# A boundary value problem associated with the second painlevé transcendent and the Korteweg-de Vries equation

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## Abstract

The differential equation considered is \(y'' - xy = y|y|^\alpha \). For general positive It is shown that there is a unique such solution, and that it is, in a certain sense, the boundary between solutions which exist on the whole real line and solutions which, while tending to zero at plus infinity, blow up at a finite

*α*this equation arises in plasma physics, in work of De Boer & Ludford. For α=2, it yields similarity solutions to the well-known Korteweg-de Vries equation. Solutions are sought which satisfy the boundary conditions (1) y(∞)=0 (2)$$y{\text{(}}\infty {\text{)}} = {\text{0}}$$

(1)

$$y{\text{(}}x{\text{) \~( - }}\tfrac{{\text{1}}}{{\text{2}}}x{\text{)}}^{{{\text{1}} \mathord{\left/ {\vphantom {{\text{1}} \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} {\text{ as }}x \to - \infty $$

(2)

*x.*More precisely, any solution satisfying (1) is asymptotic at plus infinity to some multiple*kA i(x)*of Airy's function. We show that there is a unique k^{*}(α) such that when k=k^{*}(α) the condition (2) is also satisfied. If*0<k<k*^{*}, the solution exists for all*x*and tends to zero as x→-∞, while if*k>k*^{*}then the solution blows up at a finite*x.*For the special case α=2 the differential equation is classical, having been studied by Painlevé around the turn of the century. In this case, using an integral equation derived by inverse scattering techniques by Ablowitz & Segur, we are able to show that k^{*}=1, confirming previous numerical estimates.## Keywords

Boundary Condition Differential Equation Neural Network Integral Equation Complex System
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