Integrable hydrodynamic chains

A new approach for derivation of Benney-type moment chains and integrable hydrodynamic type systems is presented. New integrable hydrodynamic chains are constructed; all their hydrodynamical reductions are described and integrated. New (2+1) integrable hydrodynamic type systems are found.

[1]  A. Aksenov Symmetries and relations between solutions of the class Euler-Poisson-Darboux equations , 2001 .

[2]  M. Pavlov,et al.  Tri-Hamiltonian Structures of Egorov Systems of Hydrodynamic Type , 2003 .

[3]  V. Zakharov On the Benney equations , 1981 .

[4]  John Gibbons,et al.  Collisionless Boltzmann equations and integrable moment equations , 1981 .

[5]  Vladimir E. Zakharov,et al.  Benney equations and quasiclassical approximation in the method of the inverse problem , 1980 .

[6]  B. Dubrovin Hamiltonian formalism of Whitham-type hierarchies and topological Landau-Ginsburg models , 1992 .

[7]  Ima Program on Nonlinear Waves,et al.  Solitons in Physics, Mathematics, and Nonlinear Optics , 1990 .

[8]  Luther Pfahler Eisenhart,et al.  Transformations of surfaces , 1934, The Mathematical Gazette.

[9]  J. Gibbons,et al.  REDUCTIONS OF THE BENNEY EQUATIONS , 1996 .

[10]  E. V. Ferapontov,et al.  Hamiltonian systems of hydrodynamic type and their realization on hypersurfaces of a pseudo-Euclidean space , 1991 .

[11]  N. N. Yanenko,et al.  Systems of Quasilinear Equations and Their Applications to Gas Dynamics , 1983 .

[12]  Classical R-matrix theory of dispersionless systems: II. (2 + 1) dimension theory , 2002, nlin/0211008.

[13]  G. Darboux Leçons sur les systémes orthogonaux et les coordonnées curvilignes , 1910 .

[14]  S. Alber Associated integrable systems , 1991 .

[15]  Lei Yu Waterbag reductions of the dispersionless discrete KP hierarchy , 2000 .

[16]  S. P. Tsarëv THE GEOMETRY OF HAMILTONIAN SYSTEMS OF HYDRODYNAMIC TYPE. THE GENERALIZED HODOGRAPH METHOD , 1991 .

[17]  Y. Manin,et al.  Conservation laws and Lax representation of Benney's long wave equations , 1979 .

[18]  E. Ferapontov Laplace transformations of hydrodynamic-type systems in Riemann invariants , 1997, solv-int/9705017.

[19]  V. Mikhalev On the Hamiltonian formalism for Korteweg—de Vries type hierarchies , 1992 .

[20]  E. Ferapontov,et al.  Reciprocal transformations of Hamiltonian operators of hydrodynamic type: nonlocal Hamiltonian formalism for linearly degenerate systems , 2002, nlin/0212026.

[21]  B. Dubrovin,et al.  Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory , 1989 .

[22]  The Symmetry Structure of the Heavenly Equation , 1996, hep-th/9604085.

[23]  M. Pavlov Hamiltonian formalism of weakly nonlinear hydrodynamic systems , 1987 .

[24]  L. Eisenhart Triply Conjugate Systems with Equal Point Invariants , 1919 .

[25]  J. Gibbons,et al.  Conformal maps and reductions of the Benney equations , 1999 .

[26]  Y. Nutku On a new class of completely integrable nonlinear wave equations. II. Multi‐Hamiltonian structure , 1987 .

[27]  E. Ferapontov,et al.  Boyer–Finley equation and systems of hydrodynamic type , 2002 .

[28]  Y. Kodama Solutions of the dispersionless Toda equation , 1990 .

[29]  A. Fordy Isospectral Flows: Their Hamiltonian Structures, Miura Maps and Master Symmetries , 1990 .

[30]  E. Ferapontov,et al.  Quasiclassical limit of coupled KdV equations: Riemann invariants and multi-Hamiltonian structure , 1991 .

[31]  E. Ferapontov Integration of weakly nonlinear hydrodynamic systems in Riemann invariats , 1991 .

[32]  Marcel Jaulent,et al.  Nonlinear evolution equations associated with ‘enegry-dependent Schrödinger potentials’ , 1976 .

[33]  The Gamma Function in the Integral Calculus , 1918 .

[34]  D. J. Benney Some Properties of Long Nonlinear Waves , 1973 .

[35]  Y. Manin,et al.  Equations of long waves with a free surface. II. Hamiltonian structure and higher equations , 1978 .