Nonlinear dispersion equations: Smooth deformations,compactions, and extensions to higher orders |
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| Authors: | V A Galaktionov |
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| Institution: | (1) Department of Mathematical Sciences, University of Bath, Math, BA2 7AY, UK |
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| Abstract: | The third-order nonlinear dispersion PDE, as the key model,
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((0.1)) |
is studied. Two Riemann’s problems for (0.1) with the initial data S
∓(x) = ∓ sgn.x create shock (u(x, t) ≡ S
−(x)) and smooth rarefaction (for the data S
+) waves (see 16]). The concept of “δ-entropy” solutions and others are developed for establishing the existence and uniqueness
for (0.1) by using stable smooth δ-deformations of shock-type solutions. These are analogous to entropy theory for scalar
conservation laws such as u
t
+ uu
x
= 0, which were developed by Oleinik and Kruzhkov (in x ∊ ℝ
N
) in the 1950s–1960s. The Rosenau-Hyman K(2, 2) (compacton) equation which has a special importance for applications, is studied. Compactons as compactly supported travelling wave solutions
are shown to be δ-entropy. Shock and rarefaction waves are discussed for other NDEs such as .
This article was submitted by the author in English.
Dedicated to the memory of Professors O.A. Oleinik and S.N. Kruzhkov |
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| Keywords: | Odd-order quasi-linear PDE shock and rarefaction waves entropy solutions self-similar patterns |
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