Note on Breather Type Solutions of the NLS as Models for Freak-Waves

Transcription

1 Physica Scripta. Vol. T8, 48^5, 1999 Note on Breather Type Solutions of the NLS as Models for Freak-Waves Kristian B. Dysthe 1 and Karsten Trulsen y 1 Department of Mathematics, University of Bergen, Johs.Brunsgt.1, 5008 Bergen, Norway Instituto Pluridisciplinar, Universidad Complutense de Madrid, Paseo Juan XXIII 1, E^8040 Madrid, Spain, Received March 1, 1999; accepted March 1, 1999 pacs ref: i, 9.10.Hm Abstract Some breather type solutions of the NLS equation have been suggested by Henderson et al. [to appear in Wave Motion] as models for a class of `freak' wave events seen in D-simulations on surface gravity waves. In this paper we rst take a closer look on these simple solutions and compare them with some of the simulation data [Henderson et al. to appear in Wave Motion]. Our ndings tend to strengthen the idea of Henderson et al. Especially the Ma breather and the so called Peregrine solution may provide useful and simple analytical models for `freak' wave events. 1. Introduction The present note is a result of our research on surface gravity waves. To lowest nonlinear order the NLS equation provides a model for these waves. We are interested in the so called `freak' waves, which are (loosely) a single wave or a very short- and shortlived group with a signi cantly larger steepness than in the surrounding wave record. Henderson, Peregrine and Dold [1] have named these events SWE (steep wave event). The same authors were also the rst to relate SWE's to the breather type solutions of the NLS. They investigated SWE's from simulations of the full nonlinear equations (see also Dold and Peregrine []) starting from a periodically perturbed plane wave. Due to the modulational instability of surface gravity waves the wave train evolves into deep modulations. Choosing the initial steepness ka to be small (< 0:1) and the perturbation period to contain many waves, they observed SWE's that had a striking similarity to a breather solution of the NLS, speci cally the solution of Peregrine [3]. We carry their analysis a little further, investigating whether the SWE's can be modeled by the space- periodic breather type solution of Ahkmediev et al. [4] and the time-periodic solution of Ma [5], as well as the solution found by Peregrine. Since Zakharov and Shabat [6,7] demonstrated how the NLS equation could be solved using the inverse scattering method, the bulk of knowledge accumulated about this equation has become very large. Together with several other integrable nonlinear PDF's the equation has been studied for more than two decades and the probability that some interesting simple results should have been overlooked appears very slim. In the litterature there are published a number of soliton and breather type solutions. It is not always easy to see whether these are unrelated or just equivalent forms of the same solution. The rst part of this note y Now at SINTEF Applied Mathematics, P.O. Box 14 Blindern, N-0314 Oslo, Norway Physica Scripta T8 is an attempt on a simple-minded presentation of some of the known solutions of the so called focusing NLS equation with special emphasis on the non-steady or breather type solutions. We shall only consider the solutions that are simple enough to be written on one line! Of course there are `composits' like the N-soliton solution and similar ones of the breather type, some of which can be written on a beautiful symmetric form (see e.g. Ablowitz and Herbst [8]). These we do not consider.. Solutions of stationary envelope In the following we shall only be concerned with the so-called focussing NLS (also referred to as the NLS of the anomalous dispersion regime) which is the relevant one for surface gravity waves. It is written on the standard form iq t q xx q q ˆ 0: 1 Perhaps the simplest and most well known solutions of this equation are the plane wave and the soliton, which can be written respectively as e it e it and cosh x : Remark that for aesthetical reasons we do not write them in their most general form. This form can always be achieved by noting that the following two (one parameter) families of transformations leave the NLS equation invariant: The `gauge' transformation: t! t; x! x kt; q! qe i kx k t k 3 where k and k are arbitrary real constants, and the `scaling' transformation: t! a t; x! ax; q! q=a: 4 This means that if q x; t is a solution of (1) then so are q x kt; t exp i kx k t k and aq ax; a t : While () are examples of solutions whose envelope q are stationary in time, they are not the only ones. They can all be derived from one of the following two (one parameter) families of solutions e i m t dn x; m and me i m 1 t cn x; m 5 We refer also to solutions that can be transformed into this form by a suitable gauge transformation (3).

