Classification of polynomial integrable systems of mixed scalar and vector evolution equations: I

We perform a classification of integrable systems of mixed scalar and vector evolution equations with respect to higher symmetries. We consider polynomial systems that are homogeneous under a suitable weighting of variables. This paper deals with the KdV weighting, the Burgers (or potential KdV or modified KdV) weighting, the Ibragimov–Shabat weighting and two unfamiliar weightings. The case of other weightings will be studied in a subsequent paper. Making an ansatz for undetermined coefficients and using a computer package for solving bilinear algebraic systems, we give the complete lists of second-order systems with a third-order or a fourth-order symmetry and third-order systems with a fifth-order symmetry. For all but a few systems in the lists, we show that the system (or, at least a subsystem of it) admits either a Lax representation or a linearizing transformation. A thorough comparison with recent work of Foursov and Olver is made.

[1]  J. Sanders,et al.  On the integrability of systems of second order evolution equations with two components , 2004 .

[2]  A. Das,et al.  Bosonic Reduction of Susy Generalized Harry Dym Equation , 2004, nlin/0404017.

[3]  Thomas Wolf Applications of CRACK in the Classification of Integrable Systems , 2003, nlin/0301032.

[4]  S. Sakovich,et al.  Integrability of Kersten–Krasil’shchik coupled KdV–mKdV equations: singularity analysis and Lax pair , 2002, nlin/0206046.

[5]  Peter H. van der Kamp On proving integrability , 2002 .

[6]  Thomas Wolf,et al.  Size Reduction and Partial Decoupling of Systems of Equations , 2002, J. Symb. Comput..

[7]  Thomas Wolf,et al.  Classification of integrable polynomial vector evolution equations , 2001, nlin/0611038.

[8]  Ying Wu,et al.  Algebraic structure of the Lie algebra so(2,1) for a quantized field in a vibrating cavity , 2001 .

[9]  Alexey Borisovich Shabat,et al.  Symmetry Approach to the Integrability Problem , 2000 .

[10]  V. E. Adler On the relation between multifield and multidimensional integrable equations , 2000, nlin/0011039.

[11]  P. Kersten,et al.  Complete integrability of the coupled KdV-mKdV system , 2000, nlin/0010041.

[12]  Mikhail V. Foursov,et al.  Classification of certain integrable coupled potential KdV and modified KdV-type equations , 2000 .

[13]  Mikhail V. Foursov,et al.  On integrable coupled Burgers-type equations , 2000 .

[14]  S. Sakovich,et al.  Symmetrically coupled higher-order nonlinear Schrödinger equations: singularity analysis and integrability , 2000, nlin/0006004.

[15]  Mikhail V. Foursov,et al.  On integrable coupled KdV-type systems , 2000 .

[16]  Mikhail V Foursov On integrable coupled KdV-type systems , 2000 .

[17]  京都大学数理解析研究所 Lie groups, geometric structures and differential equations : one hundred years after Sophus Lie , 2000 .

[18]  J. Sanders,et al.  Almost integrable evolution equations , 2002 .

[19]  M. Wadati,et al.  Complete integrability of derivative nonlinear Schrödinger-type equations , 1999, solv-int/9908006.

[20]  Willy Hereman,et al.  Algorithmic computation of generalized symmetries of nonlinear evolution and lattice equations , 1999, Adv. Comput. Math..

[21]  Xianguo Geng,et al.  A generalized Hirota–Satsuma coupled Korteweg–de Vries equation and Miura transformations , 1999 .

[22]  V. Sokolov,et al.  A symmetry test for quasilinear coupled systems , 1999, nlin/0611037.

[23]  Takayuki Tsuchida,et al.  Complete integrability of derivative nonlinear Schrödinger-type equations , 1999 .

[24]  S. Yu,et al.  Coupled KdV Equations of Hirota-Satsuma Type , 1999 .

[25]  J. Sanders,et al.  On Integrability of Systems of Evolution Equations , 2001 .

[26]  P. Olver,et al.  Non-abelian integrable systems of the derivative nonlinear Schrödinger type , 1998 .

[27]  Jan A. Sanders,et al.  ON THE INTEGRABILITY OF HOMOGENEOUS SCALAR EVOLUTION EQUATIONS , 1998 .

[28]  Jan A. Sanders,et al.  One symmetry does not imply integrability , 1998 .

[29]  M. Wadati,et al.  The Coupled Modified Korteweg-de Vries Equations , 1998, solv-int/9812003.

[30]  Peter J. Olver,et al.  Integrable Evolution Equations on Associative Algebras , 1998 .

[31]  W. Hereman,et al.  Algorithmic Computation of Higher-order Symmetries for Nonlinear Evolution and Lattice Equations , 1998, solv-int/9802004.

[32]  M. Gurses,et al.  Integrable coupled KdV systems , 1997, solv-int/9711015.

[33]  A. Karasu,et al.  Painlevé classification of coupled Korteweg-de Vries systems , 1997 .

[34]  S. I. Svinolupov,et al.  Vector-matrix generalizations of classical integrable equations , 1994 .

[35]  V. E. Adler Nonlinear superposition principle for the Jordan NLS equation , 1994 .

[36]  A. S. Fokas,et al.  Generalized conditional symmetries and exact solutions of non-integrable equations , 1994 .

[37]  Wen-Xiu Ma,et al.  A hierarchy of coupled Burgers systems possessing a hereditary structure , 1993 .

[38]  N. V. Ustinov,et al.  Darboux transforms, deep reductions and solitons , 1993 .

[39]  Xing-Biao Hu,et al.  Generalized Hirota's bilinear equations and their soliton solutions , 1993 .

