By Francesco Calogero, A. Degasperis
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Additional info for Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations
W ( ~the ) two-soliton expression (with parameters p , and p 2 resp. p I and p s ) ; and the process can then be continued (soliton rudder). -9). 3. -1) is the existence of an infinite sequence of local conservation laws. Each of these conservation laws yields, for the class of asymptotically (X-+& 0 0 ) vanishing solutions to which our consideration is confined, a conserved quantity expressed as the integral over all space of an appropriate nonlinear (polynomial) combination of u( x, t ) and its derivatives.
16, since this difficulty similarly arises in the context of nonlinear evolution equations. Our motivation for reviewing in this section some well-known facts concerning the solution of linear evolution equations by Fourier transform is because of the close similarity of this approach to the method of solution, via the spectral transform, of (certain classes of) nonlinear evolution equations, that constitutes our main interest. The correspondence applies also to the extensions that we have just mentioned (equations with t-dependent and linearly x-dependent coefficients).
1) contains two (real) parameters, f o and p . -1); and its relation to the spectral transform is given by (3). -1) one is considering. -4) corresponding to a(z ) = -z , the speed of the soliton is (8) o=4p2; thus all solitons of the KdV equation move to the right (the fact that this does not correspond to the behaviour of the waves in a canal, that should clearly be panty invariant, need not worry the reader; the KdV equation in . -4) is only appropriate to describe waves travelling in one direction; moreover it describes the behaviour of long waves in shallow canals as seen in a reference frame moving with an appropriately chosen constant speed).