Rogue Waves and Large Deviations in Deep

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APPLIED MATHEMATICS 1/2 2 2 X ikn x ˆ ˆ −kn /(2∆ ) 1 ? 2 ? u0(x) = e (2Cn ) θn , Cn = Ae , [2] I (z(λ)) = ku (λ)k , z(λ) = F (u(T , u (λ))). [9] T 2 0 C 0 n∈Z ? where u0 (λ) denotes the minimizer obtained in Eq. 8. It is easy where kn = 2πn/L, θn are complex Gaussian variables with ¯ ¯ ¯ to see from Eqs. 6 and 8 that ST (λ) is the Legendre transform of mean zero and covariance Eθn θm = δm,n , Eθn θm = Eθn θm = 0. IT (z) since: This guarantees that u0(x) is a Gaussian field with mean zero and 0 0 P ikn (x−x ) (u0(x)¯u0(x )) = 2 e Cˆn . To make contact with 1 2 E n∈Z ST (λ) = sup(λz − IT (z)) = sup λz − inf ku0kC , u ∈Ω(z) the observational data, the amplitude A and the width ∆ in Eq. z∈R z∈R 2 0 ˆ 2 are picked so that Cn has the same height and area as the [10] JONSWAP spectrum (31, 32); see Supporting Information for details. Because the initial data for Eq. 1 are random, so is the solution 3. Results at time t > 0, and our aim is to compute We considered two sets of parameters. In set 1, we took −5 −2 A = 5.4 · 10 k0 and ∆ = 0.19k0. Converting back into dimen- PT (z) ≡ P F (u(T )) ≥ z , [3] −1 sional units by using k0 = 36 m consistent with the JONSWAP where P denotes probability over the initial data and F is a spectrum (31, 32), this implies a significant wave height u T > 0 p scalar functional depending on at time . Even though Hs = 4 C (0) = 3.3 m classified as a “rough sea” (33). It also our method is applicable to more general observables, here we p yields a Benjamin–Feir index (BFI) = 2 2C (0)/∆ = 0.34, (32, will focus on 34), meaning that the modulational instability of a typical ini- F (u(T )) = max |u(T , x)|. [4] tial condition is of medium intensity. In set 2, we took A = 3.4 · x∈[0,L] −4 −2 10 k0 and ∆ = 0.19k0, for which Hs = 8.2 m is that of a “high sea” and the BFI is 0.85, meaning that the modulational instabil- 2. LDT Approach ity of a typical initial condition is stronger. A brute-force approach to calculate Eq. 3 is Monte Carlo sam- Fig. 1, Upper shows the time evolution of |u(t, x)| start- pling: Generate random initial conditions u0(x) by picking ran- ing from an initial condition from set 1 optimized so that dom θn ’s in Eq. 2, evolve each of these u0(x) deterministically via maxx |u(T , x)| = 8 m at T = 20 min. For comparison, Fig. 1, Eq. 1 up to time t = T to get u(T , x), and count the proportion Lower shows |u(t, x)| for a typical initial condition drawn from its that fulfill F (u(T )) ≥ z. While this method is simple, and will Gaussian distribution. To illustrate what is special about the ini- be used below as benchmark, it loses efficiency when z is large, tial conditions identified by our optimization procedure, in Fig. 2, which is precisely the regime of interest for the tails of the distri- we show snapshots of the surface elevation η(t, x) at three differ- bution of F (u(T )). In that regime, a more efficient approach is ent times, t = 0, 10, 20 min (black lines), using the constraint that to rely on results from LDT which assert that Eq. 3 can be esti- maxx |u(T , x)| ≥ 4.8 m at T = 20 min. Additionally, we aver- mated by identifying the most likely initial condition that is con- age all Monte Carlo samples achieving maxx |u(t, x)| ≥ 4.8 m, sistent with F (u(T )) ≥ z. To see how this result comes about, translated to the origin. Snapshots of this mean configuration are recall that the probability density of u0 is formally proportional shown in Fig. 2 (blue lines). They agree well with those of the 1 2 2 to exp(− 2 ku0kC ), where ku0kC is given by optimized solution (black lines). The one SD spread around the 2 Z L mean Monte Carlo realization (light blue) is reasonably small, 2 X |aˆn | 1 −ik x ku k = , aˆ = e n u (x)dx. [5] especially around the rogue wave at final time. This indicates 0 C ˆ n 0 Cn L 0 that the event maxx |u(T , x)| ≥ 4.8 m is indeed realized with n∈Z probability close to 1 by starting from the most likely initial To calculate Eq. 3 we should integrate this density over the set condition consistent with this event, as predicted by LDT. The Ω(z) = {u0 : F (u(T , u0)) ≥ z}, which is hard to do in practice. Instead, we can estimate the integral by Laplace’s method. As shown in Materials and Methods, this is justified for large z, when the probability of the set Ω(z) is dominated by a single u0(x) that contributes most to the integral and can be identified via the constrained minimization problem

