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It is known through the inverse scattering transform that certain nonlinear differential equations can be solved via linear integral equations. Here it is demonstrated ’’directly,’’ i.e., without the Jost-function formalism that the solution of the linear integral equation actually solves the nonlinear differential equation. In particular, this extends the scope of inverse scattering methods to ordinary differential equations which are found to be of Painlevé type. Some global properties of these nonlinear ODE’s are obtained rather easily by this approach.
Ablowitz et al. (Thu,) studied this question.