2 Note on Breather Type Solutions of the NLS as Models for Freak-Waves 49 where 0 4 m 4 1 and dn x; m and cn x; m are Jacobi elliptic functions (see [9]). In the limit m! 1 both solutions tend to the soliton solution (). In the limit m! 0the former of the solutions in (5) tends to the plane wave solution while the latter tends to zero. Other published solutions of stationary envelope can be found by suitable translations x! x x 0 of (5). 3. Breather type solutions The solutions above represent propagating waves and solitons. There is always a reference frame where the the envelope q is stationary. This is not so for the family of solutions to be discussed next. These solutions that are of the ``breather'' type are genuinely unsteady. The rst breather type solution for the NLS equation (1) was found by Ma [5]. He solved the initial value problem for the NLS equation where the initial state was a perturbed plane wave solution with boundary conditions q x; t! q0 as jxj!1: He found that the asymptotic state is a series of periodically ``pulsating'' solitary waves or breathers (Ma-breathers) plus a residual of small amplitude dispersive waves. Earlier Zakharov and Shabat [7] solved the corresponding problem for the non-focusing NLS iq t q xx q q ˆ 0: They showed that the asymptoic state is a series of `dark' solitons plus a similar residual of dispersive waves Space periodic solution Akhmediev et al. [4] found a one parameter family of space-periodic solutions with the property that they approach the plane wave solution when t!1 (If considered as a breather, it `breathes' only once). Later Ablowitz and Herbst [8] independently derived the same solution. With some simpli cation compared to these latter papers the Akhmediev solution q A can be written it cosh Xt ij cos j cos px q A ˆ e 6 cosh Xt cos j cos px where p ˆ sin j and X ˆ sin j 7 for j real. The solution is periodic in space with period p=p and tends to the plane wave solution (see ()) in the limits t!1. The largest modulation occurs for t ˆ 0; with the imum of the envelope at x ˆ 0. Then the imum, q A ; is given by ˆ cos j 1: q A Taking the limit j! 0(i.e. when the spatial period!1 we obtain q P lim q A ˆ e it 4 1 4it 1 j!0 1 4x 16t : 8 This is the solution found by Peregrine [3]. He derived it as a limiting case of the Ma breather i.e. when the `breathing' period tends to in nity. It is interesting (but not very surprising) to note that it is also the limiting case of the Akhmediev solution when the spatial period tends to in nity. Since the NLS equation is invariant to a translation in time t! t t 0, one can choose a reference time t 0 (negative) such that exp Xt 0 m1. For small values of t (6) can be linearized in m to give (except for an unimportant constant phase) q A ˆ e it 1 m sin j e i p j cos px e Xt O m : 9 It is well known that the plane wave solution exp it is unstable to the modulational instability. With an initial small periodic perturbation of the form q x; 0 ˆ1 ee ic cos px 10 where e is a small real number and c an arbitrary phase, one obtains by linearization of the NLS equation the solution q ˆ e it 1 e cos px h cos c j e Xt e i p j sin j 11 cos c j e Xt e i p j ii : Here X, j and p are as given above. Thus X is to be interpreted as the growth rate of the modulational instability corresponding to the perturbation wavenumber p. Comparing (9) and (11) we see that q A can be considered to evolve from a periodically perturbed plane wave. As pointed out by Ablowitz and Herbst [8] the solution (6) is a