[40]  W. Strampp,et al.  Multicomponent integrable reductions in the Kadomtsev–Petviashvilli hierarchy , 1993 .

[41]  Paul C. Bressloff,et al.  Low firing-rates in a compartmental model neuron , 1993 .

[42]  A. Yu. Zharkov,et al.  Computer Classification of the Integrable Coupled Kdv-like Systems with Unit Main Matrix , 1993, J. Symb. Comput..

[43]  S. I. Svinolupov Jordan algebras and integrable systems , 1993 .

[44]  Boris Konopelchenko,et al.  New reductions of the Kadomtsev–Petviashvili and two‐dimensional Toda lattice hierarchies via symmetry constraints , 1992 .

[45]  R. Hirota,et al.  Hierarchies of Coupled Soliton Equations. I , 1991 .

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

[47]  Francesco Calogero,et al.  Why Are Certain Nonlinear PDEs Both Widely Applicable and Integrable , 1991 .

[48]  Vladimir P. Gerdt,et al.  Computer Classification of Integrable Coupled KdV-like Systems , 1990, J. Symb. Comput..

[49]  V. Popkov,et al.  Completely integrable systems of brusselator type , 1989 .

[50]  B. Kupershmidt MODIFIED KORTEWEG-DE VRIES EQUATIONS ON EUCLIDEAN LIE ALGEBRAS , 1989 .

[51]  A. Fujimoto,et al.  Polynomial evolution equations of not normal type admitting nontrivial symmetries , 1989 .

[52]  S. I. Svinolupov On the analogues of the Burgers equation , 1989 .

[53]  A. Mikhailov,et al.  Extension of the module of invertible transformations. Classification of integrable systems , 1988 .

[54]  Athanassios S. Fokas,et al.  Symmetries and Integrability , 1987 .

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

[56]  W. Eckhaus,et al.  Nonlinear evolution equations, rescalings, model PDEs and their integrability: II , 1987 .

[57]  F. Calogero The evolution partial differential equation ut=uxxx+3(uxxu2 +3u2xu)+3uxu4 , 1987 .

[58]  F Calogero,et al.  Nonlinear evolution equations, rescalings, model PDES and their integrability: I , 1987 .

[59]  Solovey I. Razboinik Vector extensions of modified water wave equations , 1986 .

[60]  G. Quispel,et al.  Linear integral equations and multicomponent nonlinear integrable systems I , 1986 .

[61]  N. Bogolyubov,et al.  Complete integrability of the nonlinear ito and Benney-Kaup systems: Gradient algorithm and lax representation , 1986 .

[62]  J. Ward,et al.  factorization of operators on , 1986 .

[63]  A. Shabat,et al.  Integrability conditions for systems of two equations of the form ut=A(u)uxx+F(u, ux). I , 1985 .

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

[65]  V. V. Sokolov,et al.  Lie algebras and equations of Korteweg-de Vries type , 1985 .

[66]  A. Shabat,et al.  Integrability conditions for systems of two equations of the form ut = A(u)uxx + F(u, ux). II , 1985 .

[67]  J. Leon,et al.  A recursive generation of local higher‐order sine–Gordon equations and their Bäcklund transformation , 1984 .

[68]  M. Boiti,et al.  On a new hierarchy of Hamiltonian soliton equations , 1983 .

[69]  G. R. W. Quispel,et al.  On some linear integral equations generating solutions of nonlinear partial differential equations , 1983 .

[70]  T. Gui-zhang A new hierarchy of coupled degenerate hamiltonian equations , 1983 .

[71]  V. K. Mel’nikov On equations for wave interactions , 1983 .

[72]  B. Konopelchenko Nonlinear Transformations and Integrable Evolution Equations , 1983 .

[73]  B. Fuchssteiner The Lie Algebra Structure of Degenerate Hamiltonian and Bi-Hamiltonian Systems , 1982 .

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

[75]  G. Wilson,et al.  The affine lie algebra C(1)2 and an equation of Hirota and Satsuma , 1982 .

[76]  Allan P. Fordy,et al.  On the integrability of a system of coupled KdV equations , 1982 .

[77]  L. Alonso,et al.  Modified Hamiltonian systems and canonical transformations arising from the relationship between generalized Zakharov–Shabat and energy‐dependent Schrödinger operators , 1981 .

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

[79]  Allan P. Fordy,et al.  Factorization of operators.II , 1981 .

[80]  A. Fordy,et al.  Factorization of operators I. Miura transformations , 1980 .

[81]  David J. Kaup,et al.  On the Inverse Scattering Problem for Cubic Eigenvalue Problems of the Class ψxxx + 6Qψx + 6Rψ = λψ , 1980 .

[82]  Athanassios S. Fokas,et al.  A symmetry approach to exactly solvable evolution equations , 1980 .

[83]  N. Kh. Ibragimov,et al.  Infinite Lie-Beklund algebras , 1980 .

[84]  Alexey Borisovich Shabat,et al.  Evolutionary equations with nontrivial Lie - Bäcklund group , 1980 .

[85]  Vladimir E. Zakharov,et al.  The Inverse Scattering Method , 1980 .

[86]  Francesco Calogero,et al.  Nonlinear evolution equations solvable by the inverse spectral transform , 1978 .

[87]  Antonio Degasperis,et al.  Nonlinear evolution equations solvable by the inverse spectral transform.— II , 1977 .

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

[89]  N. Yajima,et al.  A Class of Exactly Solvable Nonlinear Evolution Equations , 1975 .

[90]  M. Wadati,et al.  On the Extension of Inverse Scattering Method , 1974 .

[91]  M. Ablowitz,et al.  Nonlinear-evolution equations of physical significance , 1973 .

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