Smoothness and Exponential Decay of L 2-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 1 solution of the critical or subcritical generalized KdV equations which is close to a soliton and satisfies a property of uniform localization of the L 2 norm (we call such solution an L 2-compact solution) is exactly a soliton. This result is a key tool in Martel and Merle (2000, 2001), as well as [Martel, Y., Merle, F. (2002a). Stability of the blow up profile and lower bounds on the blow up rate for the critical generalized KdV equation. Ann. Math. 155:235–280; Martel, Y., Merle, F. (2002b). 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] 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 paper, 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 the results of Martel and Merle (2001) and the decomposition for large time of any smooth and decaying solution into the sum of N solitons [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 (2001) by removing the assumption of closeness to a soliton.