RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
General information
Latest issue
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



TMF:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


TMF, 2011, Volume 166, Number 3, Pages 350–365 (Mi tmf6616)  

This article is cited in 3 scientific papers (total in 3 papers)

Remark on the phase shift in the Kuzmak–Whitham ansatz

S. Yu. Dobrokhotov, D. S. Minenkov

Ishlinskii Institute for Problems in Mechanics, RAS, Moscow, Russia; Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Oblast, Russia

Abstract: We consider one-phase (formal) asymptotic solutions in the Kuzmak–Whitham form for the nonlinear Klein–Gordon equation and for the Korteweg–de Vries equation. In this case, the leading asymptotic expansion term has the form $X(S(x,t)/h+\Phi(x,t),I(x,t),x,t)+O(h)$, where $h\ll1$ is a small parameter and the phase $S(x,t)$ and slowly changing parameters $I(x,t)$ are to be found from the system of “averaged” Whitham equations. We obtain the equations for the phase shift $\Phi(x,t)$ by studying the second-order correction to the leading term. The corresponding procedure for finding the phase shift is then nonuniform with respect to the transition to a linear (and weakly nonlinear) case. Our observation, which essentially follows from papers by Haberman and collaborators, is that if we incorporate the phase shift $\Phi$ into the phase and adjust the parameter $\tilde{I}$ by setting $\widetilde{S}=S+h\Phi+O(h^2)$, $\tilde{I}=I+hI_1+O(h^2)$, then the functions $\widetilde{S}(x,t,h)$ and $\tilde{I}(x,t,h)$ become solutions of the Cauchy problem for the same Whitham system but with modified initial conditions. These functions completely determine the leading asymptotic term, which is $X(\widetilde{S}(x,t,h)/h,\tilde{I}(x,t,h),x,t)+O(h)$.

Keywords: rapidly oscillating one-phase asymptotic solution, nonlinear equation, Whitham method, phase shift

DOI: https://doi.org/10.4213/tmf6616

Full text: PDF file (494 kB)
References: PDF file   HTML file

English version:
Theoretical and Mathematical Physics, 2011, 166:3, 303–316

Bibliographic databases:

Document Type: Article
Received: 06.09.2010

Citation: S. Yu. Dobrokhotov, D. S. Minenkov, “Remark on the phase shift in the Kuzmak–Whitham ansatz”, TMF, 166:3 (2011), 350–365; Theoret. and Math. Phys., 166:3 (2011), 303–316

Citation in format AMSBIB
\Bibitem{DobMin11}
\by S.~Yu.~Dobrokhotov, D.~S.~Minenkov
\paper Remark on the~phase shift in the~Kuzmak--Whitham ansatz
\jour TMF
\yr 2011
\vol 166
\issue 3
\pages 350--365
\mathnet{http://mi.mathnet.ru/tmf6616}
\crossref{https://doi.org/10.4213/tmf6616}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3165817}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2011TMP...166..303D}
\transl
\jour Theoret. and Math. Phys.
\yr 2011
\vol 166
\issue 3
\pages 303--316
\crossref{https://doi.org/10.1007/s11232-011-0025-y}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000293733500003}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79955093965}


Linking options:
  • http://mi.mathnet.ru/eng/tmf6616
  • https://doi.org/10.4213/tmf6616
  • http://mi.mathnet.ru/eng/tmf/v166/i3/p350

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Andrei Ya. Maltsev, “Whitham's Method and Dubrovin–Novikov Bracket in Single-Phase and Multiphase Cases”, SIGMA, 8 (2012), 103, 54 pp.  mathnet  crossref
    2. Maltsev A.Ya., “The Multi-Dimensional Hamiltonian Structures in the Whitham Method”, J. Math. Phys., 54:5 (2013), 053507  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. Maltsev A.Ya., “On the Minimal Set of Conservation Laws and the Hamiltonian Structure of the Whitham Equations”, J. Math. Phys., 56:2 (2015), 023510  crossref  mathscinet  zmath  adsnasa  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
    Number of views:
    This page:364
    Full text:71
    References:40
    First page:18

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019