
SIGMA 13 (2017), 053, 14 pages arXiv:1704.00043
https://doi.org/10.3842/SIGMA.2017.053
Contribution to the Special Issue on Symmetries and Integrability of Difference Equations
Symmetries of the Hirota Difference Equation
Andrei K. Pogrebkov ^{ab}
^{a)} Steklov Mathematical Institute of Russian Academy of Science, Moscow, Russia
^{b)} National Research University Higher School of Economics, Moscow, Russia
Received March 31, 2017, in final form July 02, 2017; Published online July 07, 2017
Abstract
Continuous symmetries of the Hirota difference equation, commuting with shifts of independent variables, are derived by means of the dressing procedure. Action of these symmetries on the dependent variables of the equation is presented. Commutativity of these symmetries enables interpretation of their parameters as ''times'' of the nonlinear integrable partial differentialdifference and differential equations. Examples of equations resulting in such procedure and their Lax pairs are given. Besides these, ordinary, symmetries the additional ones are introduced and their action on the Scattering data is presented.
Key words:
Hirota difference equation; symmetries; integrable differentialdifference and differential equations; additional symmetries.
pdf (378 kb)
tex (20 kb)
References

Bogdanov L.V., Konopelchenko B.G., Generalized KP hierarchy: Möbius symmetry, symmetry constraints and CalogeroMoser system, Phys. D 152/153 (2001), 8596, solvint/9912005.

Boiti M., Pempinelli F., Pogrebkov A.K., CauchyJost function and hierarchy of integrable equations, Theoret. and Math. Phys. 185 (2015), 15991613, arXiv:1508.02229.

Dryuma V.S., Analytic solution of the twodimensional Kortewegde Vries (KdV) equation, JETP Lett. 19 (1974), 387388.

Fioravanti D., Nepomechie R.I., An inhomogeneous Lax representation for the Hirota equation, J. Phys. A: Math. Theor. 50 (2017), 054001, 14 pages, arXiv:1609.06761.

Grinevich P.G., Orlov A.Yu., Virasoro action on Riemann surfaces, Grassmannians, $\det \overline\partial_J$ and SegalWilson $\tau$function, in Problems of Modern Quantum Field Theory (Alushta, 1989), Res. Rep. Phys., Springer, Berlin, 1989, 86106.

Hirota R., Nonlinear partial difference equations. II. Discretetime Toda equation, J. Phys. Soc. Japan 43 (1977), 20742078.

Hirota R., Discrete analogue of a generalized Toda equation, J. Phys. Soc. Japan 50 (1981), 37853791.

Kadomtsev B.B., Petviashvili V.I., On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl. 192 (1970), 539541.

Krichever I., Wiegmann P., Zabrodin A., Elliptic solutions to difference nonlinear equations and related manybody problems, Comm. Math. Phys. 193 (1998), 373396, hepth/9704090.

Orlov A.Yu., Shul'man E.I., Additional symmetries of the nonlinear Schrödinger equation, Theoret. and Math. Phys. 64 (1985), 862866.

Pogrebkov A.K., On time evolutions associated with the nonstationary Schrödinger equation, in L.D. Faddeev's Seminar on Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 201, Amer. Math. Soc., Providence, RI, 2000, 239255, mathph/9902014.

Pogrebkov A.K., Hirota difference equation and a commutator identity on an associative algebra, St. Petersburg Math. J. 22 (2011), 473483.

Pogrebkov A.K., Hirota difference equation: inverse scattering transform, Darboux transformation, and solitons, Theoret. and Math. Phys. 181 (2014), 15851598, arXiv:1407.0677.

Pogrebkov A.K., Commutator identities on associative algebras, the nonAbelian Hirota difference equation and its reductions, Theoret. and Math. Phys. 187 (2016), 823834.

Saito S., Octahedral structure of the HirotaMiwa equation, J. Nonlinear Math. Phys. 19 (2012), 539550.

Zabrodin A.V., Hirota difference equations, Theoret. and Math. Phys. 113 (1997), 13471392, solvint/9704001.

Zabrodin A.V., Bäcklund transformation for the Hirota difference equation, and the supersymmetric Bethe ansatz, Theoret. and Math. Phys. 155 (2008), 567584.

Zakharov V.E., Manakov S.V., Construction of multidimensional nonlinear integrable systems and their solutions, Funct. Anal. Appl. 19 (1985), 89101.

Zakharov V.E., Shabat A.B., A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I, Funct. Anal. Appl. 8 (1977), 226235.

