Traveling waves without mass resonance

Determine whether nontrivial traveling wave solutions of the three-component quadratic nonlinear Schrödinger system exist when the mass-resonance condition \(\sigma_1+\sigma_2=\sigma_3\) is not satisfied.

Background

The paper studies the three-component quadratic nonlinear Schrödinger system with nonzero dispersion parameters σ1,σ2,σ3\sigma_1,\sigma_2,\sigma_3. When σ1+σ2=σ3\sigma_1+\sigma_2=\sigma_3, the system is invariant under a Galilean transformation, so traveling wave solutions can be generated from standing waves. The cited results establish traveling waves for related two-component systems and for the three-component system under specified frequency and velocity conditions.

For the non-mass-resonant case, the Galilean invariance used to construct traveling waves is unavailable. The authors explicitly state that the existence of traveling wave solutions is not clear in this setting, leaving the question unresolved.

References

On the other hand, if the mass resonance condition {\rm (\ref{massres})} is not satisfied, then the existence of traveling wave solutions is not clear.

— On solitary wave solutions with two-frequency parameters to the three-component system of quadratic nonlinear Schrödinger equations  (2608.12983 - Hirayama et al., 13 Aug 2026) in Section 1, subsection “Previous works”