The purpose of this article is to survey recent developments of applied analysis being based upon the theory of nonlinear integrable systems. A historical background and an aim of this new methodology are discussed with several examples. First, an application to matrix eigenvalue problems is described. Two classes of Lax-type nonlinear dynamical systems are presented whose solutions isospectrally converge to diagonal matrices. Secondly, it is proved that Karmarkar's projective scaling trajectory in linear programming admits a Lax pair representation and is regarded as an integrable dynamical system. Finally, it is shown how completely integrable gradient systems appear in the information geometry, the differential geometry of parameter spaces of probability distributions.
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