The nonlinear partial differential equations considered here arise from the conservation laws of linear momentum and energy, and describe structural phase transitions (martensitic transformations) in one-dimensional shape memory alloys (SMA) with non-convex Landau-Ginzburg free energy potentials. This system is formally written as a nonlinear abstract Cauchy problem in an appropriate Hilbert space. A quasilinearization-based algorithm for parameter identification in this type of Cauchy problem is proposed. Sufficient conditions for the convergence of the algorithm are derived in terms of the regularity of the solutions with respect to the parameters. Numerical examples are presented in which the algorithm is applied to recover the non-physical parameters describing the free energy potential in SMA, from both exact and noisy data.