A connection between nonlinear evolution equations and ordinary differential equations of P-type. I

Abstract

We develop here two aspects of the connection between nonlinear partial differential equations solvable by inverse scattering transforms and nonlinear ordinary differential equations (ODE) of P-type (i.e., no movable critical points). The first is a proof that no solution of an ODE, obtained by solving a linear integral equation of a certain kind, can have any movable critical points. The second is an algorithm to test whether a given ODE satisfies necessary conditions to be of P-type. Often, the algorithm can be used to test whether or not a given nonlinear evolution equation may be completely integrable.

Meromorphy and topology of localized solutions in the Thomas–MHD model

“…In a more recent study, Passot and Pouquet (1986) have discussed the integrability of the model on the basis of the nature of the complex singularities, the so-called Painlevé test (Ablowitz et al 1980;Weiss et al 1983;Passot 1986). They pointed to the possibility that in subcases (e.g.…”

The Painlevé Approach to Nonlinear Ordinary Differential Equations

“…After the extension of the Fourier transform to nonlinear PDEs [57], called inverse spectral transform (IST), Ablowitz and Segur [2] noticed a link between those "IST-integrable PDEs" and the theory of Painlevé, link expressed by Ablowitz, Ramani and Segur [4] as the conjecture : "Every ODE obtained by an exact reduction of a nonlinear PDE solvable by the IST method has the Painlevé property". For more details, see the book by Ablowitz and Clarkson [1] and [84].…”

“…The first attempt is due to Hoyer in 1879 [64] with the system d dt 4) under the restriction that neither the determinant nor any of its first or second order diagonal minors vanishes; he even generalized the assumption to the Puiseux series…”

“…One must prove that the radius R of the punctured disk |x − x 0 | < R in which the series converges is nonzero. 4. As indicated by Gambier [56] p. 50, there is no need to expand the coefficients g(x) of the equation around x 0 .…”

“…We have retained the classical vocabulary ("famille" is used by Gambier, p. 38 of his thesis [56], "indices" is used by Gambier, Chazy [22] and Bureau [13]), rather than the one more recently introduced [3,4] ("branch", "resonances"). Indeed, "branch" has another meaning in classical analysis, where it denotes a determination of a multivalued application, which may create some confusion.…”