Differential Resultant, Computer Algebra and Completely Integrable Dynamical Systems

For a pair of differential operators A and B with periodic coefficients we construct their differential resultant and derive condition for their commutativity. By considering this condition as a stationary Lax representation we are able to treat completely integrable dynamical systems. As special cases we obtain Henon-Heiles dynamical systems. We propose algorithms to do this by using the powerful methods of computer algebra and performing symbolic calculations in Maple13 and Reduce4.

[1]  P. Lax.  INTEGRALS OF NONLINEAR EQUATIONS OF EVOLUTION AND SOLITARY WAVES. , 1968 .

[2]  J. D. Gibbon,et al.  The prolongation structure of a higher order Korteweg-de Vries equation , 1978, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[3]  Evgenii V. Vorozhtsov,et al.  Computer-Aided Analysis of Difference Schemes for Partial Differential Equations , 1996 .

[4]  Fritz Gesztesy,et al.  Floquet theory revisited , 1995 .

[5]  Yuji Komada,et al.  Nearly integrable systems , 1985 .

[6]  J. Verdier,et al.  Revêtements tangentiels et sommes de $4$ nombres triangulaires , 1990 .

[7]  Goldstein,et al.  Differential Equations with Applications in Biology, Physics, and Engineering , 1991 .

[8]  N. A. Kostov,et al.  On the geometry of elliptic solitons , 1994 .

[9]  Igor Krichever,et al.  METHODS OF ALGEBRAIC GEOMETRY IN THE THEORY OF NON-LINEAR EQUATIONS , 1977 .

[10]  Igor Krichever,et al.  Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles , 1980 .

[11]  Fred J. Hickernell,et al.  The Evolution of Large-Horizontal-Scale Disturbances in Marginally Stable, Inviscid, Shear Flows. II. Solutions of the Boussinesq Equation , 1983 .

[12]  V. Sokolov,et al.  The Symmetry Approach to Classification of Integrable Equations , 1991 .

[13]  William F. Ames,et al.  Differential equations with applications to mathematical physics , 1993 .

[14]  Igor Krichever,et al.  Nonlinear equations and elliptic curves , 1985 .

[15]  N. A. Kostov,et al.  Spectral characteristics of elliptic solitons , 1993 .

[16]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[17]  L. M. Berkovich,et al.  Algorithm of solving second order linear ordinary differential equations and its implementation in REDUCE(Theory and applications in computer algebra) , 1993 .

[18]  E. Belokolos,et al.  Algebro-geometric approach to nonlinear integrable equations , 1994 .

[19]  Ryogo Hirota,et al.  Resonance of Solitons in One Dimension , 1983 .

[20]  Allan P. Fordy,et al.  Some remarkable nonlinear transformations , 1980 .

[21]  Willy Hereman,et al.  Complete integrability of a modified vector derivative nonlinear Schrödinger equation , 1994, solv-int/9407002.

[22]  Willy Hereman,et al.  Conservations laws and solitary wave solutions for generalized Schamel equations , 1994 .

[23]  Bernard Deconinck,et al.  Computing Riemann theta functions , 2002, Math. Comput..

[24]  Vladimir P. Gerdt.  Computer Algebra, Symmetry Analysis And Integrability Of Nonlinear Evolution Equations , 1993 .

[25]  Fritz Gesztesy,et al.  Treibich-Verdier potentials and the stationary (m)KDV hierarchy , 1995 .

[26]  Vladimir P. Gerdt,et al.  Computer generation of necessary integrability conditions for polynomial-nonlinear evolution systems , 1990, ISSAC '90.

[27]  E. Belokolos,et al.  Reduction of theta functions and elliptic finite-gap potentials , 1994 .

[28]  Victor G. Ganzha,et al.  Computer-Aided Analysis of Difference Schemes for Partial Differential Equations: Ganzha/Computer-Aided , 1996 .

[29]  B. Kupershmidt,et al.  A coupled Korteweg-de Vries equation with dispersion , 1985 .

[30]  Björn Birnir,et al.  Singularities of the complex korteweg‐de vries flows , 1986 .

