Summary: We consider special cases of the second, third and fourth Painlevé equation given by (1) \(d^2\eta/d\xi^2= 2\eta^3+ \xi\eta\), (2) \(d^2\eta/d\xi^2= 2e^\xi\text{sinh }\eta\), (3) \(d^2\eta/d\xi^2= 3\eta^5+ 2\xi\eta^3+ ({1\over 4} \xi^2- \nu- {1\over 2})\eta\), respectively. We seek solutions \(\eta(\xi)\) satisfying the boundary condition (4) \(\eta(\infty)= 0\).
Equations (1)–(3) arise as similarity reductions of the modified Korteweg-de Vries, Sine-Gordon and derivative nonlinear Schrödinger equations, respectively, which are completely integrable soliton equations solvable by inverse scattering techniques. Solutions of equations (1)–(3), satisfying (4), are expressed in terms of the solutions of linear integral equations obtained from the inverse scattering formalism for the associated soliton equations. We discuss several properties of solutions of equations (1)–(3), in particular connection formulae, which can be derived using the integral equation representations.
For the entire collection see [Zbl 0846.00007].