TY - JOUR

T1 - Shock waves and compactons for fifth-order non-linear dispersion equations

AU - Galaktionov, Victor

PY - 2010/2

Y1 - 2010/2

N2 - The following first problem is posed - to justify that the standing shock wave S-(x) = -sign x ={-1 for x < 0, 1 for x > 0 F6 r x < 0, is a correct 'entropy solution' of the Cauchy problem for the fifth-ordcr degenerate non-linear dispersion equations (NDEs), as for the classic Euler one u(1) + uu(x) = 0, u(1) = -(uu(x))(xxxx) and u(1) = -(uu(xxx))(x) m R x R+. These two quasi-linear degenerate partial differential equations (PDEs) arc chosen as typical representatives, so other (2m + 1)th-order NDEs of non-divergent form admit such shocks waves. As a related second problem, the opposite initial shock S+(x) = -S-(x) = sign x is shown to be a non-entropy solution creating a rarefaction wave, which becomes C-infinity for any t > 0 Formation of shocks leads to non-uniqueness of any 'entropy solutions'. Similar phenomena are studied for a fifth-order in time NDE u(uuu) = (uu(x))(xxxx) in normal form On the other hand, related NDEs, such as u(t) = -(|u|u(x))(xxxx) + |u|u(x) in R x R+, are shown to admit smooth compactons, as oscillatory travelling wave solutions with compact support. The well-known non-negative compactons, which appeared in various applications (first examples by Dcy 1998, Phys Rev E, vol. 57, pp 4733-4738, and Rosenau and Levy, 1999, Phys Lett. A, vol. 252, pp 297-306), are non-existent in general and are not robust relative to small perturbations of parameters of the PDE.

AB - The following first problem is posed - to justify that the standing shock wave S-(x) = -sign x ={-1 for x < 0, 1 for x > 0 F6 r x < 0, is a correct 'entropy solution' of the Cauchy problem for the fifth-ordcr degenerate non-linear dispersion equations (NDEs), as for the classic Euler one u(1) + uu(x) = 0, u(1) = -(uu(x))(xxxx) and u(1) = -(uu(xxx))(x) m R x R+. These two quasi-linear degenerate partial differential equations (PDEs) arc chosen as typical representatives, so other (2m + 1)th-order NDEs of non-divergent form admit such shocks waves. As a related second problem, the opposite initial shock S+(x) = -S-(x) = sign x is shown to be a non-entropy solution creating a rarefaction wave, which becomes C-infinity for any t > 0 Formation of shocks leads to non-uniqueness of any 'entropy solutions'. Similar phenomena are studied for a fifth-order in time NDE u(uuu) = (uu(x))(xxxx) in normal form On the other hand, related NDEs, such as u(t) = -(|u|u(x))(xxxx) + |u|u(x) in R x R+, are shown to admit smooth compactons, as oscillatory travelling wave solutions with compact support. The well-known non-negative compactons, which appeared in various applications (first examples by Dcy 1998, Phys Rev E, vol. 57, pp 4733-4738, and Rosenau and Levy, 1999, Phys Lett. A, vol. 252, pp 297-306), are non-existent in general and are not robust relative to small perturbations of parameters of the PDE.

UR - http://www.scopus.com/inward/record.url?scp=77951101677&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1017/S0956792509990118

U2 - 10.1017/s0956792509990118

DO - 10.1017/s0956792509990118

M3 - Article

VL - 21

SP - 1

EP - 50

JO - European Journal of Applied Mathematics

JF - European Journal of Applied Mathematics

SN - 0956-7925

IS - 1

ER -