Smoothness and Exponential Decay of L²-Compact Solutions of the Generalized KdV Equations

Abstract

It is proved in Martel and Merle (Martel, Y., Merle, F. ([2000]). A Liouville theorem for the critical generalized Korteweg-de Vries equation. J. Math. Pures Appl.79:339–425; Martel, Y., Merle, F. ([2001]). Asymptotic stability of solitons for the subcritical generalized Korteweg-de Vries equation. Arch. Ration. Mech. Anal.157:219–254.) that any H ¹solution of the critical or subcritical generalized KdV equations which is close to a soliton and satisfies a property of uniform localization of the L ²norm (we call such solution an L ²-compact solution) is exactly a soliton. This result is a key tool in Martel and Merle (Martel, Y., Merle, F. ([2000]). A Liouville theorem for the critical generalized Korteweg-de Vries equation, J. Math. Pures Appl.79:339–425; Martel, Y., Merle, F. ([2001]). Asymptotic stability of solitons for the subcritical generalized Korteweg-de Vries equation. Arch. Ration. Mech. Anal.157:219–254; Martel, Y., Merle, F. ([2002a]). Blow up in finite time and dynamics of blow up solutions for the L ²-critical generalized KdV equation. J. Amer. Math. Soc.15:617–664; Martel, Y., Merle, F. ([2002b]). Stability of the blow up profile and lower bounds on the blow up rate for the critical generalized KdV equation. Ann. of Math.155:235–280.) and Merle (Merle, F. ([2001]). Existence of blow-up solutions in the energy space for the critical generalized Korteweg-de Vries equation. J. Amer. Math. Soc.14:555–578.) for studying the asymptotic behavior of global solutions close to the family of soliton and building a large class of blow up solutions in the critical case. In this article, we study properties of L ²-compact solutions for the generalized KdV equations (without restriction of subcriticality) that are not necessarily close to solitons. We prove C∞ regularity and uniform exponential decay properties for these solutions. For the KdV equation, as a consequence of this result, using Martel and Merle (Martel, Y., Merle, F. ([2001]). Asymptotic stability of solitons for the subcritical generalized Korteweg-de Vries equation. Arch. Ration. Mech. Anal.157:219–254.) and the decomposition for large time of any smooth and decaying solution into the sum of Nsolitons (Eckhaus, W., Schuur, P. (1983). The emergence of solutions of the Korteweg-de Vries equation from arbitrary initial conditions. Math. Meth. Appl. Sci.5:97–116.), we prove that any L ²-compact solution is a soliton. This extends the result of Martel and Merle (Martel, Y., Merle, F. ([2001]). Asymptotic stability of solitons for the subcritical generalized Korteweg-de Vries equation. Arch. Ration. Mech. Anal.157:219–254.) by removing the assumption of closeness to a soliton.

Acknowledgment

The authors would like to thank Professor F. Merle for suggesting the idea of this work.