Dear Dr. Karelin,

According to Maiden et al. (2016) (Phys. Rev. Let. DOI: 10.1103/PhysRevLett.116.174501), we can use the term DSW (Dispersive Shock Wave) in order to characterize the waves described in your question. Their general characteristics are summarized succinctly:

"A DSW is an expanding, oscillatory train of amplitude-ordered nonlinear waves composed of a large amplitude solitonic wave adjacent to a monotonically decreasing wave envelope that terminates with a packet of small amplitude dispersive waves"

Sincerely, Janusz

The study of waves can be traced back to antiquity where philosophers, such as Pythagoras (c. 560-480 BC), studied the relation of pitch and length of string in musical instruments. However, it was not until the work of Giovani Benedetti (1530-90), Isaac Beeckman (1588-1637) and Galileo (1564-1642) that the relationship between pitch and frequency was discovered. This started the science of acoustics, a term coined by Joseph Sauveur (1653-1716) who showed that strings can vibrate simultaneously at a fundamental frequency and at integral multiples that he called harmonics. Isaac Newton (1642-1727) was the first to calculate the speed of sound in his Principia. However, he assumed isothermal conditions so his value was too low compared with measured values. This discrepancy was resolved by Laplace (1749-1827) when he included adiabatic heating and cooling effects. The first analytical solution for a vibrating string was given by Brook Taylor (1685-1731). After this, advances were made by Daniel Bernoulli (1700-82), Leonard Euler (1707-83) and Jean d'Alembert (1717-83) who found the first solution to the linear wave equation, see section (The linear wave equation). Whilst others had shown that a wave can be represented as a sum of simple harmonic oscillations, it was Joseph Fourier (1768-1830) who conjectured that arbitrary functions can be represented by the superposition of an infinite sum of sines and cosines - now known as the Fourier series. However, whilst his conjecture was controversial and not widely accepted at the time, Dirichlet subsequently provided a proof, in 1828, that all functions satisfying Dirichlet's conditions (i.e. non-pathological piecewise continuous) could be represented by a convergent Fourier series. Finally, the subject of classical acoustics was laid down and presented as a coherent whole by John William Strutt (Lord Rayleigh, 1832-1901) in his treatise Theory of Sound. The science of modern acoustics has now moved into such diverse areas as sonar, auditoria, electronic amplifiers, etc.

The study of hydrostatics and hydrodynamics was being pursued in parallel with the study of acoustics. Everyone is familiar with Archimedes (c. 287-212 BC) eureka moment; however he also discovered many principles of hydrostatics and can be considered to be the father of this subject. The theory of fluids in motion began in the 17th century with the help of practical experiments of flow from reservoirs and aqueducts, most notably by Galileo's student Benedetto Castelli. Newton also made contributions in the Principia with regard to resistance to motion, also that the minimum cross-section of a stream issuing from a hole in a reservoir is reached just outside the wall (the vena contracta). Rapid developments using advanced calculus methods by Siméon-Denis Poisson (1781-1840), Claude Louis Marie Henri Navier (1785-1836), Augustin Louis Cauchy (1789-1857), Sir George Gabriel Stokes (1819-1903), Sir George Biddell Airy (1801-92), and others established a rigorous basis for hydrodynamics, including vortices and water waves, see section (Physical wave types). This subject now goes under the name of fluid dynamics and has many branches such as multi-phase flow, turbulent flow, inviscid flow, aerodynamics, meteorology, etc.

http://www.scholarpedia.org/article/Linear_and_nonlinear_waves

Dear Sir

This paper is good

Asymptotic behavior of nonlinear waves in elastic media with dispersion and dissipation

This Book also

Linear and Nonlinear Waves

by G.B. Whitham

Solitary waves in media with dispersion and dissipation

http://link.springer.com/article/10.1007/BF02831423