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
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
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.
- Mark J. Ablowitz and Harvey Segur, Solitons and the inverse scattering transform, SIAM Studies in Applied Mathematics, vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981. MR 642018
J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contene dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl., Ser. 2, 17, 55–108 (1872)
- Francesco Calogero and Antonio Degasperis, Spectral transform and solitons. Vol. I, Studies in Mathematics and its Applications, vol. 13, North-Holland Publishing Co., Amsterdam-New York, 1982. Tools to solve and investigate nonlinear evolution equations; Lecture Notes in Computer Science, 144. MR 680040
- C. M. Dafermos and J. A. Nohel, Energy methods for nonlinear hyperbolic Volterra integrodifferential equations, Comm. Partial Differential Equations 4 (1979), no. 3, 219–278. MR 522712, DOI https://doi.org/10.1080/03605307908820094
- J. G. Dix and R. M. Torrejón, A quasilinear integrodifferential equation of hyperbolic type, Differential Integral Equations 6 (1993), no. 2, 431–447. MR 1195392
- Tai-Ping Liu, Nonlinear hyperbolic-dissipative partial differential equations, Recent mathematical methods in nonlinear wave propagation (Montecatini Terme, 1994) Lecture Notes in Math., vol. 1640, Springer, Berlin, 1996, pp. 103–136. MR 1600912, DOI https://doi.org/10.1007/BFb0093708
- Bernard J. Matkowsky and Edward L. Reiss, On the asymptotic theory of dissipative wave motion, Arch. Rational Mech. Anal. 42 (1971), 194–212. MR 350169, DOI https://doi.org/10.1007/BF00250485
- John W. Miles, Solitary waves, Annual review of fluid mechanics, Vol. 12, Annual Reviews, Palo Alto, Calif., 1980, pp. 11–43. MR 565388
- P. I. Naumkin and I. A. Shishmarëv, Nonlinear nonlocal equations in the theory of waves, Translations of Mathematical Monographs, vol. 133, American Mathematical Society, Providence, RI, 1994. Translated from the Russian manuscript by Boris Gommerstadt. MR 1261868
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)
- Pasquale Renno, Un problema di perturbazione singolare per una classe di fenomeni ondosi dissipativi, Ricerche Mat. 23 (1974), no. 2, 223–254 (Italian). MR 374630
- O. V. Rudenko and S. I. Soluyan, Teoreticheskie osnovy nelineĭ noĭ akustiki, Izdat. “Nauka”, Moscow, 1975 (Russian). MR 0670102
- Ricardo Torrejón and Jiong Min Yong, On a quasilinear wave equation with memory, Nonlinear Anal. 16 (1991), no. 1, 61–78. MR 1086826, DOI https://doi.org/10.1016/0362-546X%2891%2990131-J
- M. M. Vaĭnberg and V. A. Trenogin, Teoriya vetvleniya resheniĭ nilineĭ nykh uravneniĭ, Izdat. “Nauka”, Moscow, 1969 (Russian). MR 0261416
- V. V. Varlamov, On the fundamental solution of an equation describing the propagation of longitudinal waves in a dispersive medium, Zh. Vychisl. Mat. i Mat. Fiz. 27 (1987), no. 4, 629–633, 640 (Russian). MR 889890
- V. V. Varlamov, The Cauchy problem for an equation that describes nonstationary waves in a medium with relaxation, Dokl. Akad. Nauk SSSR 304 (1989), no. 4, 793–795 (Russian); English transl., Soviet Math. Dokl. 39 (1989), no. 1, 145–148. MR 988989
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 behavior for large time values of the solution of the Boussinesq equation with dissipation, Zh. Vychisl. Mat. i Mat. Fiz. 34 (1994), no. 8-9, 1194–1205 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys. 34 (1994), no. 8-9, 1033–1043. MR 1300394
- Vladimir Varlamov, On the Cauchy problem for the damped Boussinesq equation, Differential Integral Equations 9 (1996), no. 3, 619–634. MR 1371712
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)
- Vladimir V. Varlamov, On spatially periodic solutions of the damped Boussinesq equation, Differential Integral Equations 10 (1997), no. 6, 1197–1211. MR 1608069
- G. B. Whitham, Linear and nonlinear waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0483954
M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981
J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contene dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl., Ser. 2, 17, 55–108 (1872)
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
R. Torrejón and J. Yong, On a quasilinear wave equation with memory, Nonlinear Analysis 16, 61–78 (1991)
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
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
35L80,
35B40
Retrieve articles in all journals
with MSC:
35L80,
35B40
Additional Information
Article copyright:
© Copyright 2000
American Mathematical Society
