The principal aim of the book is to give a comprehensive account of the variety of approaches to such an important and complex concept as Integrability. Dev- oping mathematical models, physicists often raise the following questions: whether the model obtained is integrable or close in some sense to an integrable one and whether it can be studied in depth analytically. In this book we have tried to c- ate a mathematical framework to address these issues, and we give descriptions of methods and review results. In the Introduction we give a historical account of the birth and development of the theory of integrable equations, focusing on the main issue of the book – the concept of integrability itself. A universal de nition of Integrability is proving to be elusive despite more than 40 years of its development. Often such notions as “- act solvability” or “regular behaviour” of solutions are associated with integrable systems. Unfortunately these notions do not lead to any rigorous mathematical d- inition. A constructive approach could be based upon the study of hidden and rich algebraic or analytic structures associated with integrable equations. The requi- ment of existence of elements of these structures could, in principle, be taken as a de nition for integrability. It is astonishing that the nal result is not sensitive to the choice of the structure taken; eventually we arrive at the same pattern of eq- tions.
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1 Symmetries of Differential Equations and the Problem of Integrability
2 Number Theory and the Symmetry Classification of Integrable Systems
Discretization and Integrability Discrete Spectral Symmetries
4 Symmetries of Spectral Problems
5 Normal Form and Solitons
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A.B. Shabat A.V. Mikhailov apply arbitrary asymptotic bilinear classiﬁcation of integrable coefﬁcients commutative conservation laws conserved density conserved quantities corresponding Darboux transformations deﬁned deﬁnition derivative differential equations differential polynomial discrete dispersion relation dynamical system equa evolution equations example existence expansion ﬁnd ﬁnite ﬁrst formal recursion operator formal series formula function Hamiltonian hierarchy higher order higher symmetries Hirota inﬁnite integrable equations integrable systems inverse scattering inverse scattering transform J.P. Wang KdV equation Korteweg–de Vries equation Lax pair leading order Lemma Lie algebra linear Math matrix method mKdV nonlinear nonlinear Schr¨odinger nontrivial normal form theory Novikov obtained ODEs Painlev´e property Painlev´e test parameter PDEs perturbed KdV equation Phys polynomial recursion operator reduction representation resonance S.P. Novikov satisﬁes scalar Schr¨odinger operator Sect solitary wave soliton symbolic symmetries symmetry approach Theorem tions V.E. Zakharov V.V. Sokolov vector ﬁeld wave equation