Perturbative symmetry approach

The aim of our paper is to formulate a perturbative version of the symmetry approach in the symbolic representation and to generalize it in order to make it suitable for the study of nonlocal and non-evolution equations. Our formalism is the development and incorporation of the perturbative approach of Zakharov and Schulman, the symbolic method of Sanders and Wang and the standard symmetry approach of Shabat et al. We apply our theory to describe integrable generalizations of the Benjamin-Ono type equations and to isolate integrable cases of the Camassa-Holm type equations.

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

[2]  P. Olver,et al.  Classification of Integrable One‐Component Systems on Associative Algebras , 2000 .

[3]  V. Zakharov,et al.  Integrability of Nonlinear Systems and Perturbation Theory , 1991 .

[4]  Darryl D. Holm,et al.  An integrable shallow water equation with peaked solitons. , 1993, Physical review letters.

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

[6]  Darryl D. Holm,et al.  A New Integrable Equation with Peakon Solutions , 2002, nlin/0205023.

[7]  A. Mikhailov,et al.  Towards classification of -dimensional integrable equations. Integrability conditions I , 1998 .

[8]  A. Mikhailov,et al.  Classification of Integrable Benjamin—Ono-Type Equations , 2003 .

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

[10]  W. Strampp,et al.  On the correspondence between symmetries and conservation laws of evolution equations , 1982 .

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

[12]  I. Gel'fand,et al.  ASYMPTOTIC BEHAVIOUR OF THE RESOLVENT OF STURM-LIOUVILLE EQUATIONS AND THE ALGEBRA OF THE KORTEWEG-DE VRIES EQUATIONS , 1975 .

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

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