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Oscillations of the zero dispersion limit of the korteweg‐de vries equation
Author(s) -
Tian Fei Ran
Publication year - 1993
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160460802
Subject(s) - mathematics , cusp (singularity) , limit (mathematics) , mathematical analysis , monotone polygon , korteweg–de vries equation , zero (linguistics) , burgers' equation , mathematical physics , partial differential equation , physics , nonlinear system , quantum mechanics , geometry , linguistics , philosophy
We attack the multiphase averaged systems for the zero dispersion limit of the KdV equation. Attention is paid to the most important case—the single phase oscillations. A scheme is developed to solve the Whitham averaged system (single phase averaged system). This system, under our scheme, is transformed to a linear over‐determined system of Euler‐Poisson‐Darboux type whose solution can be written down explicitly. We show that, for any smooth initial data which has only one hump or is a nontrivial monotone function, the weak limit has single‐phase oscillations within a cusp in the x ‐ t plane for a short time after the breaking time for the corresponding Burgers equation. Outside the cusp, the limit satisfies the Burgers equation. More surprisingly, we also show that the weak limit has global single‐phase oscillations within a cusp for any smooth nontrivial monotone initial data with only one inflection point. © 1993 John Wiley & Sons, Inc.