A. Inverse Scattering Method for the Initial Value Problem of the Nonlinear Equation of Evolution

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Abstract
The inverse spectral problem is outlined exclusively from the viewpoint of applying it to the initial value problem of the nonlinear equation of evolution. Lax's conjecture on the integral of the equation and the spectrum of the associated linear operator is introduced as the basis of the whole discussion. The interrelation between the measure (or weight) function in the completeness relationship of the operator and the corresponding scattering operator is examined. The importance of the discrete spectrum is stressed especially in connection with the soliton. The Gelfand-Levitan equation and Zakharov-Shabat's method of analytic continuation are presented and the time development of the measure function is examined. All discussions are performed on the example of the Kortweg-de Vries equation. An example of the initial value problem for this equation is solved.