Electron. J. Diff. Eqns., Vol. 2007(2007), No. 49, pp. 1-13.

Positive periodic solutions for the Korteweg-de Vries equation

Svetlin Georgiev Georgiev

Abstract:
In this paper we prove that the Korteweg-de Vries equation
$$ \partial_t u+\partial_x^3 u+u\partial_x u=0 $$
has unique positive solution $u(t, x)$ which is $\omega$-periodic with respect to the time variable $t$ and $u(0, x)\in {\dot B}^{\gamma}_{p, q}([a, b])$, $\gamma$ greater than 0$, $\gamma\notin \{1, 2, \dots\}$, p greater than 1, $q\geq 1$, a less than b are fixed constants, $x\in [a, b]$. The period $\omega$ greater than 0 is arbitrary chosen and fixed.

Submitted January 18, 2006. Published April 4, 2007.
Math Subject Classifications: 35Q53, 35Q35, 35G25
Key Words: Nonlinear evolution equation; Kortewg de Vries equation; periodic solutions.

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Svetlin Georgiev Georgiev
Department of Differential Equations
University of Sofia, Sofia, Bulgaria
email: sgg2000bg@yahoo.com

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