Modulational instabilities and Fermi Pasta Ulam recurrence. in a coupled long wave short wave system, with a mismatch in.

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1 * Manuscrit Mdulatinal instabilities and Fermi Pasta Ulam recurrence in a culed lng wave shrt wave system, with a mismatch in gru velcity by C. K. Pn *, R. H. J. Grimshaw #, K. W. Chw * (Crresnding authr: Dr. K. W. Chw) *: Deartment f Mechanical Engineering, University f Hng Kng, Pkfulam, Hng Kng Phne: (85) Fax (85) kwchw@hkusua.hku.hk #: Deartment f Mathematical Sciences Lughbrugh University, Lughbrugh LE 3U, United Kingdm Phne: (44) Fax: (44) r.h.j.grimshaw@lbr.ac.uk Submissin date: January, 006 Keywrds: Mdulatinal instabilities; lng wave shrt waves; Fermi Pasta Ulam recurrence. PACS: i; Bb; 47.35Pq

2 January 7, 006 ABSRAC he resnance f tw enveles f shrt (caillary) waves with a cmmn lng (gravity) wave cmnent is cnsidered. A mismatch in gru velcity is incrrated and the resnance cnditins need nt be satisfied exactly. his slight detuning ermits a wider chice f mdes and cnsequently, a much richer set f dynamics. he linear instability f lane waves is studied, and the dminant unstable wave numbers are identified. he subsequent fully nnlinear evlutin f these erturbed lane waves is investigated by direct numerical simulatins. he Fermi Pasta Ulam recurrence henmenn is bserved.

3 3 January 7, 006. Intrductin Mdulatinal instabilities f lane waves and their subsequent evlutin may have rfund influence n the dynamics f many nnlinear systems (e.g. Fujimura et al, 988). hese studies are relevant in many fields, e.g., anharmnic lattices (Burlakv et al, 996), fluid dynamics (Janssen, 98; Xu and Wei, 99), tics (Camburnac et al, 00; Simaeys et al, 00) and lasma hysics (Sharma et al 005). In many instances, this linear, mdulatinal instability will lead t a time-eridic return f the erturbed system int a state very clse t the riginal, rimary state. his henmenn is cmmnly termed the Fermi Pasta Ulam recurrence (FPU) in the literature. A cmrehensive analytical arach is lacking, but a first attemt restricts attentin t the lwest rder Furier mdes. Reasnable arximatins f the erid f recurrence can then be btained (Infeld, 98). Fluid dynamics is rbably the field where FPU and related questins have been studied mst intensively. Surface waves n dee water cnstitute the latfrm where a cmbinatin f theretical, cmutatinal and exerimental techniques rvides a enetrating investigatin f the interlay amng triad and quartet resnances, FPU, and the lng time dynamics (Yuen and Lake, 98). he fcus f the resent wrk is n the dynamics f interacting lng wave with shrt waves, which will be seen t rvide anther latfrm fr studying FPU. Earlier wrks have demnstrated that the necessary cnditin fr such lng shrt resnance is that the gru velcity f the shrt wave is equal t the hase velcity f the lng wave; such a resnance can be treated as a degenerate case f a triad resnance (Benney, 977; Grimshaw, 977; Grimshaw, 98). Mre recisely, we shall study culed systems where tw shrt waves are in resnance with a cmmn lng waves. Sme secial cases, e.g., mdes with equal gru velcity, are amenable t analytical treatment (Ma, 98). A main gal f the resent study is

4 4 January 7, 006 t remve this cnstraint. he inclusin f a difference in gru velcity allws a wider chice in the selectin f mdes, and ermits a richer set f dynamics. Mdulatinal instability fr a single cmnent, r unculed, lng shrt system has been studied earlier. he lane wave will generally be unstable fr even fairly small amlitudes f the rimary wave, if the disersin cefficient is sufficiently small, a cmmn ccurrence in stratified fluid flws in the envirnment (Ma and Redek, 979). hese rerties were further exlred by K and Redek (98). he instability is indeendent f the initial magnitude f the lng wave, and a lng wave can be generated by an unstable shrt wave. Such instability is unidirectinal in the sense that a lng wave cannt generate the shrt wave if the shrt wave is absent initially. Exerimental asects were als cnsidered in that wrk. Yshinaga et al (99) erfrmed numerical studies n the assciated recurrence henmena using a slightly mdulated shrt wave as an initial cnditin. Bth recurrence and chatic mtins are ssible, deending n the magnitude f certain cntrl arameters f the gverning system.. hery. Backgrund By chsing the resnance cnditins in an arriate manner, the nnlinear evlutin equatins take the frm ia ib ivax + AXX = λla, () + ivbx + BXX = λlb, () L = µ ( A ) + µ ( B ). (3) X X A, B, L are the tw shrt waves and the lng wave resectively, and V measures the mismatch between the gru velcity f the shrt waves, and the hase velcity f the lng wave. and X are the rerly scaled slw time and sace crdinates in the averaged

