Nonlinear Geometrical Optics

Using asymptotic methods, one can reduce complicated systems of equations to simpler model equations. The model equation for a single, genuinely nonlinear, hyperbolic wave is Burgers equation. Reducing the gas dynamics equations to a Burgers equation, leads to a theory of nonlinear geometrical acoustics. When diffractive effects are included, the model equation is the ZK or unsteady transonic small disturbance equation. We describe some properties of this equation, and use it to formulate asymptotic equations that describe the transition from regular to Mach reflection for weak shocks. Interacting hyperbolic waves are described by a system of Burgers or ZK equations coupled by integral terms. We use these equations to study the transverse stability of interacting sound waves in gas dynamics.

[1]  M. Lighthill.  The diffraction of blast. II , 1949, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[2]  R. Timman.  Unsteady motion in transonic flow , 1964 .

[3]  Donald Ludwig,et al.  Uniform asymptotic expansions at a caustic , 1966 .

[4]  E. Zabolotskaya,et al.  Quasi-plane waves in the nonlinear acoustics of confined beams , 1969 .

[5]  B. Kadomtsev,et al.  On the Stability of Solitary Waves in Weakly Dispersing Media , 1970 .

[6]  G. Whitham,et al.  Linear and Nonlinear Waves , 1976 .

[7]  G. Zahalak,et al.  Conical flow near singular rays , 1974, Journal of Fluid Mechanics.

[8]  Bradford Sturtevant,et al.  The focusing of weak shock waves , 1976 .

[9]  Joseph B. Keller,et al.  Rays, waves and asymptotics , 1978 .

[10]  A. R. Seebass,et al.  Focusing of weak shock waves at an arête , 1978 .

[11]  D. G. Crighton,et al.  MODEL EQUATIONS OF NONLINEAR ACOUSTICS , 1979 .

[12]  R. S. Johnson.  Water waves and Korteweg–de Vries equations , 1980, Journal of Fluid Mechanics.

[13]  A. Nayfeh.  A Comparison of Perturbation Methods for Nonlinear Hyperbolic Waves , 1980 .

[14]  Mark J. Ablowitz,et al.  Solitons and the Inverse Scattering Transform , 1981 .

[15]  和達 三樹.  M. J. Ablowitz and H. Segur: Solitons and the Inverse Scattering Transform, Society for Industrial and Applied Mathematics, Philadelphia, 1981, x+425ページ, 23.5×16.5cm, $54.40 (SIAM Studies in Applied Mathematics). , 1982 .

[16]  John K. Hunter,et al.  Weakly nonlinear high frequency waves , 1983 .

[17]  D. Kinderlehrer,et al.  Oscillation theory, computation, and methods of compensated compactness , 1986 .

[18]  Andrew J. Majda,et al.  Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws , 1986 .

[19]  J. Nimmo,et al.  Geometrical and diffusive effects in nonlinear acoustic propagation over long ranges , 1986, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[20]  Hans G. Hornung,et al.  Regular and Mach Reflection of Shock Waves , 1986 .

[21]  Andrew J. Majda,et al.  Resonantly interacting, weakly nonlinear hyperbolic waves.II. Several space variables , 1986 .

[22]  M. Hamilton,et al.  FUNDAMENTALS AND APPLICATIONS OF NONLINEAR ACOUSTICS. , 1986 .

[23]  J. K. Hunter,et al.  Caustics of nonlinear waves , 1987 .

[24]  E. Harabetian.  Diffraction of a weak shock by a wedge , 1987 .

[25]  Björn Engquist,et al.  Parabolic wave equation approxi-mations in heterogeneous media , 1988 .

[26]  Some explicit resonating waves in weakly nonlinear gas dynamics , 1988 .

[27]  A. Majda,et al.  A Canonical System of lntegrodifferential Equations Arising in Resonant Nonlinear Acoustics , 1988 .

[28]  Y. Kodama.  Exact solutions of hydrodynamic type equations having infinitely many conserved densities , 1989 .

[29]  L. Hsiao,et al.  The Riemann problem and interaction of waves in gas dynamics , 1989 .

[30]  Y. Kodama,et al.  A method for solving the dispersionless KP hierarchy and its exact solutions. II , 1989 .