[31]  G. Wilson,et al.  Modifying Lax equations and the second Hamiltonian structure , 1980 .

[32]  Igor Krichever,et al.  Elliptic solutions of nonlinear integrable equations and related topics , 1994 .

[33]  Nakao Hayashi,et al.  On the Modified Korteweg–De Vries Equation , 2001 .

[34]  Willi-Hans Steeb,et al.  Nonlinear Evolution Equations and Painleve Test , 1988 .

[35]  Ryogo Hirota,et al.  Direct method of finding exact solutions of nonlinear evolution equations , 1976 .

[36]  Alexey Borisovich Shabat,et al.  The symmetry approach to the classification of non-linear equations. Complete lists of integrable systems , 1987 .

[37]  Armando Treibich.  New elliptic potentials , 1994 .

[38]  Willy Hereman,et al.  Review of Symbolic Software for the Computation of Lie Symmetries of Differential Equations , 1994 .

[39]  Igor Krichever,et al.  Integration of nonlinear equations by the methods of algebraic geometry , 1977 .

[40]  E. Belokolos,et al.  Verdier elliptic solitons and the Weierstrass theory of reduction , 1989 .

[41]  Masaaki Ito,et al.  SYMCD—a REDUCE package for finding symmetries and conserved densities of systems of nonlinear evolution equations , 1994 .

[42]  Fritz Gesztesy,et al.  Spectral deformations and soliton equations , 1993 .

[43]  N. A. Kostov,et al.  Quasi-periodic solutions of the integrable dynamical systems related to Hill's equation , 1989 .

[44]  P. Olver.  Applications of Lie Groups to Differential Equations , 1986 .

[45]  Masaaki Ito,et al.  Symmetries and conservation laws of a coupled nonlinear wave equation , 1982 .

[46]  Ziming Li,et al.  Factoring systems of linear PDEs with finite-dimensional solution spaces , 2003, J. Symb. Comput..

[47]  J. Verdier,et al.  Revêtements exceptionnels et sommes de 4 nombres triangulaires , 1992 .

[48]  Fujio Kako,et al.  A reduce program for finding conserved densities of partial differential equations with uniform rank , 1985 .

[49]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[50]  松野 好雅,et al.  Bilinear transformation method , 1984 .

[51]  Fritz Gesztesy,et al.  On Picard potentials , 1995, Differential and Integral Equations.

[52]  Masaaki Ito,et al.  An Extension of Nonlinear Evolution Equations of the K-dV (mK-dV) Type to Higher Orders , 1980 .

[53]  Helge Holden,et al.  Soliton Equations and Their Algebro-Geometric Solutions: The AKNS Hierarchy , 2003 .

[54]  Allan P. Fordy,et al.  The He´non-Heiles system revisited , 1991 .

[55]  Fritz Gesztesy,et al.  Lamé potentials and the stationary (m)KdV hierarchy , 1995 .

[56]  Willy Hereman,et al.  Symbolic Computation of Conserved Densities for Systems of Nonlinear Evolution Equations , 1997, J. Symb. Comput..

[57]  Björn Birnir,et al.  Complex hill's equation and the complex periodic korteweg‐de vries equations , 1986 .

[58]  Alan C. Newell,et al.  The Hirota conditions , 1986 .

[59]  I. V. Cherednik,et al.  Differential equations for the Baker—Akhiezer functions of algebraic curves , 1978 .

[60]  P. Deift,et al.  Nonlinear wave equations and constrained harmonic motion , 1980, Proceedings of the National Academy of Sciences of the United States of America.

[61]  Willy Hereman,et al.  The Painleve Test for Nonlinear Ordinary and Partial Differential Equations , 1989 .

[62]  R. Hirota,et al.  Soliton solutions of a coupled Korteweg-de Vries equation , 1981 .

[63]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[64]  Gilbert Weinstein,et al.  Differential Equations and Mathematical Physics , 2000 .

[65]  C. S. Gardner,et al.  Korteweg‐de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion , 1968 .

[66]  A. O. Smirnov.  Finite-gap elliptic solutions of the KdV equation , 1994 .