The matrix Lax representation of the generalized Riemann equations and its conservation laws. (English) Zbl 1252.34101
Summary: It is shown that the generalized Riemann equation is equivalent with the multicomponent generalization of the Hunter-Saxton equation. New matrix and scalar Lax representation are presented for this generalization. New class of the conserved densities, which depends explicitly on the time are obtained directly from the Lax operator. The algorithm, which allows us to generate a big class of the non-polynomial conservation laws of the generalized Riemann equation is presented. Due to this new series of conservation laws of the Hunter-Saxton equation is obtained.
MSC:
| 34M05 | Entire and meromorphic solutions to ordinary differential equations in the complex domain |
| 34C14 | Symmetries, invariants of ordinary differential equations |
References:
| [1] | Whitham, G. B., Linear and Nonlinear Waves (1974), Willey-Interscience: Willey-Interscience New York · Zbl 0373.76001 |
| [2] | Tsarev, S. P., Izv. Akad. Nauk SSSR Ser. Mat., 54, 5, 1048 (1990) |
| [3] | Ablowitz, M. J.; Segur, H., Solitons and the Inverse Scattering Transform (1981), SIAM: SIAM Philadelphia · Zbl 0299.35076 |
| [4] | Popowicz, Z.; Prykarpatsky, A. K., Nonlinearity, 23, 2517 (2010) · Zbl 1202.35127 |
| [5] | Artemovych, O. D.; Pavlov, M.; Popowicz, Z.; Prykarpatsky, A. K., J. Phys. A: Math. Theor., 43, 295205 (2010) · Zbl 1197.35187 |
| [6] | Pavlov, M.; Golenia, J.; Popowicz, Z.; Prykarpatsky, A., SIGMA, 6, 002 (2010) |
| [7] | Gurevich, A. V.; Zybin, K. P., Sov. Phys. JET, 67, 1 (1988) |
| [8] | Gurevich, A. V.; Zybin, K. P., Sov. Phys. Usp., 38, 687 (1995) |
| [9] | Brunelli, J. C.; Das, A., J. Math. Phys., 45, 2633 (2004) · Zbl 1071.37041 |
| [10] | Hunter, J. K.; Saxton, R., SIAM J. Appl. Math., 51, 1498 (1991) · Zbl 0761.35063 |
| [11] | Hunter, J. K.; Zheng, Y., Physica D, 79, 361 (1994) · Zbl 0900.35387 |
| [12] | Sakovich, S., SIGMA, 5, 101 (2009) · Zbl 1189.35290 |
| [13] | Pavlov, M. V., J. Phys. A: Math. Gen., 38, 3823 (2005) |
| [14] | Wang, J. P., Nonlinearity, 23, 2009 (2010) · Zbl 1209.37071 |
| [15] | Dai, H. H.; Pavlov, M., J. Phys. Soc. Japan, 67, 3655 (1998) |
| [16] | Beals, R.; Sattinger, D. H.; Szmigielski, J., Appl. Anal., 78, 255 (2001) · Zbl 1020.35092 |
| [17] | Hunter, J. K.; Zheng, Y., Arch. Ration. Mech. Anal., 129, 305 (1995) |
| [18] | Bressan, A.; Constantin, A., SIAM J. Math. Anal., 37, 996 (2005) · Zbl 1108.35024 |
| [19] | Antonowicz, M.; Fordy, A., Comm. Math. Phys., 124, 465 (1989) · Zbl 0696.35172 |
| [20] | Aratyn, H.; Gomes, J. F.; Zimerman, A. H., J. Phys. A, 39, 1099 (2006) · Zbl 1329.37061 |
| [21] | Antonowicz, M.; Fordy, A.; Liu, Q. P., Nonlinearity, 4, 669 (1991) · Zbl 0739.58052 |
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.
