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The small dispersion limit of the korteweg‐de vries equation. ii
Author(s) -
Lax Peter D.,
Levermore C. David
Publication year - 1983
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.3160360503
Subject(s) - mathematics , korteweg–de vries equation , limit (mathematics) , interval (graph theory) , mathematical analysis , zero (linguistics) , mathematical physics , section (typography) , function (biology) , initial value problem , combinatorics , physics , nonlinear system , quantum mechanics , linguistics , philosophy , evolutionary biology , advertising , business , biology
In Part I* we have shown, see Theorem 2.10, that as the coefficient of u xxx tends to zero, the solution of the initial value problem for the KdV equation tends to a limit u in the distribution sense. We have expressed u by formula (3.59), where ψ x is the partial derivative with respect to x of the function ψ* defined in Theorem 3.9 as the solution of the variational problem formulated in (2.16), (2.17). ψ* is uniquely characterized by the variational condition (3.34); its partial derivatives satisfy (3.51) and (3.52), where I is the set I o defined in (3.36). In Section 4 we show that for t
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