Directional Soliton and Breather Beams

Solitons and breathers are nonlinear modes that exist in a wide range of physical systems. They are fundamental solutions of a number of nonlinear wave evolution equations, including the uni-directional nonlinear Schr\"odinger equation (NLSE). We report the observation of slanted solitons and breathers propagating at an angle with respect to the direction of propagation of the wave field. As the coherence is diagonal, the scale in the crest direction becomes finite, consequently, a beam dynamics forms. Spatio-temporal measurements of the water surface elevation are obtained by stereo-reconstructing the positions of the floating markers placed on a regular lattice and recorded with two synchronized high-speed cameras. Experimental results, based on the predictions obtained from the (2D+1) hyperbolic NLSE equation, are in excellent agreement with the theory. Our study proves the existence of such unique and coherent wave packets and has serious implications for practical applications in optical sciences and physical oceanography. Moreover, unstable wave fields in this geometry may explain the formation of directional large amplitude rogue waves with a finite crest length within a wide range of nonlinear dispersive media, such as Bose-Einstein condensates, plasma, hydrodynamics and optics.


INTRODUCTION
Ocean waves are complex two-dimensional dynamical structures that cannot be easily modelled in their full complexity. Variations of depth, wind strength, wave breaking, randomness and large amplitude waves add tremendously to this complexity [1]. Despite these complications, the research on water waves is important and significant advances have been made so far [2]. The progress is mainly due to simplified models that are used to analyse their dynamics [3]. Moreover, validity of these models can be confirmed in down-scaled experiments in water wave facilities that do exist in many research laboratories around the world. These experiments are crucially significant to build our understanding of larger scaled oceanic waves. Evolution equations and their solutions are essential for water wave modeling, while computerized equipment is a key for their accurate generation.
One of the essential complications in ocean wave dynamics is the unavoidable existence of two horizontal spatial coordinates. Directional behaviors of the surface waves in nature are of principal importance for practical applications ranging from wave forecast through modeling air-sea interactions and to, most importantly, environmental and optical sciences.
In a simplified way, such wave field consists of many waves crossing each other at various angles implying at a linear level that the water surface is a mere interference of shortand long-crested waves coming from different directions [4][5][6]. Here, we leave aside these complexities. Instead, we start with a simple question: what does the second coordinate add to the dynamics when the waves are mostly uni-directional? This simple question must be answered before considering more complicated cases.
Indeed, uni-directional nonlinear wave dynamics on the water surface in deep-water, that is, assuming that the water depth is significantly larger than the waves' wavelength, can be described by the nonlinear Schrödinger equation (NLSE) that takes into account dispersion and nonlinearity [7]. Being an integrable evolution equation, it allows for the study of particular and localized coherent wave patterns, such as solitons and breathers [8][9][10]. The latter are of major relevance to study the fundamental wave dynamics in nonlinear dispersive media with a wide range of applications [11][12][13]. While the NLSE has been formulated for planar waves and wave packets propagating in the same direction as the underlying carrier waves, there is also a generalization of the framework, the so-called directional NLSE, which allows the envelope and homogeneous planar carrier wave to propagate at an angle to each other. This possibility adds novel and unexpected features to well-known nonlinear and coherent wave propagation motions as we examine in this work. Unfortunately, from the theoretical perspective, the directional deep-water NLSE is not integrable. As a consequence, these nontrivial nonlinear solutions are not easy to identify. Early attempts to generate some nonlinear states were based on symmetry considerations [14]. It has been shown [15,16] that each uni-directional solution of the NLSE has a family counterpart solutions for which the packet beam propagates obliquely to the short-crested carrier wave.
These type of wave processes are directly relevant in oceanography [17][18][19]. However, taking into account many areas in physics for which the NLSE is the fundamental governing equation, our ideas can be bluntly expanded to these fields such as Bose-Einstein condensates, plasma and optics [20][21][22][23][24].
In the present study, we report an experimental framework and observations of hydrodynamic diagonal solitons and breathers in a deep-water wave basin. Our results confirm and prove the existence of such unique and coherent beams of quasi-one-dimensional and short-crested wave group in a nonlinear dispersive medium.

