Topological control of extreme waves

From optics to hydrodynamics, shock and rogue waves are widespread. Although they appear as distinct phenomena, transitions between extreme waves are allowed. However, these have never been experimentally observed because control strategies are still missing. We introduce the new concept of topological control based on the one-to-one correspondence between the number of wave packet oscillating phases and the genus of toroidal surfaces associated with the nonlinear Schrödinger equation solutions through Riemann theta functions. We demonstrate the concept experimentally by reporting observations of supervised transitions between waves with different genera. Considering the box problem in a focusing photorefractive medium, we tailor the time-dependent nonlinearity and dispersion to explore each region in the state diagram of the nonlinear wave propagation. Our result is the first realization of topological control of nonlinear waves. This new technique casts light on shock and rogue waves generation and can be extended to other nonlinear phenomena.


MAIN ISSUES
1. The genus. How can the authors guarantee that the structures they observe have certain genus? In other words, how much of the actual control over the wave "topology" do they have? The fact that the intensity profile they observe can be reasonably well fitted by the shape of a certain NLS breather solution (like in Fig 3 (Akhmediev) or Fig. 5 (Peregrine)) does not necessarily imply that the observed coherent structure is described by this particular NLS solution as the amplitude profile is only a "half" of the complex wave field. To be able to make the identification like this, the phase comparison is necessary along with the amplitude/intensity comparison. Can the authors compare the phase of the experimentally observed structures with that of the theoretical solution to support their claim?
2. Modulational instability. This issue is related to the first one. It is known that the breather structures naturally arise due to the development of modulational instability of a plane wave perturbed by a small noise (D. Agafontsev and V. Zakharov, Nonlinearity, 28 (2015) 2791). Thus, the coherent structures observed in the experiment reported in the manuscript could in principle arise as a result of the development of the noise induced modulational instability in the central part of the box. If the authors insist that the dynamics they observe are dominated by the "genuine" NLS box evolution, they should discuss the influence of the noise, which is inevitable in a physical experiment, and present the corresponding estimates supporting their claims. This issue is crucial, since if the effects the authors observe are dominated by modulational instability, there is no question of any "topological control".
3. Dissipation. Similar to modulational instability, inevitable effects of dissipation should be discussed and estimated. In fact, the dissipation can significantly affect the "local" genus of the breather structure, see S. Randoux et al. Phys. Rev. E 98 (2018) 022219 4. DSW evolution and the "topology" control. The wave topology control in the manuscript is verified against the simple formula (5) describing the motion of DSW boundaries (the so-called "breaking curves") separating the regions with genus 0 and 1, with the point of intersection being associated with the formation of a rogue wave (genus 2). For the comparison of the theoretical velocities (6) with experiment the authors refer to Fig 3.b) showing perfect agreement but I don't think they explain how the speeds are measured in the experiment. This is important as the breaking curves are not seen by eye on the experimental space-time diagram in Fig  3c. Furthermore, the result presented in Fig. 5 1b is misleading). However, the present manuscript implies that any solution with g>2 is not a rogue wave but a "soliton gas" (see Fig. 2). This statement is, again, misleading and, strictly speaking, incorrect. In any case, since the authors never explain what they mean by soliton gas it is difficult to meaningfully discuss this issue. At least short discussion should be present in the manuscript, in particular, because the concept of soliton gas is much less developed than that of a DSW or a rogue wave with just a handful of references following the seminal Zakharov's paper in JETP 1971. Bottom line: the formation of a soliton gas in the box problem at large evolution times is indeed predicted by the semi-classical theory but is it what is actually observed in the experiment? If yes, then the authors' claim should be supported by some quantitative arguments.

SMALLER ISSUES
1. There are some incomprehensible statements in the manuscript like: "DSWs regularize catastrophic discontinuities by mean of rapidly oscillating undular bores" (p.1, second paragraph), or "single phase DSWs are able to generate undular bores" (p. 2, 1 st paragraph). In fact, "DSW" and "undular bore" are two different names for the same phenomenon, the latter one being more often used in the fluids context.

