Polynomial-in-time upper bounds for the orbital instability of subcritical generalized      Korteweg-de Vries equations

Brian Pigott

doi:10.3934/cpaa.2014.13.389

Article Contents

2014, Volume 13, Issue 1: 389-418. Doi: 10.3934/cpaa.2014.13.389

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Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations

Brian Pigott^1,

1.
Department of Mathematics, Wake Forest University, Winston-Salem, NC 27109

Received: March 31, 2013

Revised: April 30, 2013

Published: December 2013

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Abstract

We prove polynomial-in-time upper bounds for the orbital instability of solitons for subcritical generalized Korteweg-de Vries equations in $H_x^s R$ with $s < 1$. By combining coercivity estimates of Weinstein with the $I$-method as developed by Colliander, Keel, Staffilani, Takaoka, and Tao, we construct a modified energy functional which is shown to be almost conserved while providing us with an estimate of the deviation of the solution from the ground state curve. The iteration of the almost conservation law for the modified energy functional over time intervals of uniform length yields the polynomial upper bound.

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Mathematics Subject Classification: Primary: 35Q53, 42B35; Secondary: 37K10.

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