On the modified Korteweg-De Vries equation

Nakao Hayashi, Pavel Naumkin

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30 Citations (Scopus)


We consider the large time asymptotic behavior of solutions to the Cauchy problem for the modified Korteweg-de Vries equation ut + a(t)(u3)x + 1/3uxxx = 0, (t, x) ε R × R, with initial data u(0, x) = u0(x), x ε R. We assume that the coefficient a(t) ε C1(R) is real, bounded and slowly varying function, such that |a′(t)| ≤ C〈t〉-7/6, where 〈t〉 = (1 + t2)1/2. We suppose that the initial data are real-valued and small enough, belonging to the weighted Sobolev space H1,1 = {φ ε L2; || √1+x2 √1-∂x2φ|| < ∞}. 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 u0 is zero. We prove the time decay estimates 3√t2 3√〈t〉||u(t)ux(t)|| ≤ Cε and 〈t〉1/3-1/3β||u(t)||β ≤ Cε for all t ε R, where 4 < β ≤ ∞. We also find the asymptotics for large time of the solution in the neighborhood of the self-similar solution.

Original languageEnglish
Pages (from-to)197-227
Number of pages31
JournalMathematical Physics Analysis and Geometry
Issue number3
Publication statusPublished - 2001


  • Large time asymptotics
  • Modified korteweg-de vries equation


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