Integrability of Exceptional Hydrodynamic-Type Systems
Tóm tắt
We consider non-diagonalizable hydrodynamic-type systems integrable by the extended hodograph method. We restrict the analysis to non-diagonalizable hydrodynamic reductions of the three-dimensionalMikhalev equation. We show that families of these hydrodynamictype systems are reducible to the heat hierarchy. Then we construct new particular explicit solutions for the Mikhalev equation.
Tài liệu tham khảo
E. V. Ferapontov, “Integration of weakly nonlinear hydrodynamic systems in Riemann invariants,” Phys. Lett. A 158 (3–4), 112–118 (1991).
E. V. Ferapontov and K. R. Khusnutdinova, “On the integrability of (2 + 1)-dimensional quasilinear systems,” Commun. Math. Phys. 248 (1), 187–206 (2004).
E. V. Ferapontov and K. R. Khusnutdinova, “The characterization of two-component (2 + 1)-dimensional integrable systems of hydrodynamic type,” J. Phys. A: Math. Gen. 37 (8), 2949–2963 (2004).
E. V. Ferapontov and J. Moss, “Linearly degenerate partial differential equations and quadratic line complexes,” Commun. Anal. Geom. 23 (1), 91–127 (2015).
J. Gibbons and Y. Kodama, “Integrable quasilinear systems: generalized hodograph transformation,” in Nonlinear Evolutions: Proc. Workshop, Balaruc-les-Bains, 1987 (World Scientific, Singapore, 1988), pp. 97–107.
Y. Kodama, “A method for solving the dispersionless KP equation and its exact solutions,” Phys. Lett. A 129 (4), 223–226 (1988).
Y. Kodama, “A solution method for the dispersion-less KP equation,” Prog. Theor. Phys. Suppl. 94, 184–194 (1988).
Y. Kodama, “Exact solutions of hydrodynamic type equations having infinitely many conserved densities,” Phys. Lett. A 135 (3), 171–174 (1989).
Y. Kodama, “Solutions of the dispersionless Toda equation,” Phys. Lett. A 147 (8–9), 477–482 (1990).
Y. Kodama and J. Gibbons, “A method for solving the dispersionless KP hierarchy and its exact solutions. II,” Phys. Lett. A 135 (3), 167–170 (1989).
Y. Kodama and J. Gibbons, “Integrability of the dispersionless KP hierarchy,” in Nonlinear World: Proc. Workshop, Kiev, 1989 (World Scientific, Singapore, 1990), Vol. 1, pp. 166–180.
Y. Kodama and B. G. Konopelchenko, “Confluence of hypergeometric functions and integrable hydrodynamictype systems,” Theor. Math. Phys. 188 (3), 1334–1357 (2016) [transl. from Teor. Mat. Fiz. 188 (3), 429–455 (2016)].
B. G. Konopelchenko and G. Ortenzi, “Parabolic regularization of the gradient catastrophes for the Burgers–Hopf equation and Jordan chain,” arXiv: 1711.01087 [math-ph].
V. G. Mikhalev, “On the Hamiltonian formalism for Korteweg–de Vries type hierarchies,” Funct. Anal. Appl. 26 (2), 140–142 (1992) [transl. from Funkts. Anal. Appl. 26 (2), 79–82 (1992)].
M. V. Pavlov, “Hamiltonian formalism of weakly nonlinear hydrodynamic systems,” Theor. Math. Phys. 73 (2), 1242–1245 (1987) [transl. from Teor. Mat. Fiz. 73 (2), 316–320 (1987)].
M. V. Pavlov, “Integrable hydrodynamic chains,” J. Math. Phys. 44 (9), 4134–4156 (2003).
M. V. Pavlov, “Integrability of the Egorov systems of hydrodynamic type,” Theor. Math. Phys. 150 (2), 225–243 (2007) [transl. from Teor. Mat. Fiz. 150 (2), 263–285 (2007)].
M. V. Pavlov, “Integrable dispersive chains and energy dependent Schrödinger operator,” J. Phys. A: Math. Theor. 47 (29), 295204 (2014).
S. P. Tsarev, “On Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type,” Sov. Math., Dokl. 31, 488–491 (1985) [transl. from Dokl. Akad. Nauk SSSR 282 (3), 534–537 (1985)].
S. P. Tsarev, “The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method,” Math. USSR, Izv. 37 (2), 397–419 (1991) [transl. from Izv. Akad. Nauk SSSR, Ser. Mat. 54 (5), 1048–1068 (1990)].