With the help of the Zakharov–Manakov \( \overline{\partial} \) -dressing method, new classes of exact solutions with functional parameters of a two-dimensional integrable generalization of the Kaup–Kupershmidt equation have been constructed. It is shown that the constructed solutions contain soliton solutions and a subclass of new periodic solutions. Nonsingular solutions are also present among the constructed periodic solutions.
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S. P. Novikov, V. E. Zakharov, S. V. Manakov, and L. V. Pitaevskii, Theory of Solitons: The Inverse Scattering Method (Monographs in Contemporary Mathematics), Springer-Verlag, New York (1984).
M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Lecture Note Series), Cambridge University Press, Cambridge (1991).
B. G. Konopelchenko, Introduction to Multidimensional Integrable Equations: The Inverse Spectral Transform in 2+1 Dimensions, Plenum Press, New York (1992).
B. G. Konopelchenko, Solitons in Multidimensions: Inverse Spectral Transform Method, World Scientific, Singapore (1993).
A. S. Fokas and M. J. Ablowitz, Nonlinear Phenomena. Lecture Notes in Physics, 189, 137–183 (1983).
R. Beals and R. R. Coifman, Inverse Problems, 5, No. 2, 87–130 (1988).
V. E. Zakharov, in: Inverse Methods in Action, P. C. Sabatier, ed., Springer-Verlag, New York (1990), pp. 602–623.
S. V. Manakov, Physica, D3, Nos. 1–2, 420-427 (1981).
R. Beals and R. R. Coifman, Physica, D18, Nos. 1–3, 242–249 (1986).
V. E. Zakharov and S. V. Manakov, Funkts. Analiz Ego Prilozh., 19, No. 2, 11 (1985).
V. E. Zakharov, Plasma Theory and Nonlinear and Turbulent Processes in Physics, Vol. 1, N. S. Erokhin, V. E. Zakharov, A. G. Sitenko, V. M. Chernousenko, and V. G. Bar'yakhtar, eds., Naukova Dumka, Kiev (1988), p. 152.
L. V. Bogdanov and S. V. Manakov, J. Phys., A21, No. 10, 537–544 (1988).
B. G. Konopelchenko and V. G. Dubrovsky, Phys. Lett., A102, Nos. 1–2, 15–17 (1984).
T. Miva, E. Date, M. Jimbo, and M. Kashiwara, J. Phys. Soc. Jpn., 50, No. 11, 3806 (1981).
A. B. Shabat and V. E. Zakharov, Funkts. Analiz Ego Prilozh., 8, No. 3, 45– 53 (1974).
A. B. Shabat and V. E. Zakharov, Funkts. Analiz Ego Prilozh., 13, No. 3, 13–22 (1979).
Ya. V. Lisitsyn and V. G. Dubrovsky, Phys. Lett., A295, No. 4, 198–207 (2002).
V. G. Dubrovsky and A. V. Gramolin, Teor. Mat. Fiz., 160, No. 1, 35–48 (2009).
V. G. Dubrovsky, A. V. Topovsky, and M. Yu. Basalaev, Teor. Mat. Fiz., 167, No. 3, 377–393 (2011).
X.-B. Hu, D.-L. Wang, and X.-M. Qian, Phys. Lett., A262, No. 6, 409–415 (1999).
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Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 53–61, July, 2015.
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Dubrovsky, V.G., Topovsky, A.V. & Basalaev, M.Y. Solutions with Functional Parameters of 2+1-Dimensional Integrable Nonlinear Equations. Two Dimensional Integrable Generalization of the Kaup–Kupershmidt Equation. Russ Phys J 58, 930–940 (2015). https://doi.org/10.1007/s11182-015-0592-8
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DOI: https://doi.org/10.1007/s11182-015-0592-8