On reconstruction problems in inverse problems of determining unknown coefficients for non-linear partial differential equations

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(1) We give the existence of outgoing eigen-function and its scattering amplitude for the two dimensional stationary wave equation with a friction term. We also prove in two dimensions that the friction coefficient is uniquely reconstructed from the scattering amplitude at a fixed low energy. (2) We study the inverse boundary value problem of determining a field-dependent coefficient for the non-linear wave equation in one space dimension. We prove that a linear part and a quadratic part of the field-dependent coefficient are uniquely reconstructed from the boundary measurements.