very special one in that it represents an analogue of a homoclinic orbit for a dynamical system, starting out along the unstable manifold. The same authors solved the NLS equation (1) numerically with periodic boundary conditions to generate this special solution. With an initial condition of the form (10), the proper choice of the real phase c to obtain q A is seen by comparison with (9) and (11) to be c ˆ p j: A sequence of time-periodic solutions (breathers) were obtained. As e gets smaller they are approaching the homoclinic orbit (6) which is reached in the limit e! 0. It should be noted that Ablowitz and Herbst [8] showed how one could derive more general solutions with the property that they tend to a plane wave solution when t!1. Starting from the N-dark-hole soliton solution of the so called defocusing NLS equation, they used the fact that any solution of that equation that is even and analytic on the real axis, will be a solution of (1) under the transformation x! ix and t! t. This way space periodic solutions can be constructed that `breath' more than once. As can be appreciated from their paper, however, they have rather lengthy analytic expressions. 3.. Time periodic solution The Ma-solution,, is periodic in time and tends to the plane wave solution as x!1. It is found from (6) by choosing j imaginary. The transformation j! ij implies p! ip and X! ix where now p ˆ sinh j and X ˆ sinh j : 1 From (6) we then obtain it cos Xt ij cosh j cosh px ˆ e : 13 cos Xt cosh j cosh px Physica Scripta T8

3 50 Kristian B. Dysthe and Karsten Trulsen Fig. 1. The Akhmediev solution (6) for j ˆ 0:5. The envelope q is shown as a function of x and t in (a), and the time evolution of one period p=p of the spatial pro le is shown in (b). Fig.. The Ma-breather (13) for j ˆ 1:. The envelope q is shown as a function of x and t in (a), and the spacial pro le of the envelope through aperiodp=x is shown in (b). Again the Peregrine solution is found in the limiting case when j! 0. The spacial imum of the modulation envelope occurs at x ˆ 0. It oscillates between a imum value when t ˆ 0given by ˆ cosh j 1 and a minimum value of cosh j 1whenXt ˆ p (see Figure b).the spatial shape of the central part of the envelope is qualitatively quite like that of the Akhmediev solution shown in gure 1 (for one spatial period), except of course that! 1asx!1while qa is periodic. Comparing with the Akhmediev solution it is seen that 3 while q A 3: Again the Peregrine solution is the limiting case of both with ˆ 3. q P 3.3. Double-periodic solution Akmediev et al. [4] also found a one parameter family of solutions of the breather type, periodic both in space and time. In time it oscillates periodically between a shallowand a deep spatially periodic modulation. It is given by Physica Scripta T8 the somewhat unpleasant expression q Ap ˆ p k 1 k cd x ; cn t; k i 1 k sn t; k exp it 1 k cd x ; dn t; k 1 k 14 where sn, cn, dn and cd are Jacobi elliptic functions, and the parameter k is restricted by 0 k 1: It is rather interesting to look at the limiting cases k! 0 and k! 1 where the Jacobi elliptic functions tend to elementary functions. For the limit k! 0we have lim k!0 q Ap ˆ e it u eit cosh x cosh x where the transformation p into the form () is achieved by the scaling (4) with a ˆ 1= : In the limit k! 1wehave lim q Ap ˆ p 1 cos x i sinh t e it u k!1 cosh t cos x cosh t i p p 1 15 cos x e it cosh t p 1 cos x where the form (6) of the Akhmediev solution p with j ˆ p=4 is achieved by the scaling (4) with a ˆ.

