On the Modified Korteweg–De Vries Equation

Abstract

We consider the large time asymptotic behavior of solutions to the Cauchy problem for the modified Korteweg–de Vries equation $u_t + a\left( t \right)\left( {u^3 } \right)_x + \frac{1}{3}u_{xxx} = 0,\left( {t,x} \right) \in R \times R$, with initial data $u\left( {0,x} \right) = u_0 \left( x \right),x \in R$. We assume that the coefficient $a\left( t \right) \in C^1 \left( R \right)$ is real, bounded and slowly varying function, such that $\left| {a'\left( t \right)} \right| \leqslant C\left\langle t \right\rangle ^{ - \frac{7}{6}}$, where $\left\langle t \right\rangle = \left( {1 + t^2 } \right)^{\frac{1}{2}}$. We suppose that the initial data are real-valued and small enough, belonging to the weighted Sobolev space $H^{1,1} = \left\{ {\phi \in L^2 ;\left\| {\sqrt {1 + x^2 } \sqrt {1 - \partial _x^2 } \phi } \right\| < \infty } \right\}$. In comparison with the previous paper (Internat. Res. Notices 8 (1999), 395–418), here we exclude the condition that the integral of the initial data u ₀ is zero. We prove the time decay estimates $\sqrt[3]{{t^2 }}\sqrt[3]{{\left\langle t \right\rangle }}\left\| {u\left( t \right)u_x \left( t \right)} \right\|_\infty \leqslant C\varepsilon$ and $\left\langle t \right\rangle ^{\frac{1}{3} - \frac{1}{{3\beta }}} \left\| {u\left( t \right)} \right\|_\beta \leqslant C\varepsilon$ for all $t \in R$, where $4 < \beta \leqslant \infty$. We also find the asymptotics for large time of the solution in the neighborhood of the self-similar solution.