Full Family of Flattening Solitary Waves for the Critical Generalized KdV Equation

For the critical generalized KdV equation partial differential tu+ partial differential x( partial differential x2u+u5)=0$$\partial _t u + \partial _x (\partial _x<^>2 u + u<^>5)=0$$\endwe construct a full family of flattening solitary wave solutions. LetQbe the unique even positive solution ofQ ''+Q5=Q. For any nu is an element of(0,13) there exist global (fort >= 0 solutions of the equation with the asymptotic behavioru(t,x)=t-nu 2Q mml:mfenced close=")" open="("t-nu(x-x(t))+w(t,x)\$$\begin{aligned} u(t,x)= t<^>{-\frac{\nu }{2}} Q\left( t<^>{-\nu } (x-x(t))\right) +w(t,x) \end{aligned}$$\end{document}and||w(t)||H1(x>12x(t))-> 0ast ->+infinity.$$\begin{aligned} x(t)\sim c t<^>{1-2\nu } \quad \text{ and }\quad \Vert w(t)\Vert _{H<^>1(x>\frac{1}{2} x(t))} \rightarrow 0\quad \text{ as } t\rightarrow +\infty . \end{aligned}$$\end{document}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.