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Izv. Akad. Nauk SSSR Ser. Mat., 1989, Volume 53, Issue 2, Pages 243–257 (Mi izv1239)  

This article is cited in 39 scientific papers (total in 39 papers)

Overturning solitons in new two-dimensional integrable equations

O. I. Bogoyavlenskii

Abstract: Two two-dimensional nonlinear equations are constructed which are integrable by means of a one-dimensional inverse scattering problem. Soliton and $N$-soliton solutions are indicated which are smooth in one coordinate and in the other possess the same overturning property as the classical Riemann wave.
Bibliography: 9 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1990, 34:2, 245–259

Bibliographic databases:

UDC: 539.2
MSC: Primary 35Q20, 76B25; Secondary 35R30
Received: 26.12.1988

Citation: O. I. Bogoyavlenskii, “Overturning solitons in new two-dimensional integrable equations”, Izv. Akad. Nauk SSSR Ser. Mat., 53:2 (1989), 243–257; Math. USSR-Izv., 34:2 (1990), 245–259

Citation in format AMSBIB
\by O.~I.~Bogoyavlenskii
\paper Overturning solitons in new two-dimensional integrable equations
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1989
\vol 53
\issue 2
\pages 243--257
\jour Math. USSR-Izv.
\yr 1990
\vol 34
\issue 2
\pages 245--259

