One-phase solutions to the three-wave equation

One-phase solutions to the three-wave equation are constructed using the monodromy matrix method. If we construct a monodromy matrix based on the original Lax pair, then the solutions will depend on only one variable. This is due to the fact that the spectral curve has genus 1 and the corresponding to the second variable Abelian integral of the second kind is a meromorphic function. Therefore, an auxiliary Lax pair is used to construct solutions, which corresponds to Abelian integrals that are different from meromorphic functions. Since the curve has genus 1, the arguments of the solution are linear combinations of both auxiliary and main variables. Examples in elliptic and hyperbolic functions are given.