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Integral equations and exact solutions for the fourth Painlevé equation. (English) Zbl 0751.34002
Summary: We consider a special case of the fourth Painlevé equation given by (1) $d\sp 2\eta/d\xi\sp 2=3\eta\sp 5+2\xi\eta\sp 3+\left({1\over 4}\xi\sp 2- \nu-{1\over 2}\right)\eta$, with $\nu$ a parameter, and seek solutions $\eta(\xi;\nu)$ satisfying the boundary condition (2) $\eta(\infty)=0$. Equation (1) arises as a symmetry reduction of the derivative nonlinear Schrödinger (DNLS) equation, which is a completely integrable soliton equation solvable by inverse scattering techniques. Solutions of equation (1), satisfying (2), are expressed in terms of the solutions of linear integral equations obtained from the inverse scattering formalism for the DNLS equation. We obtain exact `bound state’ solutions of equation (1) for $\nu=n$, a positive integer, using the integral equation representation, which decay exponentially as $\xi\to\pm\infty$ and are the first example of such solutions for the Painlevé equations. Additionally, using Bäcklund transformations for the fourth Painlevé equation, we derive a nonlinear recurrence relation (commonly referred to as a Bäcklund transformation in the context of soliton equations) for equation (1) relating $\eta(\xi;\nu)$ and $\eta(\xi;\nu+1)$.

##### MSC:
 34M55 Painlevé and other special equations; classification, hierarchies 34A05 Methods of solution of ODE 34A25 Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.) 45A05 Linear integral equations 81U40 Inverse scattering problems (quantum theory)
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