Abstract
With the help of the Zakharov–Manakov \(\overline \partial \)-dressing method, the scheme of obtaining exact rational solutions to the well-known two-dimensional Veselov–Novikov integrable nonlinear equation and exact rational potentials for the two-dimensional Schrödinger stationary equation corresponding to wave functions with multiple poles is developed. As an example, new exact rational nonsingular and singular solutions to the Veselov–Novikov equation and the corresponding exact rational potentials for the two-dimensional Schrödinger stationary equation with multiple second-order poles are obtained.
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REFERENCES
V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons [in Russian], Nauka, Moscow (1980).
M. J. Ablowitz and P. A. Clarkson, Solitons Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society Lecture Notes Series, Vol. 14, 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/London (1992). 426
B. G. Konopelchenko, Solitons in Multidimensions: Inverse Spectral Transform Method, World Scientific, Singapore (1993).
S. V. Manakov, Physica, D3, 420 (1981).
R. Beals and R. R. Coifman, Physica, D18, 242 (1986).
V. E. Zakharov and S. V. Manakov, Funct. Anal. Pril., 19, No. 2, 11 (1985).
V. E. Zakharov, in: Proc. Third International Workshop, Vol. I, Naukova Dumka, Kiev (1988), p. 152.
L. V. Bogdanov and S. V. Manakov, J. Phys., A21, 537 (1988).
A. S. Fokas and V. E. Zakharov, J. Nonlin. Sci., 2, No. 1, 109 (1992).
R. Beals and R. R. Coifman, Inv. Probl., 5, 87–130 (1989).
A. S. Fokas and M. J. Ablowitz, in: Proc. CIFMO School and Workshop on Nonlinear Phenomena, K. B. Wolf, ed., Mexico (1983), pp. 37–183.
V. E. Zakharov, in: Inverse Methods in Action, P. C. Sabatier, ed., Springer, Berlin (1990), p. 602.
A. P. Veselov and S. P. Novikov, Dokl. Akad. Nauk SSSR, 279, 20 (1984).
P. G. Grinevich and S. V. Manakov, Funct. Anal. Pril., 20, No. 7, 14 (1986).
P. G. Grinevich and R. G. Novikov, Proc. Int. Workshop on Plasma Theory and Nonlinear and Turbulent Processes in Physics, Vol. 1, V. G. Bar'yakhtar, V. M. Chernousenko, N. S. Erokhin, A. G. Sitenko, and V. E. Zakharov, eds., Singapore (1988), p. 58.
P. G. Grinevich and R. G. Novikov, Funct. Anal. Pril., 22, No. 1, 3 (1988).
V. B. Matveev and M. A. Salle, Darbu Transformations and Solitons. Springer Series in Nonlinear Dynamics, Springer, Berlin/Heidelberg (1991).
P. G. Grinevich, Author's Abstract Doct. Phys.-Math. Sci. Dissert., Chernogolovka (1999).
P. G. Grinevich, Russ. Math. Surv., 55, No. 6, 1015 (2000).
V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP, 34, 62 (1972).
E. Olmedilla, Physica, D25, 330 (1987).
H. Tsuru and M. Wadati, Jpn. J. Phys., 51, 2029 (1981).
C. Pope, Physica, D9, 103 (1983).
M. Wadati and M. Sakagami, J. Phys. Soc. Jpn., 53, 1933 (1984).
R. Ward, Phys. Lett., A208, 203 (1995).
T. Inonniou, J. Math. Phys., 37, 3422 (1996).
M. J. Ablowitz and J. Villaroel, Phys. Rev. Lett., 78, 570 (1997).
M. Manas and P. M. Santini, Phys. Lett, A227, 325 (1997).
V. G. Dubrovsky, J. Phys., A32, 369 (1999).
V. G. Dubrovsky and I. B. Formusatik, J. Phys., A34, 1837 (2001).
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Dubrovskii, V.G., Formusatik, I.B. Construction of New Exact Rational Solutions to the Veselov–Novikov Equation and New Exact Rational Potentials for the Two-Dimensional Schrödinger Stationary Equation by the \(\overline \partial \)-Dressing Method. Russian Physics Journal 46, 414–426 (2003). https://doi.org/10.1023/A:1025736228444
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DOI: https://doi.org/10.1023/A:1025736228444