Breather-like solitons, rogue waves, quasi-periodic/chaotic states for the surface elevation of water waves

Abstract

In this paper, we investigate the surface elevation of water waves $$\eta $$ via an NLS equation under the coefficient constraint $$\omega ^{2}=gk$$ , where g is the gravity acceleration, k and $$\omega $$ are the carrier wave number and cyclic frequency, respectively. Through the Euler formula, we derive the breather-like solitons and rogue waves on the periodic background for $$\eta $$ . Breather-like solitons and rogue waves for $$\eta $$ are related with g and k. Effects of k on the breather-like solitons for $$\eta $$ are discussed analytically and graphically: Increase of k makes the decrease in the amplitudes of the breather-like solitons for $$\eta $$ . Amplitudes of the rogue waves for $$\eta $$ increase with k increasing in the position $$X=0$$ , where X denotes the spatial coordinate. Rogue waves on the periodic background for $$\eta $$ are more stable than breather-like solitons for $$\eta $$ . Multiple rogue waves of $$\eta $$ are generated from the initial rogue waves and periodic waves for $$\eta $$ with the noise perturbations in the baseband MI regime. Quasi-periodic state for $$\eta $$ is seen with $$0.523<k<0.544$$ . Chaotic state for $$\eta $$ appears with $$k<0.523$$ or $$k>0.544$$ .