2.
A very minor, "cosmetic" comment. In the phase diagram in Fig. 1, the sketch of a rogue wave looks more like a fundamental soliton. The presence of two (at least two) side lobes is a "signature" of a rogue wave.
Overall, I believe, the manuscript requires a major revision along the above lines, to be considered for publication in Nature Communications.
Reviewer #2: Remarks to the Author: The authors of this work used the connection between topology and the optical beam propagation in nonlinear regime for classification of processes in the beam evolution. This is an interesting approach in nonlinear optics which deserves to be presented to community. The work is reasonably well written, the approach is clear and well illustrated The paper can be published after the authors will take into account a few comments that are given below. 2. The following sentence is confusing: ` However, the latter equation is solvable by IST only when the number of degrees of freedom in the IST description is limited." It is well known that the NLSE describes the system with infinite number of degrees of freedom. Equivalently, IST description also has an infinite number of degrees of freedom. The number of eigenvalues including those responsible for dispersive radiation is always infinite. This point should be clarified.
3. The difference between `small dispersion' NLSE (1) and the ordinary NLSE is only in rescaling the variables \zeta and \xi. The parameter \epsilon can be eliminated completely from Eq.(1). The only consequence would be then using the `box' (2) of larger size and longer evolution scale.
There is no any point in stressing the use of SDNLSE instead of the normal NLSE. The equivalence of the two approaches should be explained in order to avoid confusing the potential reader. , thanks to the one-to-one correspondence between the number of wave packet oscillating phases and the genus of toroidal surfaces associated with the nonlinear Schroedinger equation solutions by the Riemann theta function.
The Authors claim to be the first ones to experimentally observe controlled transitions between extreme waves with different genera, varying from dispersive shock waves to Akhmediev Breathers, Peregrine soliton and Soliton gas.
They use a parametric time-dependent nonlinearity to shape the asymptotic wave genus. They consider the box problem in a focusing Kerr-like photorefractive medium and tailor timedependent propagation coefficients, to explore all the dynamic phases in the nonlinear wave propagation.
The paper contains original results, it is clear and very well written, and may be of interest to different scientific communities. Therefore I am in principle favorable to its publication in Nature Communications.
Nevertheless, a few points need a clarification in order to definitely validate and/or strengthen the conclusions.
The Authors should address the following major points: 1) The manuscript is based on a rather mathematical paper [31] centered on the one-dimensional focusing nonlinear Schroedinger equation (SDNLSE=small dispersion nonlinear Schroedinger equation, corresponding to eq. (1) in the submitted manuscript).
Here the propagation (longitudinal) coordinate, appearing in the first derivative, is called \zeta, while the transverse coordinate, appearing in the second derivative, is called \csi. A coefficient called \epsilon appears in the equation in front of the two derivatives (as a linear and quadratic term). In the original paper [31], this coefficient is "a free parameter defining the modulation scale", but what is its physical meaning?
The Authors call \epsilon "dispersion parameter". Mathematically, this may be equivalent, but considering the physical problem presented in the paper here I would rather call it diffraction parameter. Am I right or did I miss something? I think this point is really crucial and needs clarification.
2) In what sense the reported solutions of the NLSE are extreme or rogue waves?
As reported in literature (see for example the same ref [31] here): "Rogue waves represent the waves of unusually large amplitude |\psi|_m, whose appearance statistics deviates from the Gaussian distribution. The conventional "rogue wave criterion" is |\psi|_m / |\psi|_s > 2, where |\psi|_s is the significant wave height computed as the average wave height of the largest 1/3 of waves. For the random wave field with Gaussian statistics one has |\psi|_s = 2 |\psi|_0, where |\psi|_0 is the background (mean field) amplitude, leading to the criterion |\psi|_m ^2 > 8 |\psi|_0 ^2 ." Would it be possible for the Authors to make a statistical analysis to obtain the PDF of the intensity values (both experimental and numerical), and check which among these waves may be classified as extreme, by the usual methods? This part could be added as supplementary material in a separated file.
3) I have a fundamental problem in my understanding of the paper. Spatial solitons are waves that remain localized in the transverse coordinate during propagation, thanks to a balance between nonlinearity and diffraction. Here nonlinearity is changing during propagation, due to time dependent photorefractive effect. The present paper is exactly based on this nonlinearity change, responsible of the transitions between the different solution.
Looking at the experimental and numerical Figures, without the mathematical frame given by ref.
[31], I see an almost plane wave undergoing a phenomenon of modulational instability/filamentation/pattern formation. We thank the Referee for considering our manuscript attractive and promising. We report in the following our detailed answers to the issues that the Referee kindly highlighted and the list of significant changes we made to address all of them. We are trustful that our new results will convince the Referee to suggest acceptance of this deeply revised version of our work. To answer the Referee question, we analyzed phase evolution in the small waist regime, that is, in the Peregrine-like soliton generation, inspired by G. Xu et al. paper, cited above. Through interference between the crystal output and a constant signal, we measured the phase shift along the propagation and the transverse direction. These new experimental results profoundly improve our paper. We proved experimentally that the observed light propagation is in remarkable agreement with the one predicted by theory, and in particular that our topological control allows supervising the transition between genus-zero to genus-two, and therefore to design the resulting genus outcome by tailoring the time of detection. To illustrate these our new experimental observation, we change Fig. 5, removing results at W0=50µm and adding 4 new panels showing phase evolution in the transition from g=0 to g=2. Numerical simulations are reported as well. Action Taken: Page 3, line 32, LHS → We radically changed the third paragraph, regarding the description of Fig. 5, in order to address the modification to the related figure and to describe our experimental and numerical results on phase evolution. Fig. 5 → We removed intensity outline at W0=50µm and added 4 new panels showing phase evolution in the transition from g=0 to g=2, that is, in the Peregrine-like soliton generation, the first three experimentally, while the fourth numerically.  Fig. 1 reports the experimental outcomes, while simulations are in Suppl. Figs. 1a-e. MI generates transversal periodic waves, whereas DSWs occur in strongly nonlinear regimes and present fast non-periodic oscillation. From our further analysis, it turns out that MI affects light propagation only for very large beam waists, much larger than the ones analyzed in the main manuscript. This is detailed in Suppl. Fig. 1 and discussed in the Supplementary Information: the MI has a characteristic period much longer than the DSW and occurs in the central part of the box, in correspondance of the flat beam profile. Our new experiments and theoretical analysis clearly enables to discriminate MI dynamics and the box evolution. Reviewer #1: 4. DSW evolution and the "topology" control. The wave topology control in the manuscript is verified against the simple formula (5) describing the motion of DSW boundaries (the socalled "breaking curves") separating the regions with genus 0 and 1, with the point of intersection being associated with the formation of a rogue wave (genus 2). For the comparison of the theoretical velocities (6) with experiment the authors refer to Fig 3.b) showing perfect agreement but I don't think they explain how the speeds are measured in the experiment. This is important as the breaking curves are not seen by eye on the experimental space-time diagram in Fig 3c. Furthermore, the result presented in Fig. 5 looks very rough. I do not see how it can be compared with Fig. 4 to support the topology control claim. Authors: We thank the Referee for such a comment and apologize for having been unprecise. For a fixed instant t1, the theoretical shock velocity v expressed in Eq. (6) is proportional to the width Δx=W0*t1/t0 of the plateau between the two DSW in Figs. 2a,3c. Indeed, Δx=2*t1*v. We measured Δx/2 at t1=(30±2)s varying the initial power P, so we found v1=v/v0, with v0=L/t1 and L=2.5mm the crystal length. We added this information in the main text, in the labels and the caption of Fig. 3b. About the comparison between Fig. 4a and Fig. 5b (now changed in Fig. 5a), we agree that the former Fig. 4a did not represent appropriately the output detected at small waist regime, whereas we found an excellent agreement with numerical results at W0=10µm. Now we can see in both graphics the absence of shock, and the generation of two Peregrine solitons. Action Taken 1b is misleading). However, the present manuscript implies that any solution with g>2 is not a rogue wave but a "soliton gas" (see Fig. 2 Regarding the first issue, we agree with the Referee: in giving at-a-glance picture of the topological signatures of the generated RWs we lacked precision. In order to fix it, we changed Fig. 1b, leaving g=2 when referred to the PS (we also changed the RW profile with the analytical PS outline, so the correspondence now is exact), but wrote g>>2 for SG, and in the caption we changed g=2 with g ∼ 2.

