Solution of a nonlinear singular integral equation with quadratic nonlinearity

Abstract

Using methods of the theory of boundary-value problems for analytic functions, we prove a theorem on the existence of solutions of the equation

$$u^2 \left( t \right) + \left( {\frac{1}{\pi }\int\limits_{ - \infty }^\infty {\frac{{u\left( \tau \right)}}{{\tau - t}}d\tau } } \right)^2 = A^2 \left( t \right)$$

and determine the general form of a solution by using zeros of an entire function A ² (z) of exponential type.