A method for finding coefficients of a quasilinear hyperbolic equation

The inverse problem of finding the coefficients $q(s)$ and $p(s)$ in the equation $u_{tt}=a^2u_{xx}+q(u)u_t-p(u)u_x$ is investigated. As overdetermination required in the inverse setting, two additional conditions are set: a boundary condition and a condition with a fixed value of the timelike variable. An iteration method for solving the inverse problem is proposed based on an equivalent system of integral equations of the second kind. A uniqueness theorem and an existence theorem in a small domain are proved for the inverse problem to substantiate the convergence of the algorithm.