SIGMA 5 (2009), 101, 7 pages      arXiv:0909.4455      https://doi.org/10.3842/SIGMA.2009.101

On a Whitham-Type Equation

Sergei Sakovich
Institute of Physics, National Academy of Sciences, 220072 Minsk, Belarus

Received September 27, 2009, in final form November 05, 2009; Published online November 08, 2009

Abstract
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.

Key words: nonlinear PDEs; transformations; general solutions.

pdf (176 kb)   ps (106 kb)   tex (9 kb)

References

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.
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 pages, arXiv:0705.2889.
4. Hunter J.K., Saxton R., Dynamics of director fields, SIAM J. Appl. Math. 51 (1991), 1498-1521.
5. Hunter J.K., Zheng Y., On a completely integrable nonlinear hyperbolic variational equation, Phys. D 79 (1994), 361-386.
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.
7. Pavlov M.V., The Calogero equation and Liouville-type equations, Theoret. and Math. Phys. 128 (2001), 927-932, nlin.SI/0101034.
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.
10. Beals R., Sattinger D.H., Szmigielski J., Inverse scattering solutions of the Hunter-Saxton equation, Appl. Anal. 78 (2001), 255-269.
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.
12. Bressan A., Constantin A., Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal. 37 (2005), 996-1026, math.AP/0502059.
13. Reyes E.G., The soliton content of the Camassa-Holm and Hunter-Saxton equations, in Proceedinds of Fifth International Conference "Symmetry in Nonlinear Mathematical Physics" (July 9-15, 2001, Kyiv), Editors A.G. Nikitin, V.M. Boyko and R.O. Popovych, Proceedings of Institute of Mathematics, Kyiv 43 (2002), Part 1, 201-208.
14. Lenells J., The Hunter-Saxton equation: a geometric approach, SIAM J. Math. Anal. 40 (2008), 266-277.
15. Sakovich S.Yu., On conservation laws and zero-curvature representations of the Liouville equation, J. Phys. A: Math. Gen. 27 (1994), L125-L129.
16. Olver P.J., Applications of Lie groups to differential equations, 2nd ed., Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, 1993.
17. Gurevich A.V., Zybin K.P., Nondissipative gravitational turbulence, Soviet Phys. JETP 67 (1988), 1-12.
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.
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.