By Boris G. Konopelchenko

ISBN-10: 3540175679

ISBN-13: 9783540175674

ISBN-10: 3540474595

ISBN-13: 9783540474593

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**Extra info for Nonlinear Integrable Equations: Recursion Operators, Group Theoretical and Hamiltonian Structures of Soliton Equations**

**Example text**

For the sine-Gordon equation the spectral parameter ~ is in essence a two-dimensional vector. For the massive Thirring model a spectral parameter is a two-component spinor. The possibility to formulate the corresponding spectral problems and the whole IST method in the manifestly relativistically-invariant form has been demonstrated in /556/. The general structure of the symmetry groups of completely integrable equations has been discussed in /179/. Completely integrable equations are canonically equivalent to linear equations.

Such transformations take out the class of bona fide potentials and do not conserve the restrictions on the scattering matrix mentioned above. 17) should be verified by direct calculations. 17) is, of course, rather formal. We will fellow the same line for other spectral problems tOO. 17) is the integro-differential relation between P and from the rigorous point of view. 17) as the transformation following to the already established terminology. p' Thus we have constructed the infinite-dimensional BC group for the bundle (3.

15) ( /~ = 0 , 1 , 2 , . ° ) where ~ denotes a t e n s o r p r o d u c t and O denotes a p o i n t p r o d u c t . /~6~)~ an d /~za2- are usual classical F-matrices on the whole axis /388, 389/. The matrix ~{~) is of the projection type and the third term cannot be transformed into the 56 ~ - t y p e terms. For /~ -- 0 #{o)= O. The existence of the third term is the main feature of the higher Poisson brackets 2 , }~ • Consider the simplest case ~ -- 2 as illustration. 17). So the algebra of the scattering data is defined by the classical ~ - m a t r i x only.

### Nonlinear Integrable Equations: Recursion Operators, Group Theoretical and Hamiltonian Structures of Soliton Equations by Boris G. Konopelchenko

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