×

On the full dispersion Kadomtsev-Petviashvili equations for dispersive elastic waves. (English) Zbl 1524.35632


MSC:

35Q74 PDEs in connection with mechanics of deformable solids
74B20 Nonlinear elasticity
74J35 Solitary waves in solid mechanics

References:

[1] Kadomtsev, B.; Petviashvili, V. I., On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl., 15, 539-541 (1970) · Zbl 0217.25004
[2] Lannes, D., Consistency of the KP approximation, Discrete Cont. Dyn. Syst. Suppl., 517-525 (2003) · Zbl 1066.35017
[3] Lannes, D., (The Water Waves Problem: Mathematical Analysis and Asymptotics. The Water Waves Problem: Mathematical Analysis and Asymptotics, AMS Mathematical Surveys and Monographs, vol. 188 (2013), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 1410.35003
[4] Whitham, G. B., Linear and Nonlinear Waves (1974), John Wiley: John Wiley New York · Zbl 0373.76001
[5] Lannes, D.; Saut, J.-C., Remarks on the full dispersion Kadomtsev-Petviashvili equation, Kinet. Relat. Models, 6, 989-1009 (2013) · Zbl 1292.35266
[6] Pilod, D.; Saut, J.-C.; Selberg, S.; Tesfahun, A., Dispersive estimates for full dispersion KP equations, J. Math. Fluid Mech., 23, Article 25 pp. (2021) · Zbl 1458.35083
[7] Klein, C.; Linares, F.; Pilod, D.; Saut, J.-C., On Whitham and related equations, Stud. Appl. Math., 140, 133-177 (2018) · Zbl 1393.35208
[8] Erbay, H. A.; Erbay, S.; Erkip, A., The Cauchy problem for a class of two-dimensional nonlocal nonlinear wave equations governing anti-plane shear motions in elastic materials, Nonlinearity, 24, 1347-1359 (2011) · Zbl 1211.74114
[9] Duruk, N.; Erbay, H. A.; Erkip, A., Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity, Nonlinearity, 23, 107-118 (2010) · Zbl 1253.74040
[10] Eringen, A. C., Nonlocal Continuum Field Theories (2002), Springer: Springer New York · Zbl 1023.74003
[11] Horgan, C. O., Anti-plane shear deformations in linear and nonlinear solid mechanics, SIAM Rev., 37, 53-81 (1995) · Zbl 0824.73018
[12] Destrade, M.; Pucci, E.; Saccomandi, G., Generalization of the Zabolotskaya equation to all incompressible isotropic elastic solids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 475, Article 20190061 pp. (2019) · Zbl 1472.74043
[13] Turitsyn, S. K.; Fal’kovich, G. E., Stability of magnetoelastic solitons and self-focusing of sound in antiferromagnets, Sov. Phys.—JETP, 62, 146-152 (1985)
[14] Destrade, M.; Goriely, A.; Saccomandi, G., Scalar evolution equations for shear waves in incompressible solids: a simple derivation of the Z, ZK, KZK and KP equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467, 1823-1834 (2011) · Zbl 1228.74039
[15] Erbay, S., Coupled modified Kadomtsev—Petviashvili equations in dispersive elastic media, Int. J. Non-Linear Mech., 34, 289-297 (1999) · Zbl 1342.74014
[16] Alexander, J. C.; Pego, R. L.; Sachs, R., On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation, Phys. Lett. A, 226, 187-192 (1997) · Zbl 0962.35505
[17] Fornberg, B.; Whitham, G. B., A numerical and theoretical study of certain nonlinear wave phenomena, Phil. Trans. R. Soc. A, 289, 373-404 (1978) · Zbl 0384.65049
[18] Kataoka, T.; Tsutahara, M., Instability of solitary wave solutions to long-wavelength transverse perturbations in the generalized Kadomtsev-Petviashvili equation with negative dispersion, Phys. Rev. E, 70, Article 016604 pp. (2004)
[19] Benjamin, T. B.; Bona, J. L.; Mahony, J. J., Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond. Ser. A: Math. Phys. Sci., 272, 47-78 (1972) · Zbl 0229.35013
[20] Bona, J. L.; Liu, Y.; Tom, M. M., The Cauchy problem and stability of solitary-wave solutions for RLW-KP-type equations, J. Differential Equations, 185, 437-482 (2002) · Zbl 1023.35085
[21] Klein, C.; Saut, J.-C., Numerical study of blow-up and stability of solutions to generalized Kadomtsev-Petviashvili equations, J. Nonlinear Sci., 22, 763-811 (2012) · Zbl 1253.35150
[22] Chen, M., From Boussinesq systems to KP-type equations, Can. Appl. Math. Q., 15, 367-374 (2007) · Zbl 1176.35132
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.