dimension for the generalized nonlinearity $f(s)=|s|^{p-1}s$, $p\ge 2$, and speeds $|c|>c_+(p)$ with $c_+(p)\ge \sqrt{(p-1)/4}$. The authors develop a moving-frame modulation framework together with a novel vector-valued virial analysis in mixed variables, and employ a Darboux-type transform to derive a transformed system with coercive matrix structure. They establish a sequence of virial identities (three in total) and a hierarchy of coercivity estimates for the transformed operator under mixed orthogonality, enabling control of perturbations in the energy space $H^1\times L^2$. The combination of these ingredients yields quantitative decay of the perturbation and convergence of the scaling parameter $c(t)$ to a limit $c_+$, thereby achieving asymptotic stability of the GB solitary waves. This work advances the understanding of two-direction GB dynamics and provides a rigorous framework potentially extensible to other Boussinesq-type models with similar vector-valued long-time behavior.

Abstract

We consider the generalized Good-Boussinesq (GB) model in one dimension, with subcritical power nonlinearity $1<p<5$ and data in the energy space $H^1\times L^2$. This model has solitary waves with speeds $c\in (-1,1)$. If $c^2>\frac{p-1}{4}$, Bona and Sachs showed the orbital stability of such waves. Previously, one of us proved that unstable GB standing waves can be perturbed with particular odd-even data in a suitable submanifold of the energy space, leading to the asymptotic stability property if $p\ge 2$. In this paper we prove that stable GB solitary waves are asymptotically stable in the case of general initial data placed in the energy space for any $p\ge 2$ and speeds $|c|>c_+(p)\geq \sqrt{\frac{p-1}{4}}$. The proof involves the introduction of a new set of virial estimates specifically adapted to the GB system in a moving setting. In particular, a new virial estimate with mixed variables is considered to treat arbitrary scaling and shift modulations. Another new ingredient is the understanding the corresponding linear matrix operator under mixed orthogonality conditions, a feature absent in our previous works.

Figures (2)

• Figure 1: Left. Negative eigenvalue of $\tilde{L}_0$ as a function of $p\in (1,5)$. Notice that at $p=1$ the corresponding value is zero. Right: Second eigenvalue of $\tilde{L}_0$, which is zero at $p=1$.
• Figure 2: Above left: the coefficient $a_2(p)$ (blue) for $1<p<5$. The yellow line represents the zero value. Notice that $a_2=0$ for $p\sim 6.6$. Above right: the coefficient $a_1(p)<0$ (blue), in yellow, the zero value. Below left: $m_{c,+}(p)$ (blue), in yellow the zero value. Below right: $c_+(p)$ (blue), $p\mapsto \sqrt{\frac{p-1}{4}}$ (yellow). In green the zero value, in red the value 1.

Theorems & Definitions (47)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Proposition 3.1