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Semiclassical Soliton Ensembles for the Three-Wave Resonant Interaction Equations

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Abstract

The three-wave resonant interaction equations are a non-dispersive system of partial differential equations with quadratic coupling describing the time evolution of the complex amplitudes of three resonant wave modes. Collisions of wave packets induce energy transfer between different modes via pumping and decay. We analyze the collision of two or three packets in the semiclassical limit by applying the inverse-scattering transform. Using WKB analysis, we construct an associated semiclassical soliton ensemble, a family of reflectionless solutions defined through their scattering data, intended to accurately approximate the initial data in the semiclassical limit. The map from the initial packets to the soliton ensemble is explicit and amenable to asymptotic and numerical analysis. Plots of the soliton ensembles indicate the space–time plane is partitioned into regions containing either quiescent, slowly varying, or rapidly oscillatory waves. This behavior resembles the well-known generation of dispersive shock waves in equations such as the Korteweg–de Vries and nonlinear Schrödinger equations, although the physical mechanism must be different in the absence of dispersion.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, USA

    R. J. Buckingham

  2. Department of Mathematics, University of Arizona, Tucson, AZ, USA

    R. M. Jenkins

  3. Department of Mathematics, University of Michigan, Ann Arbor, MI, USA

    P. D. Miller

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  1. R. J. Buckingham
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  2. R. M. Jenkins
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  3. P. D. Miller
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Correspondence to R. J. Buckingham.

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Communicated by P. Deift

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Buckingham, R.J., Jenkins, R.M. & Miller, P.D. Semiclassical Soliton Ensembles for the Three-Wave Resonant Interaction Equations. Commun. Math. Phys. 354, 1015–1100 (2017). https://doi.org/10.1007/s00220-017-2897-7

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