RESULTS
In order to illustrate such type of universal and directional wave packet, in Fig. 1 we show an example of the dimensional shape of an envelope soliton and a Peregrine breather, as parametrized in [25,26], with amplitude a = 0.02 m propagating at zero diagonal angle which consists of 32 sections. Each plunger has a width of 32 cm. More details on the methodology adopted for the data acquisition can be found in [27]. A picture and a sketch of the experimental set-up and the co-ordinate system adopted are depicted in Fig. 2.
We emphasize that due to the significant size of the digitally collected data, the stereoreconstruction, that includes an interpolation process, is very challenging [27].
The measured evolution of a sech-type envelope soliton [25] as well as the results obtained from the (2D+1) NLSE prediction are illustrated in Fig. 3.  our observations. This discovery proves that localized, short-crested and directional water waves, particularly rogue waves, can be also described by a nonlinear framework.
The Peregrine solution can be considered as the limit of the Akhmediev breather, the analytical and deterministic modulation instability model, when the period of the modulation tends to infinity [28,29]. Then, maxima of the periodic modulated structure are well separated and only one localized peak remains at the center. The temporal evolution of a slanted Peregrine solution measured in the experiment is shown and compared to the (2D+1) NLSE predicted wave curves in Fig. 4d and f, respectively. Again, comparison of the corresponding panel pair in Fig. 4 shows a remarkably good agreement between the measurements and the directional NLSE theory. The measured and calculated focusing dis- wave packets studied here is their finite crest length. The latter can be observed by simply watching the ocean waves. The crest length and thus, the transverse size of the waves is always limited. Now, it turns out that coherent waves with finite crest length might be a consequence of nonlinear beam dynamics. This is an important observation especially for the breather solutions, as this suggests that the nonlinearity is also a possible underlying mechanism for the actual finite-length-crested rogue wave events complementing the linear superposition and interference arguments as has been generally suggested. Further studies using a fully nonlinear hydrodynamic approach [30,31] may increase the accuracy of the description. These will characterize the ranges of accuracy of the approach, however, will not add anything substantial to the concept. Serious implications of such wave packets in oceanography is an important aspect of our results. This includes directional wave modeling, swell propagation and diffraction as well as remote sensing of waves to name few.
Moreover, investigating wave breaking processes [32,33] and prediction [34,35] of extreme directional waves is also crucial for future application purposes. Since the effect that can be explained by means of a general and universal theory for two-dimensional nonlinear wave fields in dispersive environments, its further extensions can stimulate analogous theoretical, numerical and experimental studies in two-dimensional optical surfaces, multi-dimensional plasmas, among other relevant physical media, elevating our level of understanding of these phenomena.
For a wave envelope ψ(x, y, t) with carrier wavenumber k along the x-direction and carrier where λ = ω 8k 2 , γ = ωk 2 2 and g denotes the gravitational acceleration. At the leading order, it is known that ∂ψ ∂t c g ∂ψ ∂x . This relation can be used to write the equation to express the wave packet propagation in space along the spatial x co-ordinate to give a time-(2D+1) NLSE [9] i ∂ψ ∂x As the measurements are made at fixed positions along the flume, this equation can be used for experimental investigations. Now, we introduce the following transformation with variable parameter ϑ that sets a special relation between time, t, and the spatial coordinate y. Then, the evolution equation for the new wave function, ψ(x, T ), reads with C g = c g / cos ϑ, Λ = λ(1 − 3 sin 2 ϑ)/c 3 g and Γ = γ/c g . When the angle |ϑ| < arcsin (1/3) 35.26 • [15,16], Eq. (4) is the standard (1D+1) focusing NLS equation that is known to be integrable [8,25,36]. When θ = 0 the envelope and the phase travel at a finite angle to each other.
From an experimental point of view, the boundary condition for the surface elevation η(x, y, t) at the wave maker, placed at x = 0, can be described, to the leading-order, by the following expression: η(x = 0, y, t) = 1 2 [ψ(0, T ) exp (−iωt) + c.c.] , where ψ(0, T ) is the desired solution of the one dimensional NLSE in Eq. (4) and T is given by Eq. (3). Eq. (5) is used for driving the wave maker.