RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 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

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

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:

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} 

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

 SHARE:

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.
2. Maltsev A.Ya., “The Multi-Dimensional Hamiltonian Structures in the Whitham Method”, J. Math. Phys., 54:5 (2013), 053507
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
•  Number of views: This page: 444 Full text: 121 References: 52 First page: 18