The Goursat and Cauchy Problems for the Sine-Gordon Equation
David J. Kaup, Alan C. Newell
Source abstract
The equivalence of the methods for solving both the Goursat problem and the Cauchy problem for the sine-Gordon equation is discussed. Sufficient conditions on the initial data for which each problem may be solved by the inverse scattering transform are given. It is shown that the inversion procedure for the Goursat problem is slightly more general than the inversion procedure for the Cauchy problem. In most cases, they are equivalent. Thus, except for this slight difference, given Cauchy initial data, one may use the Goursat inversion, or vice versa. We also show how our previous analysis contains the method suggested later by Faddeev and Takhtajan, and also how our method is related to the Lax approach.
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