Abstract
The dynamics of focussing of extended patches of nonlinear capillary–gravity waves within the primitive fluid dynamic equations is presented. It is found that, when the envelope has certain properties, the patch focusses initially in accordance to predictions from nonlinear Schrödinger equation, and focussing can concentrate energy to the vicinity of a point or a curve on the fluid surface. After initial focussing, other effects dominate and the patch breaks up into a complex set of localised structures–lumps and breathers–plus dispersive radiation. We perform simulations both in the inviscid regime and for small viscosities. Lastly we discuss throughout the similarities and differences between the dynamics of ripple patches and self-focussing light beams.
Highlights
IntroductionThe nonlinear Schrödinger equation (NLS) is a universal model that appears in many contexts in science, including nonlinear optics, plasma physics, and hydrodynamics [1] and provides a canonical description for the evolution of the envelope of dispersive and/or diffracting quasi-monochromatic, plane, weakly nonlinear waves
We discuss throughout the similarities and differences between the dynamics of ripple patches and self-focussing light beams
The nonlinear Schrödinger equation (NLS) is a universal model that appears in many contexts in science, including nonlinear optics, plasma physics, and hydrodynamics [1] and provides a canonical description for the evolution of the envelope of dispersive and/or diffracting quasi-monochromatic, plane, weakly nonlinear waves
Summary
The nonlinear Schrödinger equation (NLS) is a universal model that appears in many contexts in science, including nonlinear optics, plasma physics, and hydrodynamics [1] and provides a canonical description for the evolution of the envelope of dispersive and/or diffracting quasi-monochromatic, plane, weakly nonlinear waves. >0.6 by using the localised ground state of the focussing NLS (called the Townes profile [9]) to modulate a carrier wave with minimum phase speed [3,10], and occur in a variety of models where they coexist with unstable elevation waves [3,11] These smallamplitude lumps are known to be unstable, which can be seen by either recalling that the Townes Profile is unstable within the focussing NLS or by direct computation on the full equations. Certain larger-amplitude lumps occurring on the same branch of solution beyond the regime of validity of NLS have been constructed numerically, and are stable [6,10] These lumps coexist with stable breathers which appear in time dependent computations. In all water-wave-nonlinear optics analogies, one must keep in mind that while NLS is an extremely accurate model in optics, its range of applicability in water waves is more restricted—in particular by the limited range of amplitudes and modulation scales that are physically achievable
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