Integrability and Nijenhuis Tensors

P. D. Lax. Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math., 21:467–490, 1968.MATHCrossRefMathSciNetGoogle Scholar

V. E. Zakharov and L. D. Faddeev. Korteweg-de Vries equation: A completely integrable Hamiltonian system. Funct. Anal. Appl., 5(4):280–287, 1971.MATHCrossRefGoogle Scholar

S. De Filippo, G. Marmo, M. Salerno, and G. Vilasi. On the Phase Manifold Geometry of Integrable Nonlinear Field Theory. Preprint IFUSA, 1982.Google Scholar

S. De Filippo, G. Marmo, M. Salerno, and G. Vilasi. A new characterization of complete integrable systems. Nuovo Cimento B, 83:97–112, 1984.CrossRefADSGoogle Scholar

D. J. Kaup. Closure of the squared Zakharov–Shabat eigenstates. J. Math. Anal. Appl., 54(3):849–864, 1976.MATHCrossRefMathSciNetGoogle Scholar

P. J. Olver. Evolution equations possessing infinitely many symmetries. J. Math. Phys., 18:1212, 1977.MATHCrossRefADSMathSciNetGoogle Scholar

Magri, F.: A geometrical approach to the nonlinear solvable equations. In: Boiti, M., Pempinelli, F., Soliani, G. (eds.) Nonlinear Evolution Equations and Dynamical Systems: Proceedings of the Meeting Held at the University of Lecce June 20–23, 1979. Lect. Notes Phys. 120, 233–263 (1980)Google Scholar

G. Vilasi. On the hamiltonian structures of the Korteweg-de Vries and sine-Gordon theories. Phys. Lett. B, 94(2):195–198, 1980.CrossRefADSMathSciNetGoogle Scholar

S. De Filippo, G. Marmo, and G. Vilasi. A geometrical setting for the Lax representation. Phys. Lett. B, 117(6):418–422, 1983.Google Scholar

S. De Filippo, M. Salerno, and G. Vilasi. A geometrical approach to the integrability of soliton equations. Lett. Math. Phys., 9:85–91, 1985.MATHCrossRefADSMathSciNetGoogle Scholar

E. Hopf. The partial differential equation u _t + uu _x = u _xx. Comm. Pure Appl. Math., 3:201–230, 1950.Google Scholar

J. D. Cole. On a quasi-linear parabolic equation occurring in aerodynamics. Q. Appl. Math., 9(3):225–236, 1951.MATHMathSciNetGoogle Scholar

W. F. Ames. Nonlinear Partial Differential Equations in Engineering. Academic Press, New York-London edition, 1965.Google Scholar

G. Vilasi. Phase manifold geometry of burgers hierarchy. Lett. Nuovo Cimento, 37(3):105–109, 1985.MathSciNetGoogle Scholar

S. De Filippo, G. Marmo, M. Salerno, and G. Vilasi. Phase manifold geometry of burgers hierarchy. Lett. Nuovo Cimento, 37(3):105–109, 1983.Google Scholar

A. S. Mishchenko and A. T. Fomenko. Generalized Liouville method of integration of Hamiltonian systems. Funct. Anal. Appl., 12(2):113–121, 1978.MATHCrossRefGoogle Scholar

M. R. Adams and T. Ratiu. The three-point vortex problem: Commutative and noncommutative integrability. Contemp. Math., 81:245–257, 1988.MathSciNetGoogle Scholar

V. V. Trofimov and A. T. Fomenko. Algebra and Geometry of the Integrable Hamiltonian Differential Equations. Factorial, Minsk, 1995.Google Scholar

A. T. Fomenko. Symplectic Geometry. Advanced Studies in Contemporary Mathematics. Gordon & Breach Publishers, Luxembourg, 1995.Google Scholar

F. Fassò and T. Ratiu. Compatibility of symplectic structures adapted to noncommutatively integrable systems. J. Geom. Phys., 27:199–220, 1998.MATHCrossRefADSMathSciNetGoogle Scholar

O. I. Bogoyavlenskij. Extended integrability and Bi-Hamiltonian systems. Commun. Math. Phys., 196(1):19–51, 1998.MATHCrossRefADSMathSciNetGoogle Scholar

O. I. Bogoyavlenskij. Theory of tensor invariants of integrable Hamiltonian systems. I. Incompatible Poisson structures. Commun. Math. Phys., 180(3):529–586, 1996.MATHCrossRefADSMathSciNetGoogle Scholar

O. I. Bogoyavlenskij. Theory of tensor invariants of integrable Hamiltonian systems. II. Theorem on symmetries and its applications. Commun. Math. Phys., 184(2):301–365, 1997.MATHCrossRefADSMathSciNetGoogle Scholar

B. Cordani. The Kepler Problem:Group Theoretical Aspects, Regularization and Quantization, with Applications to the Study of Perturbations. Birkhauser Verlag, Boston, MA, 2003.Google Scholar

G. Marmo and G. Vilasi. When do recursion operators generate new conservation laws? Phys. Lett. B, 277(1–2):137–140, 1992.ADSMathSciNetGoogle Scholar

F. Magri and C. Morosi. A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson–Nijenhuis manifolds. Quaderni del Dipartimento di Matematica, Università di Milano, 1984.Google Scholar

G. Magnano and F. Magri. Poisson–Nijenhuis structures and Sato hierarchy. Rev. Math. Phys., 3(4):403–466, 1991.MATHCrossRefMathSciNetGoogle Scholar

Y. Kosmann-Schwarzbach and F. Magri. Lax–Nijenhuis operators for integrable systems. J. Math. Phys., 37:6173–6197, 1996.MATHCrossRefADSMathSciNetGoogle Scholar

Y. Kosmann-Schwarzbach. The Lie bialgebroid of a Poisson–Nijenhuis manifold. Lett. Math. Phys., 38(4):421–428, 1996.MATHCrossRefADSMathSciNetGoogle Scholar

V. G. Drinfeld and V. V. Sokolov. Lie Algebras and Korteweg-de Vries Type Equations. VINITI Series: Contemporary problems of mathematics. Recent developments. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985.Google Scholar

L. D. Faddeev and L. A. Takhtajan. Hamiltonian Methods in the Theory of Solitons. Springer-Verlag, Berlin, 1987.MATHGoogle Scholar

F. Magri, C. Morosi, and O. Ragnisco. Reduction techniques for infinite-dimensional Hamiltonian systems: Some ideas and applications. Commun. Math. Phys., 99(1):115–140, 1985.MATHCrossRefADSMathSciNetGoogle Scholar

Magri, F., Falqui, G., Pedroni. M.: The method of Poisson pairs in the theory of nonlinear PDEs. Direct and Inverse Methods in Nonlinear Evolution Equations. Lect. Notes Phys. 632, 85–136. Springer-Verlag, Berlin (2003)ADSMathSciNetGoogle Scholar

I. M. Gel’fand and I. Y. Dorfman. Hamiltonian operators and algebraic structures related to them. Funct. Anal. Appl., 13(4):248–262, 1979.CrossRefMathSciNetGoogle Scholar

I. M. Gel’fand and I. Y. Dorfman. The schouten bracket and hamiltonian operators. Funct. Anal. Appl., 14(3):223–226, 1980.CrossRefMathSciNetGoogle Scholar

C. Godbillion. Géométrie différentielle et méchanique analytique. Hermann, Paris, 1969.Google Scholar

B. Florko and A. B. Yanovski. On Magri’s theorem for complete integrability. Communication JINR (Joint Institute for Nuclear Research), Dubna, 1983.Google Scholar