Periodic and solitary wave solutions of the three-wave problem. A different approach (Q1121462)

For the so-called three-wave system (1) \[ (Q_ 1)_ t+c_ 1(Q_ 1)_ x)=ir_ 1Q^*_ 2 Q^*_ 3, \] \[ (Q_ 2)_ t+c_ 2(Q_ 2)_ x=ir_ 2Q^*_ 3 Q^*_ 1, \] \[ (Q_ 3)_ t+c_ 3(Q_ 3)_ x=ir_ 3Q^*_ 1 Q^*_ 2, \] which is exactly integrable by means of the inverse scattering transform (IST), a new method to obtain special explicit solutions is proposed. To that end, bilinear combinations of eigenfunctions of an auxiliary linear problem (on which the IST is based), that are analytic in the spectral parameter \(\lambda\), are assumed to be polynomials of a finite degree n. Next, it is demonstrated that the evolution of zeros \(\lambda_ 0(x,t)\) of those polynomials both in t and x is governed by a closed system of differential equations. For the case \(n=3\), equations governing the x- evolution are integrated explicitly, so that a system of algebraic equations for the zeros results. The method proposed does not require that the unknown functions \(Q_ j\) in equations (1) vanish at \(| x| =\infty\).