5 5 January 7, 006 gru velcity reference frame. he linear art f such an evlutin eratr is quite generic in many nnlinear evlutin systems (Aransn and Kramer, 00). he values fr the ther arameters will deend n the exact cnfiguratins f the fluid flw under cnsideratin (Funakshi and Oikawa, 983). Plane waves fr this system are readily btained: ex( ) L i A A λ =, ) ex( L i B B λ =, L L =, (4) where A, B, L are the initial amlitudes f the unerturbed lane waves. Linear stability is studied by intrducing small erturbatins, and searching fr mdal slutins with a factr f ex[i( x -Ω t )]. he disersin relatin is: 0 ) ( ) 4 4 ( ) ( ) ( = µ λ µ λ µ λ µ λ Ω µ λ λ µ Ω µ λ µ λ Ω Ω B B V A A V VB VA V V V B A V V V, (5) is the wave number. Ω is the angular frequency, and cmlex Ω imlies mdulatinal instability, with the imaginary art reresenting a rate f grwth f the disturbance.. Numerical simulatins Linear instability is valid rvided the nnlinear terms are negligible. study the eventual fate f the instability fr sufficiently large disturbances, we slve the system ( 3) numerically with these initial cnditins: ] cs [ X A A ε =, (6a) ] cs [ X B B ε =, (6b) = 0 L, (6c) where A, B are the initial amlitudes f the tw unerturbed lane waves. he strength f the erturbatinε is set t be 0. fr all the simulatins rerted in this study, and is the wave number f the erturbatin. he Hsctch finite difference technique will be used

6 6 January 7, 006 (aha and Ablwitz 984; sang and Chw, 004). his uses bth imlicit and exlicit schemes t march frward in time. By using a secial arrangement f the grid ints, all the intermediate variables can be cmuted exlicitly and these schemes are stable. 3. Numerical results he disersin relatin fr caillary gravity waves dictates that relatively lng waves are dminated by gravity, while relatively shrt waves are influenced mainly by caillary effects. An insectin f the relevant disersin relatin shws that nly caillary waves can satisfy the resnance cnditin with a lng gravity wave. Rescaling in () allws ne t set =, λ =, and further rescaling f A and B ermits ne t ut µ = and µ = ±, leaving V,, λ as the actual free arameters. he wave mde B can be categrized as a caillary wave if (related t the secnd derivative f the disersin relatin) is sitive, r a gravity wave if is negative. Since we nte that a gravity wave cannt satisfy the lng shrt resnance cnditin with anther lng gravity wave, we shall fcus n the case being sitive. We shall, hwever, still resent ne case f negative, since we can treat ( 3) as a generic system fr lng shrt wave interactin, withut any articular reference t surface waves in mind. Fr simlicity f resentatin we fix all the ther arameters as lus r minus ne excet fr the mismatch in gru velcity V. A fully three dimensinal search in the arameter sace can be erfrmed but is unlikely t change the main hysics. he cases we will cnsider are summarized in able. Case Wave mde A Wave Mde B λ µ λ µ I Caillary Caillary II Caillary Gravity - - -

