TY - JOUR

T1 - Nonlinear Dispersion Equations: Smooth Deformations, Compactions, and Extensions to Higher Orders

AU - Galaktionov, Victor A

N1 - ID number: 000262335000008

PY - 2008/10

Y1 - 2008/10

N2 - The third-order nonlinear dispersion PDE, as the key model,
u(t) = (uu(x))(xx) in R x R+
is studied. Two Riemann's problems for (0.1) with the initial data S--/+ (x) = -/+ sgnx create shock (u (x, t) equivalent to S-(x)) and smooth rarefaction (for the data S+) waves (see [16]). The concept of "delta-entropy" solutions and others are developed for establishing the existence and uniqueness for (0.1) by using stable smooth delta-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 is an element of R-N) in the 1950s-1960s. The Rosenau-Hyman K (2, 2) (compacton) equation
u(t) = (uu(x))(xx) + 4uu(x),
which has a special importance for applications, is studied. Compactons as compactly supported travelling wave solutions are shown to be delta-entropy. Shock and rarefaction waves are discussed for other NDEs such as
u(t) = (u(2)u(x))(xx), u(tt) = (uu(x))(xx), u(tt) = uu(x), u(ttt) = (uu(x))(xx), u(t) = (uu(x))(xxxxx), etc.

AB - The third-order nonlinear dispersion PDE, as the key model,
u(t) = (uu(x))(xx) in R x R+
is studied. Two Riemann's problems for (0.1) with the initial data S--/+ (x) = -/+ sgnx create shock (u (x, t) equivalent to S-(x)) and smooth rarefaction (for the data S+) waves (see [16]). The concept of "delta-entropy" solutions and others are developed for establishing the existence and uniqueness for (0.1) by using stable smooth delta-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 is an element of R-N) in the 1950s-1960s. The Rosenau-Hyman K (2, 2) (compacton) equation
u(t) = (uu(x))(xx) + 4uu(x),
which has a special importance for applications, is studied. Compactons as compactly supported travelling wave solutions are shown to be delta-entropy. Shock and rarefaction waves are discussed for other NDEs such as
u(t) = (u(2)u(x))(xx), u(tt) = (uu(x))(xx), u(tt) = uu(x), u(ttt) = (uu(x))(xx), u(t) = (uu(x))(xxxxx), etc.

KW - self-similar patterns

KW - solutions

KW - entropy

KW - Odd-order quasi-linear PDE

KW - shock and rarefaction waves

UR - http://www.scopus.com/inward/record.url?scp=54249107908&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1134/S0965542508100084

U2 - 10.1134/S0965542508100084

DO - 10.1134/S0965542508100084

M3 - Article

VL - 48

SP - 1823

EP - 1856

JO - Computational Mathematics and Mathematical Physics

JF - Computational Mathematics and Mathematical Physics

SN - 0965-5425

IS - 10

ER -