4 Note on Breather Type Solutions of the NLS as Models for Freak-Waves 51 Thus the double-periodic solution tends to a special case (j ˆ p=4 of the `homoclinic' Akhmediev solution (6). It is rather tempting to assume that a two parameter family of double-periodic solutions exist that tend to the more general `homoclinic' solution in the above limit. To our knowledge such a solution (on an explicit form) has not been reported. 4. Applicationto`SteepWaveEvents' Recently Henderson, Peregrine and Dold [1] (HPD hereafter) have shown that steep wave events (SWE) in their numerical calculations (based on the exact equations of motion) occurs in wave groups having a form very similar to the breathers discussed above. The simulations are of D nonlinear waves evolving from a periodically perturbed uniform wavetrain as described by Dold and Peregrine [] (see also Banner and Tian [9] and [10]). Their initial condition consists of a Stokes wavetrain of wavelength p and steepness ka 0 ; with an additional small periodic perturbation. Comparing with the NLS equation this translates into an initial condition of the form q x; 0 ˆq 0 1 ee ic x cos 16 N where e is a small number, N is the number of waves in a modulation period, c is an arbitrary constant phase. (The number is a consequence of the transformation of coordinates leading to the NLS equation). p The initial steepness ka 0 ˆ q0 is chosen su ciently large for the Stokes wave to be modulationally unstable to the above perturbation. The condition for this to happen is approximately (Dysthe [11]) ka 0 1 > 8N N 1 : The modulational instability then evolves, and if breaking does not occur, shows a tendency towards recurrence. Exact recurrence as shown by Yuen and Ferguson [1] for the NLS equation can only be expected when the higher harmonics of the initial perturbation period are modulationally stable. That requires approximately 1 4N N= 1 > ka 0 1 > 8N N 1 : Even when this is not satis ed, as was the case in the HPD simulations referred to, the `` rst period'' of wave interaction is dominated by the initially imposed sidebands as shown by Yuen and Ferguson [1]. In what follows we shall only be concerned with this `` rst period'' of the evolution. Since HPD used periodic boundary conditions it would seem most natural to compare the rst SWE to the space periodic solution of Akhmediev et al. [4]. Comparing (8), (11) and (13) there are two conditions that must be satis ed for the initial condition (13) to produce a solution of the type (7) in the limit e! 0:. The periodicity must be the same i.e. p ˆ 1 N :. The initial phase must be such as to initiate only the unstable mode i.e. c ˆ p j: The latter condition was not satis ed in the HPD simulations. The parameters were chosen such that Table I. Comparison between simulations (HPD) and quantities calculated from the Akhmediev solution (6), the Ma solution (13) and the Peregrine solution (8). Here y 0 and y are the imum surface elevations initially, and at the peak of the rst SWE. The imum elevation y happens when a wavecrest is located at the group imum. Approximately one wave period aftewards, when a wave trough is located at the group imum, the elevation imum is y 1 : The ratios y =y 0 and y =y 1 are listed for four cases with di erent initial steepness ka 0 and number of waves N in the initial modulation period. ka N y =y 0 HPD-simulation Akhmediev Peregrine Ma y =y 1 HPD-simulations Akhmediev Peregrine Ma j ' 5 in all the simulation runs we are considering. The value of c was taken to be 45 and not 65 which would satisfy the condition above. On the other hand their simulations seem to indicate that there is no drastic change in the rst SWE for the initial phase in this region. When keeping ka 0 ; N and e constant and varying the initial phase c, it was found that the largest initial growth, corresponding to the earliest (and locally largest) SWE, (see their table I) occurred for c ˆ j ' 5 in accordance with theory (see (10)). With c ˆ j p ' 115 (see (10)) the initial (linearized) growth is zero, and HPD found a large delay in the occurrence of the rst SWE. We shall compare some of the ndings of HPD with the Akhmediev solution (6) taking the space periodicity to be that of the initial perturbation. The simulations by Ablowitz and Herbst [8] using the NLS equation, produced the Akhmediev solution as a limit of time periodic solutions (breathers) when e! 0. Since the relative amplitude e of HPD's initial perturbation was taken to be rather large (typically ˆ 0.) space- and time periodic solutions would appear to be most relevant. Unfortunately the analytical form of these solutions does not seem to be known, except for a special case (14), which does not (except for the limiting case k ˆ 1) have the property of periodically returning to a plane wave. Comparisons are also made with the Peregrine solution (8) and the Ma solutions (13). These are not space periodic of course, but their ``widths'' are considerably less than the spacial period of the initial perturbation. In fact the ratios between the spacial period and the widths (taken to be the distance between the two zeroes of q p (see gure b and 3b )) are =3 pnka0 ' 4:3: Tocomparewetake the asymptotic steepness (as x!1 ) of the solutions to be that of the unperturbed wave train. For the Ma-breather the value of the parameter j is then determined by taking the time period p=x to be the recurrence time observed in the simulations. Physica Scripta T8

5 5 Kristian B. Dysthe and Karsten Trulsen Fig. 4. Illustration of the wavepacket at imum modulation, with snapshots of the wavetrain at two di erent times separated by a wave period: one with a crest and the other with a trough at the group imum. The imum elevation for these two situations are y and y 1 respectively. on the small side, though markedly better for the Peregrine solution. The Ma solution on the other hand is quite close to the simulated values. On the whole it seems that the Ma solution is the best approximation to the SWE events closely followed by the Peregrine solution. This tends to support the conjecture [1] that breather type solutions, like those of Ma and Peregrine, may be good and simple analytic models for SWE events. Acknowledgement Fig. 3. The Peregrine solution (8). The envelope q is shown as a function of x and t in (a), and the time evolution of the spacial pro le of the envelope is shown in (b). 5. Conclusion In table I we list the ratio between the imum suface elevation, y, and the unperturbed one, y 0, for a number of simulations. The imum value refers to the st SWE of each simulation, and corresponds to the situation whenthewavehasacrestattheimumofthegroup envelope. Because the group at a SWE is very narrow, the imum elevation, y 1, when the wave has a trough at the group imum (approximately one period after) is appreciably less than y (see Fig. 4): Both these quantities were recorded graphically in the paper by HPD. Their ratio is a good measure of the group width, and in table I it is compared to the corresponding ratio of the Akmediev solution (6), the Peregrine solution and the Ma solution (13). From the table it is seen that there is excellent agreement with regards to the form of the SWE as indicated by the ratio y =y 1 ; especially for the Akhmediev and Peregrine solutions. The imum elevations of these two are both This work was done when the rst author was on sabbatical leave at School of Mathematics, UNSW in Sydney with grants from The Norwegian Reseach Council. Inspiring discussions with M. Banner and W. Schief are thankfully acknowledged. We are also thankful for help from K. Henderson and H. Peregrine. K. T. acknowledges support from the European Union through a fellowship (MAS3^CT96^5016), a DGICYT grant (PB96^599) and a TMR Programme Network (ERBFNMRXCT96^0010). Additional support is acknowledged from Norsk Hydro and Statoil. References 1. Henderson, K. L., Peregrine, D. H. and Dold, J. W., Wave Motion 9, pp. 341^361.. Dold, J. W. and Peregrine, D. H., Proc. 0th. Int. Conf. Coastal Engng., Taipei. vol. 1, pp asce (1986). 3. Peregrine D. H., J. Austral. Math. Soc. Ser. B 5, 16(1983). 4. Akhmediev, N. N., Eleonskii V. M. and Kulagin., N. E., Theor. Math. Phys. (USSR) 7, 809 (1987). 5. Ma, Y. C., Stud. Appl. Math. 60, 43, (1979). 6. Zakharov, V. E. and Shabat A. B., J. Exp. Theor. Phys. 34, No.1 (197). 7. Zakharov, V. E. and Shabat A. B., J. Exp. Theor. Phys. 37, No.5 (1973). 8. Ablowitz, M. J. and Herbst B. M., SIAM J.Appl.Math. 50, 339 (1990). 9. Banner, M. and Tian, X., Phys. Rev. Lett. 77, 953 (1996). 10. Banner, M. and Tian, X., J. Fluid Mech. 367, 107 (1998). 11. Dysthe, K. B., Proc. Roy. Soc. Lond. A 369, 105, (1979). 1. Yuen, H. and Ferguson W. E., Phys. Fluids 1, 175 (1978). Physica Scripta T8