Modified KdV equation with higher order dispersion terms

  • Pavel I. NaumkinEmail author
  • Jhon J. Perez


We consider the Cauchy problem for the modified Korteweg–de Vries equation with higher order dispersion terms
$$\begin{aligned} u_{t}-\partial _{x}u^{3}-\sum _{j=1}^{n}\lambda _{j}\partial _{x}^{2j+1}u=0, \end{aligned}$$
where \(\lambda _{j}\in {\mathbb {R}},\) \(n\ge 2.\) Our purpose in this paper is to prove the large time asymptoitic behavior of solutions under the non zero mass condition \(\int u_{0}\left( x\right) dx\ne 0.\)


Modified KdV equation Higher order Large time asymptotics Critical nonlinearity Self-similar solutions 

Mathematics Subject Classification

35B40 35Q35 



We are grateful to an unknown referee for many useful suggestions and comments. The work of P.I.N. is partially supported by CONACYT and PAPIIT Project IN100616.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Centro de Ciencias MatemáticasUNAM Campus MoreliaMoreliaMexico

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