7 7 January 7, Caillary-caillary shrt waves interacting with lng gravity waves Deendence n the erturbatin wave number In the first case, we will study the interactins amng tw shrt caillary wave enveles (A and B) and ne lng wave cmnent (L), which are effectively tw degenerate resnant triads with the lng wave in cmmn. We first treat the case where the initial amlitudes fr bth waveguides A and B are the same ( A B = 0. ), and there is n gru velcity mismatch (V = 0) between A and B. here is a single interval f linear instability (Figure a), and erturbatin mdes in that range will grw initially. Here we nly shw the lng wave cmnent, since the behavir f the shrt wave enveles is similar. Nnlinear effects will sn cme int lay and a sequence f grwth and decay results (Fermi Pasta Ulam (FPU) recurrence, Figure b). Fr cmarisn urse, the results fr an initial cnditin with wave number ( = ) utside the band f instability are als rerted. Fr small times the lng wave cmnent is almst unchanged, as that wave number crresnds t stability. Hwever, discretizatin errrs f the finite difference schemes and nise in the cmutatins will eventually generate wave mdes in the unstable band, and hence the FPU rerty again manifests itself (Figure c). = he dynamics is even mre intriguing if ne intrduces a mismatch in gru velcity (V = ) between the tw shrt waves. he unstable regin fr mdulatinal instability slits int three bands (Figure a). he recurrence behavir becmes much mre cmlex as there are three lcal maxima fr the erturbatin wave number (Figure b). Observe that each hill r blb in the recurrence has a wavelength f abut 4 t 6, which is cnsistent with a wave number f abut t.5 (since wave number = π / wavelength). Deendence n the initial amlitudes When the initial amlitudes A, B are varied and a gru velcity difference (V = ) is resent, the dynamics is even mre cmlex and will deend n the relative magnitude f

8 8 January 7, 006 A, B as the intrductin f the mismatch V breaks the symmetry f the system. First we cnsider the case f A being smaller. he unstable regin (Figure 3a) diminishes dramatically bth in magnitude and extent. he imact n the dynamics is clearly illustrated in Figure 3b fr the tyical case f the amlitude f A being small (0.0) and B being large (0.). Recurrence behavir is again bserved. he erid f abut 500 time units is fairly well defined, and the fluctuatins in the lng wave cmnent are smaller, again cnsistent with the smaller magnitude f the linear instability. On the ther hand, the case f a relatively smaller amlitude B might still have a fairly large band f instability. Figure 4a illustrates the situatin fr A being 0. and B being 0.0. Numerical simulatins again shws sme recurrence henmena, with each hill r blb having a wavelength f abut 5, cnsistent with the dminant unstable wave number f abut (Figure 4b). An extreme case is the scenari where the amlitude B is almst zer initially. he remarkable fact is that the magnitude f B can still grw t rder ne thrugh this lng shrt resnance mechanism. Further details are described in sectin 3.3 belw. Deendence n (further) increase in mismatch in gru velcity As we increase the mismatch in gru velcity between the tw shrt wave enveles further (V = ), the instability bands shift t higher wave numbers, with tw clsely clustered bands, and a third, negligible ne near the rigin (Figure 5a). he dynamical imlicatin is that the redminant feature in any Fermi Pasta Ulam recurrence will have smaller wavelength. his feature is clearly revealed in Figure 5b, where the cycle f self lcalizatin int a sequence f eaks, demdulatin t the near riginal state, and self lcalizatin again can be bserved. he hills r blbs have wavelength f abut 3, a manifestatin f the maximum instability f abut =. Sme hills exhibit twin eaks, rbably as a result f the clsely clustered bands f instability. he whle structure is rather rbust as mre r less

9 9 January 7, 006 the same feature is demnstrated again with a different initial cnditin. Here we chse a smaller wave number ( = 0.5) as the starting int. Even thugh this chice is far away frm the bands f dminant instabilities, discretizatin errrs and nise, as argued abve, will eventually generate mdes with unstable wavelength. hus the recurrence behavir is again bserved after a fairly quiet erid fr small time (Figure 5c). When the mismatch in gru velcity is increased still further (V = 3), the bands f wave numbers fr instability cntinue t shift higher (Figure 6a). Again full numerical simulatins f the lng shrt interactin equatins ( 3) shw the FPU recurrence (Figure 6b). Nt surrisingly, the wave lengths f the recurring states (abut ) crrelate strngly with the unstable wave number f abut 3 (again wave number = π /wavelength). he regin r time interval f demdulatin t nearly riginal state is nw much lnger. 3. Disersive terms f different signs We nw cnsider the case where the disersive effects in the tw waveguides are f site signs, which crresnds t the signs f the secnd derivative terms in equatins () and () being different. Althugh this situatin cannt be realized fr gravity-caillary waves, it can arise in stratified fluid flws, where there are several branches fr the disersin relatin. Furthermre, we can cnsider ( 3) as a generic system f lng shrt waves, withut reference t any fluid flw setting, and then it is always cnceivable that a medium will exist with the desired cnditins. In fact, in nnlinear tics, such dual-channel systems exist and are alicable. One channel, which surts bright slitns, will act as the aylad channel, while the ther ne with an site sign f disersin, will serve as the surt channel by rviding the necessary stability rerty (sang et al, 005). Here fr simlicity we shall chse the cefficients as µ = and µ =. here is n linear instability fr V = 0, and hence we shall fcus n nnzer V.

