Asymptotic Equations for Nonlinear Hyperbolic Waves

Hunter, John K.

doi:10.1007/978-1-4615-1991-1_3

Cited by 21 publications

(18 citation statements)

References 172 publications

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“…The scalar wave equation considered in the previous section is the simplest representative of this class of equations. The weakly nonlinear geometrical optics solution of (2.17) has the form [9,12,14] …”

Section: Systems Of Variational Wave Equationsmentioning

confidence: 99%

Diffractive nonlinear geometrical optics for variational wave equations and the Einstein equations

Alì¹,

Hunter²

2007

Comm Pure Appl Math

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Section: Systems Of Variational Wave Equationsmentioning

confidence: 99%

Diffractive nonlinear geometrical optics for variational wave equations and the Einstein equations

Alì¹,

Hunter²

2007

Comm Pure Appl Math

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“…For other work in nonlinear hyperbolic waves using Chapman-Enskog expansions see Hunter (1995). The leading order solution v 0 and the correction v ′ depend on ǫ explicitly.…”

Section: An Asymptotic Derivation Of Wnlrtmentioning

confidence: 99%

“…(See Hunter, 1995). Note that the assumption |n − n * | = O(ǫ) breaks down as soon the nonlinear wavefront starts approaching a caustic region of the linear theory.…”

Section: Comparison With Other Theoriesmentioning

confidence: 99%

“…Though, ray theories for the propagation of a nonlinear wavefront were dealt by many (Keller (1954), Whitham (1956), Parker (1969Parker ( , 1971); Choquet-Bruhat (1969) presented a systematic formal derivation of it for a general hyperbolic system of quasilinear partial differential equations. Hunter and his collaborators extended Choquet-Bruhat's theory to many important situations (Hunter, 1995). The Choquet-Bruhat's perturbation procedure leads to an eikonal equation which is independent of the amplitude w and therefore, uses a transport equation for the amplitude along linear rays.…”

Section: Introductionmentioning

confidence: 99%

“…We have been able to deduce the transport equation for w with an error of the order of ǫ 2 . The method of expansion is similar to the Chapman-Enskog expansion, a discussion of which for a hyperbolic system is available in an article by Hunter (1995). A careful examination of the various terms in ray equations and the transport equation show that in practice only a few terms may be retained and this leads to the nonlinear ray theory of Prasad, which has been very successful in giving physically realistic results and also in showing that the caustic of a linear theory is resolved when nonlinear effect are included (Prasad and Sangeeta (1990) and Monica and Prasad (1989)).…”

Section: Introductionmentioning

confidence: 99%

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An asymptotic derivation of weakly nonlinear ray theory

Prasad

2000

Proc. Math. Sci.

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Using a method of expansion similar to Chapman -Enskog expansion, a new formal perturbation scheme based on high frequency approximation has been constructed. The scheme leads to an eikonal equation in which the leading order amplitude appears. The transport equation for the amplitude has been deduced with an error O(ǫ 2 ) where ǫ is the small parameter appearing in the high frequency approximation. On a length scale over which Choquet -Bruhat's theory is valid, this theory reduces to the former. The theory is valid on a much larger length scale and the leading order terms give the weakly nonlinear ray theory (WNLRT) of Prasad, which has been very successful in giving physically realistic results and also in showing that the caustic of a linear theory is resolved when nonlinear effects are included. The weak shock ray theory with infinite system of compatibility conditions also follows from this theory.

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Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities

Biello

Hunter

2009

Comm Pure Appl Math

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Waves with constant, nonzero linearized frequency form an interesting class of nondispersive waves whose properties differ from those of nondispersive hyperbolic waves. We propose an inviscid Burgers-Hilbert equation as a model equation for such waves, and give a dimensional argument to show that it models Hamiltonian surface waves with constant frequency. Using the method of multiple scales, we derive a cubically nonlinear, quasilinear, nonlocal asymptotic equation for weakly nonlinear solutions. We show that exactly the same asymptotic equation describes surface waves on a planar discontinuity in vorticity in two-dimensional inviscid, incompressible fluid flows. Thus, the BurgersHilbert equation provides an effective equation for these waves. We describe the Hamiltonian structure of the Burgers-Hilbert and asymptotic equations, and show that the asymptotic equation may be also be derived by means of a near-identity transformation. We derive a semi-classical approximation of the asymptotic equation, and show that spatially periodic, harmonic traveling waves are linearly and modulationaly stable. Numerical solutions of the Burgers-Hilbert and asymptotic equations are in excellent agreement in the appropriate regime. In particular, the lifespan of small-amplitude smooth solutions of the Burgers-Hilbert equation is given by the cubically nonlinear timescale predicted by the asymptotic equation.

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Asymptotic Equations for Nonlinear Hyperbolic Waves

Cited by 21 publications

References 172 publications

Diffractive nonlinear geometrical optics for variational wave equations and the Einstein equations

Diffractive nonlinear geometrical optics for variational wave equations and the Einstein equations

An asymptotic derivation of weakly nonlinear ray theory

Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities

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