Long-time asymptotics of solutions of the third-order nonlinear evolution equation governing wave propagation in relaxing media
Author:
Vladimir Varlamov
Journal:
Quart. Appl. Math. 58 (2000), 201-218
MSC:
Primary 35L80; Secondary 35B40
DOI:
https://doi.org/10.1090/qam/1753395
MathSciNet review:
MR1753395
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Abstract: A classical Cauchy problem for a third-order nonlinear evolution equation is considered. This equation describes the propagation of weakly nonlinear waves in relaxing media. The global existence and uniqueness of its solutions is proved and the solution is constructed in the form of a series in a small parameter present in the initial conditions. Its long-time asymptotics is calculated, which shows the presence of two solitary wave pulses traveling in opposite directions and diffusing in space. Each of them is governed by Burgers’ equation with a transfer.
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F. Calogero and A. Degasperis, Spectral Transform and Solitons, North Holland, Amsterdam, 1982
C. M. Dafermos and J. A. Nohel, Energy methods for nonlinear hyperbolic Volterra integrodifferential equations, Comm. Partial Differential Equations 4, 219–278 (1979)
J. G. Dix and R. M. Torrejón, A quasilinear integrodifferential equation of hyperbolic type, Differential and Integral Equations 6, 431–447 (1993)
T.-P. Liu, Nonlinear Hyperbolic-Dissipative Partial Differential Equations, Lecture Notes in Mathematics, Springer, 1640, 103–136 (1994)
B. J. Matkowsky and E. L. Reiss, On the asymptotic theory of dissipative wave motion, Arch. Rat. Mech. Anal. 42, 194–212 (1971)
J. W. Miles, Solitary waves, Ann. Rev. Fluid Mech. 12, 11–43 (1980)
P. I. Naumkin and I. A. Shishmarëev, Nonlinear Nonlocal Equations in the Theory of Waves, Transl. Math. Monographs, Vol. 133, Amer. Math. Soc., Providence, RI, 1994
V. N. Nikolaevskiî, Mechanics of Porous and Cracked Media, Nauka, Moscow, 1984 (Russian)
P. Renno, Sulla soluzione fondamentale di un operatore iperbolico della termochimica tridimensionale, Rend. Accad. Naz. Sci. XL, Mem. Matem. 4, 43–62 (1979–80) (Italian)
P. Renno, Un problema di perturbazione singolare per una classe di fenomeni ondosi dissipativi, Ric. Matem. 23, 223–254 (1974) (Italian)
O. V. Rudenko and S. I. Soluyan, Theoretical Foundations of Nonlinear Acoustics, Nauka, Moscow, 1975 (Russian). English transl. Consultants Bureau, New York, 1977
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M. M. Vainberg and V. A. Trenogin, Theory of Bifurcation of Solutions of Nonlinear Equations, Nauka, Moscow, 1969 (Russian)
V. V. Varlamov, On the fundamental solution of the equation describing the propagation of longitudinal waves in a medium with dispersion, Zh. Vychisl. Mat. & Mat. Fiz. (Comp. Math. & Math. Phys.) 27, 629–633 (Russian)
V. V. Varlamov, On the Cauchy problem for an equation describing time-dependent waves in a medium with relaxation, USSR Acad. Sci. Dokl. 39, 145–147 (1989)
V. V. Varlamov and A. V. Nesterov, Asymptotic representation of the solution of the problem of the propagation of acoustic waves in an inhomogeneous compressible relaxing medium, Zh. Vychisl. Mat. & Mat. Fiz. (Comp. Math. & Math. Phys.) 30, 705–715 (Russian)
V. V. Varlamov, The asymptotic form for long times of the solution of the Boussinesq equation with dissipation, Zh. Vychisl. Mat. & Mat. Fiz. (Comp. Math. & Math. Phys.) 34, 1033–1043 (1994)
V. V. Varlamov, On the Cauchy problem for the damped Boussinesq equation, Differential and Integral Equations 9 (3), 619–634 (1996)
V. V. Varlamov, Long-time asymptotics of the spatially periodic solutions of the Boussinesq equation with dissipation, Doklady Acad. Nauk 345 (4), 459–462 (1995) (Russian). English transl. Russian Acad. Sci. Dokl. 52 (3), 446–449 (1995)
V. V. Varlamov, On spatially periodic solutions of the damped Boussinesq equation, Differential and Integral Equations 10 (6), 1197–1211 (1997)
G. B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974
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