On a Whitham-Type Equation
Symmetry, integrability and geometry: methods and applications, Tome 5 (2009) Cet article a éte moissonné depuis la source Math-Net.Ru

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The Hunter–Saxton equation and the Gurevich–Zybin system are considered as two mutually non-equivalent representations of one and the same Whitham-type equation, and all their common solutions are obtained exactly.
Keywords: nonlinear PDEs; transformations; general solutions.
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Sergei Sakovich. On a Whitham-Type Equation. Symmetry, integrability and geometry: methods and applications, Tome 5 (2009). http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a100/

[1] Prykarpatsky A. K., Prytula M. M., “The gradient-holonomic integrability analysis of a Whitham-type nonlinear dynamical model for a relaxing medium with spatial memory”, Nonlinearity, 19 (2006), 2115–2122 | DOI | MR | Zbl

[2] Bogoliubov N. N. Jr., Prykarpatsky A. K., Gucwa I., Golenia J., Analytical properties of an Ostrovsky–Whitham type dynamical system for a relaxing medium with spatial memory and its integrable regularization, arXiv:0902.4395

[3] Sakovich A., Sakovich S., “On transformations of the Rabelo equations”, SIGMA, 3 (2007), 086, 8 pp., ages ; arXiv:0705.2889 | MR | Zbl

[4] Hunter J. K., Saxton R., “Dynamics of director fields”, SIAM J. Appl. Math., 51 (1991), 1498–1521 | DOI | MR | Zbl

[5] Hunter J. K., Zheng Y., “On a completely integrable nonlinear hyperbolic variational equation”, Phys. D, 79 (1994), 361–386 | MR | Zbl

[6] Dai H.-H., Pavlov M., “Transformations for the Camassa–Holm equation, its high-frequency limit and the Sinh-Gordon equation”, J. Phys. Soc. Japan, 67 (1998), 3655–3657 | DOI | MR | Zbl

[7] Pavlov M. V., “The Calogero equation and Liouville-type equations”, Theoret. and Math. Phys., 128 (2001), 927–932 ; nlin.SI/0101034 | DOI | MR | Zbl

[8] Morozov O. I., Contact equivalence of the generalized Hunter–Saxton equation and the Euler–Poisson equation, math-ph/0406016

[9] Olver P. J., Rosenau P., “Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support”, Phys. Rev. E, 53 (1996), 1900–1906 | DOI | MR

[10] Beals R., Sattinger D. H., Szmigielski J., “Inverse scattering solutions of the Hunter–Saxton equation”, Appl. Anal., 78 (2001), 255–269 | DOI | MR | Zbl

[11] Hunter J. K., Zheng Y. X., “On a nonlinear hyperbolic variational equation. I. Global existence of weak solutions”, Arch. Rational Mech. Anal., 129 (1995), 305–353 | DOI | MR | Zbl

[12] Bressan A., Constantin A., “Global solutions of the Hunter–Saxton equation”, SIAM J. Math. Anal., 37 (2005), 996–1026 ; math.AP/0502059 | DOI | MR | Zbl

[13] Reyes E. G., “The soliton content of the Camassa–Holm and Hunter–Saxton equations”, Proceedinds of Fifth International Conference “Symmetry in Nonlinear Mathematical Physics” (July 9–15, 2001, Kyiv), Proceedings of Institute of Mathematics, 43, no. 1, eds. A. G. Nikitin, V. M. Boyko and R. O. Popovych, Kyiv, 2002, 201–208 | MR | Zbl

[14] Lenells J.,, “The Hunter–Saxton equation: a geometric approach”, SIAM J. Math. Anal., 40 (2008), 266–277 | DOI | MR | Zbl

[15] Sakovich S. Yu., “On conservation laws and zero-curvature representations of the Liouville equation”, J. Phys. A: Math. Gen., 27 (1994), L125–L129 | DOI | MR | Zbl

[16] Olver P. J., Applications of Lie groups to differential equations, 2nd ed., Graduate Texts in Mathematics, 107, Springer-Verlag, New York, 1993 | MR | Zbl

[17] Gurevich A. V., Zybin K. P., “Nondissipative gravitational turbulence”, Soviet Phys. JETP, 67 (1988), 1–12 | MR

[18] Gurevich A. V., Zybin K. P., “Large-scale structure of the Universe. Analytic theory”, Soviet Phys. Usp., 38 (1995), 687–722

[19] Pavlov M. V., “The Gurevich–Zybin system”, J. Phys. A: Math. Gen., 38 (2005), 3823–3840 ; nlin.SI/0412072 | DOI | MR

[20] Davidson R. C., Methods in nonlinear plasma theory, Academic Press, New York, 1972

[21] Brunelli J. C., Das A., “On an integrable hierarchy derived from the isentropic gas dynamics”, J. Math. Phys., 45 (2004), 2633–2645 ; nlin.SI/0401009 | DOI | MR | Zbl