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Evolution equation for wave amplitude exhibiting quartic nonlinearity
Author(s) -
Shukla Triveni P.,
Sharma Vishnu D.
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5297
Subject(s) - quartic function , mathematics , quadratic equation , nonlinear system , amplitude , mathematical analysis , inflection point , partial differential equation , wave equation , hyperbolic partial differential equation , classical mechanics , physics , geometry , quantum mechanics , pure mathematics
We consider quasi‐linear hyperbolic system of partial differential equations and use the method of multiple scales to derive a transport equation governing the evolution of weakly nonlinear waves propagating through a state that lies close to the one where the quadratic and cubic nonlinearity parameters are, respectively, O ( ε 2 ) and O ( ε ); here, ε is a small parameter measuring the wave amplitude. The resultant evolution equation involves quadratic, cubic, and quartic nonlinear terms and the flux function admits two inflection points. We provide an example from gasdynamics with analytical and numerical results exhibiting a rich variety of wave phenomena and study the interaction of expansion and compression waves evolving from a rectangular pulse.