1 2 min ku0kC ≡ IT (z), [6] 2 u0∈Ω(z) which then yields the following LDT estimate for Eq. 3

PT (z) exp (−IT (z)) . [7] Here, means that the ratio of the logarithms of both sides tends to 1 as z → ∞. As discussed in Materials and Methods, a multiplication prefactor can be added to Eq. 7, but it does not affect signiﬁcantly the tail of PT (z) on a logarithmic scale. In practice, the constraint F (u(T , u0)) ≥ z can be imposed by adding a Lagrange multiplier term to Eq. 6, and it is easier to use this multiplier as control parameter and simply see a posteriori what value of z it implies. That is to say, perform for various values of λ the minimization 1 2 min ku0kC − λF (u(T , u0)) ≡ ST (λ), [8] u0 2 Fig. 1. (Upper) Time evolution of |u(t, x)| from an initial condition opti- over all u0 of the form in Eq. 2 (no constraint), then observe that mized for maxx |u(T, x)| ≥ 8 m at T = 20 min. (Lower) Same for a typical this implies the parametric representation Gaussian random initial condition.

856 | www.pnas.org/cgi/doi/10.1073/pnas.1710670115 Dematteis et al. 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APPLIED MATHEMATICS large transient excursion. In this scenario, as long as the initial probability distribution in these pockets is accurate, the dynamics will permit precise estimation of the distribution tail. In some sense, the distribution of the initial condition plays a role of the prior distribution in Bayesian inference, and the posterior can be effectively sampled by adding the additional information from the dynamics over short periods of time during which instabilities can occur. [Note in particular that the Gaussian ﬁeld in Eq. 2 is the random ﬁeld that maximizes entropy given the constraint on its covariance C (x).] In ref. 45, this picture is made predictive by using a 2D ansatz for the initial condition u0(x) to avoid having to perform sampling in high dimension over the original u0(x). What our results show is that this approximation can be avoided all together by using LDT to perform the calculations directly with the full Gaussian initial condition in Eq. 2. Interestingly, we can use the results above to calculate the probability of occurrence of rogue waves in a given time window. More precisely, the probability p(z, TI ) that a rogue wave of amplitude larger than z be observed in the domain [0, L] dur-

ing [0, TI ] [i.e., that maxt∈[0,TI ] maxx∈[0,L] |u(t, x)| ≥ z] can be estimated in terms of P(z) and τc as

