Nonlinear surface waves on a tangential discontinuity in magnetohydrodynamics
Authors:
Giuseppe Alì and John K. Hunter
Journal:
Quart. Appl. Math. 61 (2003), 451-474
MSC:
Primary 35L60; Secondary 76W05
DOI:
https://doi.org/10.1090/qam/1999831
MathSciNet review:
MR1999831
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Abstract: We derive an asymptotic equation that describes the propagation of weakly nonlinear surface waves on a tangential discontinuity in incompressible magnetohydrodynamics. The equation is similar to, but simpler than, previously derived asymptotic equations for weakly nonlinear Rayleigh waves in elasticity, and is identical to a model equation for nonlinear Rayleigh waves proposed by Hamilton et al. The most interesting feature of the surface waves is that their nonlinear self-interaction is nonlocal. As a result of this nonlocal nonlinearity, smooth solutions break down in finite time, and appear to form cusps.
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E. A. Zabolotskaya, Nonlinear propagation of plane and circular Rayleigh waves in isotropic solids, J. Acoust. Soc. Am. 91 (1992), 2569–2575.
G. Alì, J. K. Hunter, and D. F. Parker, Hamiltonian equations for scale-invariant waves, Stud. Appl. Math., 108 (2002), 305–321.
M. Artola, and A. J. Majda, Nonlinear development of instabilities in supersonic vortex sheets I. The basic kink modes, Physica D, 28 (1987), 253–281.
M. Artola, and A. J. Majda, Nonlinear development of instabilities in supersonic vortex sheets. II. Resonant interaction among kink modes, SIAM J. Appl. Math., 49 (1989), 1310–1349.
S. Benzoni-Gavage, Stability of subsonic planar phase boundaries in a van der Waals fluid, Arch. Ration. Mech. Anal. 150 (1999), 23–55.
M. F. Hamilton, Yu. A. Il’insky, and E. A. Zabolotskaya, Local and nonlocal nonlinearity in Rayleigh waves, J. Acoust. Soc. Am., 97 (1995), 882–890.
M. F. Hamilton, Yu. A. Il’insky, and E. A. Zabolotskaya, Evolution equations for nonlinear Rayleigh waves, J. Acoust. Soc. Am., 97 (1995), 891–897.
M. F. Hamilton, Yu. A. Il’insky, and E. A. Zabolotskaya, Nonlinear surface acoustic waves in crystals, J. Acoust. Soc. Am., 105 2 (1999), 639–651.
R. L. Higdon, Initial-boundary value problems for linear hyperbolic systems, SIAM Rev., 28 (1986), 177–217.
J. K. Hunter, Nonlinear surface waves, Contemporary Mathematics, 100 (1989), 185–202.
N. Kalyanasundaram, Nonlinear surface acoustic waves on an isotropic solid, Int. J. Enqng. Sci., 19 (1981), 279–286.
N. Kalyanasundaram, R. Ravindran and P. Prasad, Coupled amplitude theory of nonlinear surface acoustic waves, J. Acoust. Soc. Am., 72 (1982), 488–493.
H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 277–298.
L. D. Landau, and E. M. Lifschitz, Electrodynamics of Continuous Media, 2nd ed., Pergamon Press, New York, 1984.
R. W. Lardner, Nonlinear surface waves on an elastic solid, Int. J. Engng. Sci., 21 (1983), 1331–1342.
R. W. Lardner, Nonlinear surface acoustic waves on an elastic solid of general anisotropy, J. Elast., 16 (1986), 63–73.
A. J. Majda, The stability of multidimensional shock fronts. Mem. Amer. Math. Soc., 41 (1983), no. 275.
A. J. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc., 43 (1983), no. 281. AMS, Providence, 1983.
A. J. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, Vol. 53, Springer-Verlag, New York, 1984.
A. J. Majda, and R. R. Rosales, A theory for spontaneous Mach stem formation in reacting shock fronts. I. The basic perturbation analysis, SIAM J. Appl. Math., 43 (1983), 1310–1334.
A. J. Majda, and R. R. Rosales, A theory for spontaneous Mach stem formation in reacting shock fronts. II. Steady wave bifurcation and the evidence for breakdown, Stud. Appl. Math., 71 (1984), 117–148.
P. J. Olver, Hamiltonian and non-Hamiltonian models for water waves, Trends and Applications of Pure Mathematics to Mechanics (Palaiseau, 1983), 273–290, Lecture Notes in Phys., 195, Springer-Verlag, Berlin, 1984.
D. F. Parker, Waveform evolution for nonlinear surface acoustic waves, Int. J. Engng. Sci., 26 (1988), 59–75.
D. F. Parker, and F. M. Talbot, Analysis and computation for nonlinear elastic surface waves of permanent form, J. Elast., 15 (1985), 389–426.
R. R. Rosales, Stability theory for shocks in reacting media: Mach stems in detonation waves, Lectures in Applied Mathematics 24 (1986), 431–465.
R. Sakamoto, Mixed problems for hyperbolic equations. I. Energy inequalities, J. Math. Kyoto Univ., 10 (1970), 349–373.
R. Sakamoto, Mixed problems for hyperbolic equations. II. Existence theorems with zero initial datas and energy inequalities with initial datas, J. Math. Kyoto Univ., 10 (1970), 403–417.
D. Serre, Systems of Conservation Laws. 2. Geometric Structures, Oscillations, and Initial-Boundary Value Problems, Cambridge University Press, Cambridge, 2000.
E. Tadmor, Convergence of spectral methods for nonlinear conservation laws, SIAM J. Numer. Anal. 26 (1989), 30–44.
E. A. Zabolotskaya, Nonlinear propagation of plane and circular Rayleigh waves in isotropic solids, J. Acoust. Soc. Am. 91 (1992), 2569–2575.
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