Drifting breathers and Fermi–Pasta–Ulam paradox for water waves

Abstract

One physical mechanism that is responsible for the focusing of uni-directional water waves is the modulation instability (MI). This occurs when side-bands around the main frequency are excited either deterministically or randomly and subsequently grow exponentially. In physical space the periodically-perturbed wave group can reach significant wave amplifications and in the case of infinite modulation period even three times the initial amplitude of the regular Stokes wave train. These periodic wave groups propagate in deep-water with a group velocity half the wave's phase speed. In this experimental study, we investigate the dynamics of modulationally unstable wave groups that propagate with a velocity that is higher than the packet's group velocity in deep-water. It is shown that when this additional velocity to the wave group is marginal, a very good agreement with nonlinear Schrödinger (NLS) hydrodynamics is reached at all stage of propagation and the characteristic wave energy cascade remains symmetric and stationary. Otherwise, a significant deviation is observed only at the stage of large wave focusing of the isolated wave packet. In this case, the wave field experiences an almost perfect return to the initial conditions after the compression, a particular dynamics also known as the Fermi-Pasta-Ulam paradox.

Vector breathers in the Manakov system

“…Meanwhile, experimental observation of various aspects of the integrable scalar NLSE dynamics and statistics has been successfully performed in many different works, see, for example, Refs. 14, 17–24, 78, 79. In addition, the development of vector MI and vector dark rogue waves has been studied experimentally in a Manakov fiber system 40,80 .…”

“…The scalar NLSE breathers have been the focus of the studies for the past decades, revealing such fundamental building blocs of the breather dynamics as Kuznetsov, Akhmediev, Peregrine, and Tajiri‐Watanabe solutions; 5,11–13 as well as superregular and ghost interaction patterns, 14–17 and breather wave molecules 17 . All these scenarios of nonlinear wavefield evolution have been confirmed experimentally with optical, hydrodynamical, and plasma setups 14,17–24 . In addition, the breathers play an essential role in the formation of rational rogue waves, 25,26 modulation instability (MI) development, 11,27 and in the dynamics and statistics of complex nonlinear random wave states 28–31 …”

“…where m = 𝑚 1 = 𝑚 2 . One can see that the numerator (97) is exactly canceled at any 𝑥 and 𝑡, when 𝑞 12 = ℎ𝑞 13 , 𝑞 22 = ℎ𝑞 23 , where ℎ is an arbitrary constant. The latter happens only when 𝐶 0,1 = 0, 𝐶 0,2 = 0, so that ℎ = 𝐴 2 ∕𝐴 1 , that is, when both breathers are of type I and the vector two-breather solution is the trivial generalization of the scalar one, see transformation (40).…”

“…17 All these scenarios of nonlinear wavefield evolution have been confirmed experimentally with optical, hydrodynamical, and plasma setups. 14,[17][18][19][20][21][22][23][24] In addition, the breathers play an essential role in the formation of rational rogue waves, 25,26 modulation instability (MI) development, 11,27 and in the dynamics and statistics of complex nonlinear random wave states. [28][29][30][31] In this work, we consider the vector two-component extension of the NLSE-the Manakov system.…”

Separatrix crossing and symmetry breaking in NLSE-like systems due to forcing and damping

“…Thus we performed experiments in a wind-wave facility to corroborate our theoretical results. To accurately describe the nonlinear group velocity and asymmetries in the spectrum of water waves that arise due to the high steepness and spectral broadening [9], the higher-order version of the NLSE, the Dysthe equation [10] is required. Unlike the NLSE, there are no known analytic solutions to the Dysthe equation.…”