T /τ Fig. 4. Contour plot of the optimal trajectories from LDT for T = 10, 15, p ≡ P max max |u(t, x)| ≥ z ∼ 1 − (1 − P(z)) I c , and 20 min in set 1. The trajectories, superposed to match at t = T, coin- t∈[0,TI ] x∈[0,L] cide, which is consistent with the convergence of the probabilities PT (z) for [13] large T. where we use the fact that rogue waves can be considered independent on timescales larger than τc and assume TI τc . The of our method and its results. Notice first that this convergence function p is plotted in Fig. 5 as a function of z and TI . For exam- can be explained if we assume that the probability distribution of ple, for set 1, Eq. 13 indicates a 50% chance to observe a rogue the solutions to Eq. 1 with Gaussian initial data converges to an wave of height z = 4 m (that is, ∼8 m from crest-to-trough) after −2 invariant measure. In this case, for large T , the Monte Carlo sim- 11 h [using τc = 10 min and P(z = 4 m) = 1.1 · 10 ]; if we wait ulations will sample the value of maxx |u| on this invariant mea- 30 h, the chance goes up to 85%. Similarly, for set 2, the chance sure, and the optimization procedure based on LDT will do the to observe a wave of 11 m height is ∼50% after 3 h and ∼85% −2 same. The timescale τc over which convergence occurs depends after 8 h (τc = 3 min and P(z = 11 m) = 1.2 · 10 ). on how far this invariant measure is from the initial Gaussian measure of u0(x). Interestingly, the values we observe for τc are 5. Concluding Remarks in rough agreement with the time scales predicted by the semi- We have shown how an optimization problem building on LDT classical limit of NLS that describes high-power pulse propaga- can be used to predict the pathway and likelihood of appear- tion (35, 36). As recalled in Supporting Information, this approach ance of rogue waves in the solutions of MNLS fed by random predicts that the timescale of apparition of a focusing solution initial data consistent with observations. This setup guarantees starting from a large initial pulse of maximal amplitude Ui and √ accuracy of the core of the initial distribution, which in turn 1 2 2−1 length-scale Li is τc = TnlTlin, where Tnl = 2 ω0k0 Ui is permits the precise estimation of its tail via the dynamics. Our the nonlinear timescale for modulational instability and Tlin = results give quantitative estimate for the probabilities of observ- −1 2 2 8ω0 k0 Li is the linear timescale associated with group disper- ing high-amplitude waves within a given time window. These sion. Setting√ Ui = Hs (the size at the onset of rogue waves) and results also show that rogue waves have very specific precur- −1 Li = 2π∆ (the correlation length of the initial field) gives sors, a feature that was already noted in ref. 47 in the context τc ' 18 min for set 1 and τc ' 8 min for set 2, consistent with the of a reduced model and could potentially be used for their early convergence times of PT (z). This observation has implications in detection. terms of the mechanism of apparition of rogue waves, in particular, their connection to the so-called Peregrine soliton, that has been invoked as prototype mechanism for rogue wave creation (5, 13, 37–40), in particular for water waves (24, 25, 41), plasmas AB (42), and fiber optics (36, 43, 44). This connection is discussed in Supporting Information. Our findings also indicate that, even though the assumption that u0(x) is Gaussian is incorrect in the tail [that is, PT=0(z) is not equal to the limiting P(z) in the tail], it contains the right seeds to estimate P(z) via PT (z) if T & τc . (This convergence occurs on the timescale τc which is much smaller than the mix- ing time for the solutions of Eq. 1, i.e., the time it would take from a given initial condition, rather than an ensemble thereof, to sample the invariant measure.) Altogether, this is consistent with the scenario put forward by Sapsis and coworkers in refs. 45 and 46 to explain how extreme events arise in intermittent dynamical systems and calculate their probability: They occur Fig. 5. Contour plots of the probability to observe a wave whose amplitude when the system hits small instability pockets which trigger a exceeds z in the time window [0, TI] for sets 1 (A) and 2 (B).

