Equilibrium states of class-I Bragg resonant wave system

• The class-I Bragg resonant waves are solved analytically.
• Multiple equilibrium-state resonant wave systems with time-independent wave spectrum are found.
• Bifurcations with respect to wave propagation angle, water depth, bottom slope and nonlinearity are found.

In this paper, the class-I Bragg resonant waves are investigated in the case that a primary surface wave propagates obliquely over the bottom with ripples distributed in a very large area. Two kinds of equilibrium-state resonant wave systems with time-independent wave spectrum are found. In all cases, the primary and resonant wave components contain most of the wave energies. For the first kind, the primary and resonant wave components have the same amplitude. However, for the second kind, they contain different wave energies. Especially, the bifurcations of the equilibrium-state resonant waves with respect to the wave propagation angle, the water depth, bottom slope and nonlinearity are found for the first time. To the best of our knowledge, these two kinds of equilibrium-state class-I Bragg resonant waves and especially the bifurcations have never been reported. All of these might deepen and enrich our understanding about the Bragg wave resonance. Mathematically, unlike previous analytic approaches which regard the considered problem as an initial-value one, we search for the unknown equilibrium-state resonant waves from the viewpoint of boundary-value problem, using an analytic method that has nothing to do with small/large physical parameters at all.