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

  • S. P. Hastings
  • J. B. McLeod
Article

Abstract

The differential equation considered is \(y'' - xy = y|y|^\alpha \). For general positive 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)
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 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.

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Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • S. P. Hastings
    • 1
    • 2
  • J. B. McLeod
    • 1
    • 2
  1. 1.Department of MathematicsState University of New YorkBuffalo
  2. 2.Wadham CollegeOxford

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