858 | www.pnas.org/cgi/doi/10.1073/pnas.1710670115 Dematteis et al. 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Materials Program Award (NSF) Center Foundation Engineering Universit Science and National and Science by Politecnico semiclassi- part of in the program supported to PhD attention Math Schr our joint nonlinear drawing the for for theory reviewer cal anonymous an and B O. cussions; ACKNOWLEDGMENTS. compo- the scalar with through spectrum metric the gradient diagonal of the nents step-independent, precondition the then by we multiplication step, descent Eq. steepest for as method pseudospectral same both grating Information). (Supporting of calculation values this repeat larger also we using convergence, check shift—to phase modes overall the an by spanned Eq. space of dimensional minimization that means This the get to sampling, Carlo data representation Monte initial the the using the truncate in we above, done presented results was what with Consistent notation this in that see to easy is it and as written be can equation this b(u), u(t function Eq. in tion Procedure. 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Numerical work. this in apply we by that dominated result is deviations amplitude large its the of log the that where notation) vectorial compact (using as expressed be can gradient this step evaluat- rule, of involves gradient adaptive This (functional) (50). with the gradient descent the ing of steepest preconditioning and using search) (line minimization this perform We = hnpromn h ot al iuain,w s 10 use we simulations, Carlo Monte the performing When npatc,teeauto ftegain nEq. in gradient the of evaluation the practice, In , u I 3mdswith modes 23 T 0 hc erwieas rewrite we which 8, (z .Cletn l em ntergthn ieo h NSEq. MNLS the of side right-hand the on terms all Collecting ). J (t nEq. in ) u(t , u 0 , ) u(t x he,M oaa,adT assfritrsigcomments; interesting for Sapsis T. and Mohamad, M. uhler, ¨ = ntm,w s suopcrlscn-re exponential second-order pseudospectral a use we time, in ) Z 6 and ) δ P a eeautdb ovn h ulotmzto problem optimization dual the solving by evaluated be can (z u(t ) opromtecluain,w ov Eq. solve we calculations, the perform To u e −11 E , 0 Ω − (x (u u J (z C ˆ (t PNAS δ 1 2 0 M ∂ etakW ri n .Ooaofrhlfldis- helpful for Onorato M. and Craig W. thank We ) δ 0 ∂ n )/δ u and ) pt time to up ) |θ t sepandaoe h ag eito aefunc- rate deviation large the above, explained As , E = t J sdaoa elements. diagonal as 0 ≤ u λ) | n n ontcal ifrnei h results the in difference noticeable no find and 2 = u n=−11 = = d S n 0 X ≡ T θ | 11 δ δ P C (λ) steJcba ftetransformation the of Jacobian the is b(u), ≤ b u (z −1 1 2 aur 0 2018 30, January 23 E dne qain ..i upre ythe by supported is G.D. equation. odinger J ¨ ssboiati htcs,i h sense the in case, that in subdominant is ) ku (u , = 11—i.e., e e u ik − osa oga h boundary the as long as does 0 0 0 n , inf t k 2 1 − u(t x ihrsetto respect with λ) 24 C 2 = ˆ a |θ J (t u n λJ − 1 15 ? 0 , J spromdi h 2M the in performed is T = (z (t = ntesm rd opromthe perform To grid. same the on E T Eq. . λF −3∆ )| ˆ a (T (u , ta s h rfco a take may prefactor the is, [that k 0) 2 n 0) u ∂ n (uT conigfrivrac by invariance for accounting , 0 , 0 Ω = , = u |θ satisfies ) = 27 ,weew en h cost the define we where λ), (z 0 ? u ) ≤ 2π , | as Id. near ) (z δ δ 0 sitgae yuigthe using by integrated is u , u F 25 hsi h sec of essence the is This )|. n/L. k o.115 vol. 0 z )). u n 21 ∞. → 0 iTrn.EV-.is E.V.-E. Torino. di a spromdb inte- by performed is ` ≤ over θ u osntnecessar- not does ? 0 hs modes These 3∆: sn h chain the Using . (z 6 1 ] h rougher the )], M | elztosof realizations with o 5 no. = oevolve To 1. 12 2 − 3modes 23 vrthe over L ∂ equidis- 1 Ω = 1 u | = (z 0 into 40π [23] [28] [27] [26] [24] [25] 859 is ) → 45 z .

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