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    This publication is cited in the following articles:
    1. O. I. Bogoyavlenskii, “Breaking solitons. II”, Math. USSR-Izv., 35:1 (1990), 245–248  mathnet  crossref  mathscinet  zmath
    2. O. I. Bogoyavlenskii, “Breaking solitons. IV”, Math. USSR-Izv., 37:3 (1991), 475–487  mathnet  crossref  mathscinet  zmath  adsnasa
    3. O. I. Bogoyavlenskii, “A theorem on two commuting automorphisms, and integrable differential equations”, Math. USSR-Izv., 36:2 (1991), 263–279  mathnet  crossref  mathscinet  zmath  adsnasa
    4. O. I. Bogoyavlenskii, “Breaking solitons. III”, Math. USSR-Izv., 36:1 (1991), 129–137  mathnet  crossref  mathscinet  zmath  adsnasa
    5. O. I. Bogoyavlenskii, “Breaking solitons in $2+1$-dimensional integrable equations”, Russian Math. Surveys, 45:4 (1990), 1–89  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    6. O. I. Bogoyavlenskii, “Algebraic constructions of integrable dynamical systems-extensions of the Volterra system”, Russian Math. Surveys, 46:3 (1991), 1–64  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    7. O. I. Bogoyavlenskii, “Breaking solitons. V. Systems of hydrodynamic type”, Math. USSR-Izv., 38:3 (1992), 439–454  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    8. P A Clarkson, E L Mansfield, Nonlinearity, 7:3 (1994), 975  crossref  mathscinet  zmath  isi
    9. Peter A Clarkson, Pilar R Gordoa, Andrew Pickering, Inverse Probl, 13:6 (1997), 1463  crossref  mathscinet  zmath  isi
    10. Song-Ju Yu, Kouichi Toda, Narimasa Sasa, Takeshi Fukuyama, J Phys A Math Gen, 31:14 (1998), 3337  crossref  mathscinet  zmath
    11. Song-Ju Yu, Kouichi Toda, Takeshi Fukuyama, J Phys A Math Gen, 31:50 (1998), 10181  crossref  mathscinet  zmath
    12. Kouichi Toda, Yu Song-Ju, Takeshi Fukuyama, “The Bogoyavlenskii-Schiff hierarchy and integrable equations in (2 + 1) dimensions”, Reports on Mathematical Physics, 44:1-2 (1999), 247  crossref
    13. P G Est$eacute$vez, G A Hern$aacute$ez, J Phys A Math Gen, 33:10 (2000), 2131  crossref  zmath  adsnasa
    14. Kouichi Toda, Song-Ju Yu, “The investigation into the Schwarz–Korteweg–de Vries equation and the Schwarz derivative in (2+1) dimensions”, J Math Phys (N Y ), 41:7 (2000), 4747  crossref  mathscinet  zmath
    15. Yu. Song-Ju, K. Toda, T. Fukuyama, “A quest for the integrable equation in $3+1$ dimensions”, Theoret. and Math. Phys., 122:2 (2000), 256–259  mathnet  crossref  crossref  mathscinet  zmath  isi
    16. A.N.W. Hone, “Reciprocal link for 2 + 1-dimensional extensions of shallow water equations”, Applied Mathematics Letters, 13:3 (2000), 37  crossref
    17. Song-Ju Yu, Kouichi Toda, “Lax pairs, Painlevé Properties and Exact Solutions of the Calogero Korteweg-de Vries Equation and a New (2+1)-Dimensional Equation”, Journal of Nonlinear Mathematical Physics, 7:1 (2000), 1  crossref
    18. Kouichi Toda, Song-Ju Yu, Inverse Probl, 17:4 (2001), 1053  crossref  mathscinet  zmath  isi
    19. Kouichi Toda, Song-Ju Yu, “Note on an extension of the CDF equation to (2 + 1) dimensions”, Reports on Mathematical Physics, 48:1-2 (2001), 255  crossref
    20. Kouichi Toda, Song-Ju Yu, “The Investigation into New Equations in (2+1) Dimensions”, Journal of Nonlinear Mathematical Physics, 8:sup1 (2001), 272  crossref
    21. Kouichi Toda, “A Search for Higher-Dimensional Integrable Modified KdV Equations – The Painlevé Approach”, Journal of Nonlinear Mathematical Physics, 9:sup1 (2002), 207  crossref
    22. M. S. Bruzón, M. L. Gandarias, C. Muriel, J. Ramíres, S. Saez, F. R. Romero, “The Calogero–Bogoyavlenskii–Schiff Equation in $2+1$ Dimensions”, Theoret. and Math. Phys., 137:1 (2003), 1367–1377  mathnet  crossref  crossref  mathscinet  isi
    23. M Hamanaka, “Towards noncommutative integrable systems”, Physics Letters A, 316:1-2 (2003), 77  crossref  elib
    24. Xianguo Geng, Cewen Cao, “Explicit solutions of the 2+1-dimensional breaking soliton equation”, Chaos, Solitons & Fractals, 22:3 (2004), 683  crossref
    25. Zhenyun Qin, Ruguang Zhou, “A (2+1)-dimensional breaking soliton equation associated with the Kaup–Newell soliton hierarchy”, Chaos, Solitons & Fractals, 21:2 (2004), 311  crossref
    26. Masashi Hamanaka, “Commuting flows and conservation laws for noncommutative Lax hierarchies”, J Math Phys (N Y ), 46:5 (2005), 052701  crossref  mathscinet  zmath  isi
    27. Yan-ze Peng, “New Types of Localized Coherent Structures in the Bogoyavlenskii-Schiff Equation”, Int J Theor Phys, 45:9 (2006), 1764  crossref  mathscinet  isi
    28. Tadashi Kobayashi, Kouichi Toda, “The Painlevé Test and Reducibility to the Canonical Forms for Higher-Dimensional Soliton Equations with Variable-Coefficients”, SIGMA, 2 (2006), 063, 10 pp.  mathnet  crossref  mathscinet  zmath
    29. Masashi Hamanaka, “Non-commutative Ward's conjecture and integrable systems”, Nuclear Physics B, 741:3 (2006), 368  crossref
    30. J Ramírez, J L Romero, “New classes of solutions for the Schwarzian Korteweg–de Vries equation in (2+1) dimensions”, J Phys A Math Theor, 40:16 (2007), 4351  crossref  mathscinet  zmath  adsnasa  isi
    31. Su Ting, Geng Xian-Guo, Ma Yun-Ling, “Wronskian Form of N-Soliton Solution for the (2+1)-Dimensional Breaking Soliton Equation”, Chinese Phys Lett, 24:2 (2007), 305  crossref  adsnasa
    32. J. Ramírez, J.L. Romero, M.S. Bruzón, M.L. Gandarias, “Multiple solutions for the Schwarzian Korteweg–de Vries equation in (2+1) dimensions”, Chaos, Solitons & Fractals, 32:2 (2007), 682  crossref  elib
    33. Xuelin Yong, Zhiyong Zhang, Yufu Chen, “Bäcklund transformation, nonlinear superposition formula and solutions of the Calogero equation”, Physics Letters A, 372:41 (2008), 6273  crossref  elib
    34. Lü Zhuo-Sheng, Duan Li-Xia, Xie Fu-Ding, “Cross Soliton-Like Waves for the (2+1)-Dimensional Breaking Soliton Equation”, Chinese Phys Lett, 27:7 (2010), 070502  crossref
    35. Hong-Yan Zhi, Hui Chang, “Invariance of Painlevé property for some reduced (1+1)-dimensional equations”, Chinese Phys. B, 22:11 (2013), 110203  crossref
    36. Wei-Ting Zhu, Song-Hua Ma, Jian-Ping Fang, Zheng-Yi Ma, Hai-Ping Zhu, “Fusion, fission, and annihilation of complex waves for the (2+1)-dimensional generalized Calogero—Bogoyavlenskii—Schiff system”, Chinese Phys. B, 23:6 (2014), 060505  crossref
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    38. Yiren Chen, Rui Liu, “Some new nonlinear wave solutions for two (3+1)-dimensional equations”, Applied Mathematics and Computation, 260 (2015), 397  crossref
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