Full family of flattening solitary waves for the mass critical generalized Korteweg-de Vries equation
For the mass critical generalized KdV equation ∂tu+∂x(∂2xu+u5)=0 on R, we construct a full family of flattening solitary wave solutions.
Let Q be the unique even positive solution of Q″. For any \nu\in (0,\frac 13), there exist global (for t\geq 0) solutions of the equation with the asymptotic behavior
u(t,x)= t^{-\frac{\nu}2} Q\left(t^{-\nu} (x-x(t))\right)+w(t,x)
where, for some c>0,
x(t)\sim c t^{1-2\nu} \quad \mbox{and}\quad \|w(t)\|_{H^1(x>\frac 12 x(t))} \to 0\quad \mbox{as $t\to +\infty$.}
Moreover, the initial data for such solutions can be taken arbitrarily close to a solitary wave in the energy space. The long-time flattening of the solitary wave is forced by a slowly decaying tail in the initial data.
This result and its proof are inspired and complement recent blow-up results for the critical generalized KdV equation.
This article is also motivated by previous constructions of exotic behaviors close to solitons for other nonlinear dispersive equations such as the energy-critical wave equation.
This talk is based on a joint work with Yvan Martel (Ecole Polytechnique).