Dam break problem for the focusing nonlinear Schrödinger equation and the generation of rogue waves

Abstract

Abstract. We propose a novel, analytically tractable, scenario of the rogue wave formation in the framework of the small-dispersion focusing nonlinear Schrödinger (NLS) equation with the initial condition in the form of a rectangular barrier (a "box"). We use the Whitham modulation theory combined with the nonlinear steepest descent for the semi-classical inverse scattering transform, to describe the evolution and interaction of two counter-propagating nonlinear wave trains -the dispersive dam break flows -generated in the NLS box problem. We show that the interaction dynamics results in the emergence of modulated largeamplitude quasi-periodic breather lattices whose amplitude profiles are closely approximated by the Akhmediev and Peregrine breathers within certain spacetime domain. Our semi-classical analytical results are shown to be in excellent agreement with the results of direct numerical simulations of the small-dispersion focusing NLS equation.

Experimental observation of the emergence of Peregrine-like events in focusing dam break flows

“…This phenomenon provides a new semi-classical interpretation of that has been previously described in the spatial domain as a nonlinear Fresnel diffraction [20]. This box problem (i.e., an initial square profile) then gives rise to two such counter-propagating modulation dynamics, whose interaction may turn into a cluster of breathers [19]. In the present contribution, we confirm theoretical predictions of Refs.…”

“…The second one is ruled by DSWs that satisfy the so-called nonlinear shallow water equations [17]. However, recent theoretical works have stressed that some similar characteristics with the DSW may appear in the regime of focusing nonlinearity with weak dispersion, thus leading to the emergence of dispersive dam break flows in the NLSE box problem [18,19]. In this new scenario, the emergence of a nonlinear wave train regularizes an initial sharp transition between the uniform plane wave and the zero-intensity background.…”

Rogue waves in multiphase solutions of the focusing nonlinear Schrödinger equation

“…Indeed, as was suggested in [13] and explicitly demonstrated in [29], the evolution of generic (including random) initial conditions leads to the formation of complex coherent nonlinear wave structures that are locally well approximated by modulated finite-band solutions ψg(x, t) defined by the spectral branch points α j , j = 0, 1, . .…”

“…It is also evident that in this case the mean |ψ 2 2 | is close to one. A slow evolution of a genus 2 breather lattice from a "rogue wave free" configuration to the configuration exhibiting rogue waves was shown in [13] to naturally occur in the semi-classical fNLS with initial data in the form of a rectangular barrier (the "box" problem). Figs.…”

“…The prominent feature of this scenario is the formation of expanding chains of Peregrine breathers right beyond the gradient catastrophe point. In the second scenario, the highamplitude breather lattices are formed as a result of interaction of dispersive shock waves (dam break flows) generated in the fNLS evolution of a rectangular initial potential [13].…”