Archive for Rational Mechanics and Analysis

, Volume 73, Issue 1, pp 31–51

# 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.

## Preview

Unable to display preview. Download preview PDF.

### References

1. 1.
P. C. T. de Boer & G. S. S. Ludford, Spherical electric probe in a continuum gas. Plasma Phys. 17, 29–43 (1975).Google Scholar
2. 2.
E. L. Ince, Ordinary Differential Equations. New York: Dover 1944.Google Scholar
3. 3.
E. Hille, Ordinary Differential Equations in the Complex Domain. New York: Wiley 1976.Google Scholar
4. 4.
M. J. Ablowitz & H. Segur, Exact linearization of a Painlevé transcendent. Phys. Rev. Letters 38, 1103–1106 (1977).Google Scholar
5. 5.
G. N. Watson, A Treatise on the Theory of Bessel Functions. (2nd ed., Cambridge, 1944).Google Scholar
6. 6.
R. Rosales, The similarity solution for the Korteweg-de Vries equation and the related Painlevé transcendent. Preprint.Google Scholar
7. 7.
P. Boutroux, Recherches sur les transcendents de M. Painlevé et l'étude asymptotique des équations différentielles du second ordre. Ann. École Norm. Supér. (3) 30, 255–375 (1913);Google Scholar
8. 8.
P. Boutroux, Recherches sur les transcendents de M. Painlevé et l'étude asymptotique des équations différentielles du second ordre. Ann. École Norm. Supér. (3) 31, 99–159 (1914).Google Scholar
9. 9.
R. M. Miura, The Korteweg-de Vries equation: a survey of results. SIAM Rev. 18, 412–459 (1976).Google Scholar
10. 10.
E. C. Titchmarsh, Eigenfunctions Expansions (Part I, Oxford, 1962).Google Scholar

## 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