Solution of the equations for nonlinear interaction of three damped waves

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Abstract
Three-wave interaction is analyzed in a coherent-wave description with assumption of different linear damping (or growth) of the individual waves. It is demonstrated that when two of the coefficients of dissipation are equal, the set of equations can be reduced to a single equivalent equation, which in the nonlinearly unstable case where one wave is undamped, asymptotically takes the form of an equation defining the third Painlevé transcendent. It is then possible to find an asymptotic expansion near the time of explosion. This solution is of principal interest since it indicates that the solution of the general three-wave system, where the waves experience mutually different dissipations, belongs to a higher class of functions, which reduces to Jacobian elliptic functions only in the case where all waves experience the same damping.