Perturbative symmetry approach

By A. V. Mikhailov, V. Novikov. 2002

186 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b309f70cfe1c390bcd1fba5821a64fe321dda44f

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In Atomic and Molecular Physics, and Optics

In Numerical Analysis

In Statistical and Nonlinear Physics

Citations in the corpus

On asymptotically equivalent shallow water wave equations (2003, background): “The b-family of equations (42) was also shown in [28] to admit the symmetry conditions necessary for integrability only in the cases b = 2 for CH and b = 3 for DP.”
Global weak solutions for the Novikov equation (2011, background): “The perturbative symmetry approach [22] yields necessary conditions for a PDE to admit infinitely many symmetries.”
Equations with Peakon Traveling Wave Solutions (2013, background): “For related results about them, we refer the reader to [2, 5–15, 17–19, 21] and the references therein.”
Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions (2009, unknown): “Mikhailov and Novikov [MN02] showed that among the b-equations only the cases b = 2 and b = 3 are completely integrable as Hamiltonian systems.”
Symmetries of Differential Equations and the Problem of Integrability (2009, background): “Recently the first results on classification of integrable nonlinear integro-differential equations have been obtained [ 36 , 37].”
Cuspons and Smooth Solitons of the Degasperis–Procesi Equation Under Inhomogeneous Boundary Condition (2007, methodology): “This family is proved to be integrable when b = 2, 3 by using symmetry approach [11].”
Well-posedness and global existence for the Novikov equation (2012, background): “The perturbative symmetry approach [19] yields necessary conditions for a PDE to admit infinitely many symmetries.”
Camassa–Holm, Korteweg–de Vries-5 and other asymptotically equivalent equations for shallow water waves (2003, background): “Such scenarios for a similar, but diGerent, equation have been studied in detail in Qian and Tang (2001). Another system which admits coexistence of regular solitary waves, peakons and cuspons has been studied in Grimshaw et al.”
Autonomous Ovsyannikov theorem and applications to nonlocal evolution equations and systems (2016, background): “In this paper Novikov investigated the question of integrability for CH type equations of the form (1− ∂2x)ut = P (u, ux, uxx, uxxx, · · · ), (1.7) where P is a polynomial of u and its x-derivatives.”
A note on the Cauchy problem of the Novikov equation (2013, background): “The perturbative symmetry approach [3] yields necessary conditions for a PDE to admit infinitely many symmetries.”