RESPONSE TO THE REFEREES' REPORTS
Also in the introductory part of the paper, we changed g=2 in g ∼ 2 when defining RWs, and changed g>2 in g>>2 every time we talk about SGs. Concerning the SG definition, corrected in g>>2, we followed the treatise in [40]. G. A. El et al. associate the long-time asymptotic solution ψ with a "breather gas" and numerically observe the presence of higher-order RWs with maximum height 4 < |ψM | < 5 in the regions with g ≥ 4. Their numerical simulations suggest that the pattern of the ξ −ζ plane (splitting into the regions of different genera) persists as ζ increases, and therefore g ∼ ζ asymptotically. This leads to a disordered finitedensity soliton ensemble rather than a well-ordered modulated soliton lattice. In the KdV theory, the thermodynamic type infinite-genus limit of finite-band potentials leads to the kinetic description of a SG [36], but such a treatise is still missing for the NLSE box problem, even in [40]. In our experiments, we see the disordered finite-density soliton ensemble, but the long-time evolution does not allow further analysis, because for t>>τ the nonlinearity saturates, hence we loose integrability. These considerations are reported in Supplementary Information.  2, 1st paragraph). In fact, "DSW" and "undular bore" are two different names for the same phenomenon, the latter one being more often used in the fluids context. Authors: We see the Referee point, that is, there is no distinction between the definition of DSWs and undular bores, so we changed the sentences reporting both the definitions together. Action Taken: Page 1, line 13, LHS → We changed "DSWs regularize catastrophic discontinuities by mean of rapidly oscillating undular bores" in "DSWs regularize catastrophic discontinuities by mean of rapidly oscillations". Page 2, line 13, LHS → We changed "These are single-phase DSWs (g = 1), able to generate undular bores" in "These wave trains are single-phase DSWs (g = 1)".