Small Dispersive Shocks
In certain transport equations, dissipation governs the behavior at small length scales. As the dissipation goes to zero, the equations can become hyperbolic and develop shocks (for instance the inviscid Burgers equation). Here the Rankine-Hugoniot shock conditions typically describe the propagation of the shock front. In complete contrast, the small scale physics in other wave systems (for example in some optical systems) are described by dispersion. As a result, these dispersive regulated shock systems exhibit radically different behavior near the shock front.
Using techniques based on Whitham averaging, Gurevich and Pitaevskii, (Nonstationary structure of a collisionless shock wave, Sov. Phys. JETP 38, 291 (1974)) develop a theory for a single phase (single frequency) dispersive regulated shock. Following their work, the study of zero-limit dispersive wave systems has resulted in several very nice ideas and techniques.
A Soft Introduction to Dispersive Shocks
These notes (originally for a course project - 18.377J) contain an introduction to dispersive shocks. Some topics included are: physical background and modeling, Whitham averaging for KdV, and some basic numerical methods for evolving solutions.
This is really only the start. Some other interesting papers are:
P. D. Lax, C. D. Levermore
The small dispersion limit of the Korteweg-de Vries equation. I. II. III.
Comm. Pure Appl. Math. 36 (1983), No. 3, 253-290; No. 5, 571-593; No. 6, 809-829.
H. Flaschka, M. G. Forest, D. W. McLaughlin
Multiphase averaging and the inverse spectral solution of the Korteweg-deVries equation
Comm. Pure Appl. Math. 33, (1980), pp. 739-784.
D. McLaughlin, J. Strain
Computing the weak limit of KdV
Comm. Pur Appl. Math. 47, (1994), 1319-1364.
P. Deift, X. Zhou,
A Steepest Descent Method for Oscillatory Riemann-Hilbert Problems; Asymptotics for the MKdV Equation, Ann. of Math. 137, (1993), 295-368.
Using techniques based on Whitham averaging, Gurevich and Pitaevskii, (Nonstationary structure of a collisionless shock wave, Sov. Phys. JETP 38, 291 (1974)) develop a theory for a single phase (single frequency) dispersive regulated shock. Following their work, the study of zero-limit dispersive wave systems has resulted in several very nice ideas and techniques.
A Soft Introduction to Dispersive Shocks
These notes (originally for a course project - 18.377J) contain an introduction to dispersive shocks. Some topics included are: physical background and modeling, Whitham averaging for KdV, and some basic numerical methods for evolving solutions.
This is really only the start. Some other interesting papers are:
The small dispersion limit of the Korteweg-de Vries equation. I. II. III.
Comm. Pure Appl. Math. 36 (1983), No. 3, 253-290; No. 5, 571-593; No. 6, 809-829.
Multiphase averaging and the inverse spectral solution of the Korteweg-deVries equation
Comm. Pure Appl. Math. 33, (1980), pp. 739-784.
Computing the weak limit of KdV
Comm. Pur Appl. Math. 47, (1994), 1319-1364.
A Steepest Descent Method for Oscillatory Riemann-Hilbert Problems; Asymptotics for the MKdV Equation, Ann. of Math. 137, (1993), 295-368.