10 0 January 7, 006 With the intrductin f a mismatch in gru velcity (V = ), a band f wave numbers crresnding t mdulatinal instabilities aears (Figure 7a). Full scale numerical simulatins f the gverning equatins ( 3) again shw a frm f recurrence behavir (Figure 7b). he wavelength f in the grwth cycle crrelates very well with the redminant mde in the linear instability regime (Figure 7b). Hence we can cnclude that this FPU recurrence is fairly rbust fr this tye f evlutin equatins invlving lng wave shrt waves. Furthermre, n changing the amlitude ratis (whether A large, B small r vice versa), the unstable regins d nt change drastically, unlike the revius sectin. Recurrence behavir can again be fund, but the details will be mitted. 3.3 Generatin f a shrt wave by anther shrt wave Even if the amlitude B is negligibly small ( ) initially, a suitable set f arameters can be chsen where B can grw thrugh this resnance mechanism with a cmmn lng wave cmnent. Figure 8 illustrates this situatin where nly the amlitude A is resent initially, but the amlitude B eventually attains a magnitude f rder ne, and then affects the FPU henmena f A in a significant manner. 4. Discussins and cnclusins A system f tw shrt waves in resnance with a cmmn lng wave is cnsidered. he nvel feature here is that a mismatch in gru velcity is allwed, and hence a wider chice f mdes is ermitted. Linear, mdulatinal instability f lane waves is identified, and the fate f these instabilities is studied by full numerical simulatins. In general sme frms f Fermi Pasta Ulam recurrence (FPU) exist fr this culed system. he FPU recurrence is altered accrdingly in the sense that the wavelengths f the underlying structures shrten, as the dminant wave numbers f the instabilities shift t higher values. he erid f recurrence seems t lengthen with an increasing difference in gru velcity.

11 January 7, 006 Furthermre, such mdulatinal instabilities and FPU cntinue t exist in systems f lng shrt waves with site disersins, and thus such henmena are fairly rbust and rbably universal fr a generic lng shrt interactin system. In a single cmnent, r unculed, system, shrt wave instability can generate lng waves, but nt vice versa, i.e., shrt waves cannt be generated if they are absent initially. While this is still bradly true fr the culed system, a shrt wave can generate anther shrt wave via their cmmn lng wave artner. Such a scenari culd be esecially relevant in ceanic alicatins. One questin which has nt been rerly addressed in this wrk is the existence f chatic mtin, which has been demnstrated earlier in the literature (Yshinaga et al, 99). Fr the secial classes f initial cnditins emlyed here, n chatic mtins are bserved but the existence f such chas cannt be ruled ut. Further wrk alng these lines will be fruitful in the future. Acknwledgement Partial financial surt has been rvided by the Research Grants Cuncil grants HKU 73/05E. he authrs (CKP, KWC) have many simulating discussins with Prfessr D. J. Benney. References Aransn, I. S., Kramer, L., 00. he wrld f the cmlex Ginzburg Landau equatin. Reviews f Mdern Physics 74, Benney, D. J., 977. General thery fr interactins between shrt and lng waves. Studies in Alied Mathematics 56, 8 94.

12 January 7, 006 Burlakv, V. M., Darmanyan, S. A., V. N. Pyrkv, V. N., 996. Mdulatin instability and recurrence henmena in anharmnic lattices. Physical Review B 54, Camburnac, C., Mailltte, H., Lantz, E., Dudley, J. M., Chauvet, M., 00. Satitemral behavir f eridic arrays f satial slitns in a lanar waveguide with relaxing Kerr nnlinearity. Jurnal f the Otical Sciety f America B 9, Fujimura, K., Yanase, S., Mizushima, J., 988. Mdulatinal instability f lane waves in a tw-dimensinal jet and wake. Fluid Dynamics Research 4, 5 4. Funakshi, M., Oikawa, M., 983. he resnant interactin between a lng internal gravity and a surface gravity wave acket. Jurnal f the Physical Sciety f Jaan 5, Grimshaw, R. H. J., 977. Mdulatin f an internal gravity wave acket and resnance with mean mtin. Studies in Alied Mathematics 5, Grimshaw, R. H. J., 98. Mdulatin f an internal gravity wave acket in a stratified shear flw. Wave Mtin 3, Infeld, E., 98. Quantitative thery f the Fermi Pasta Ulam recurrence in the nnlinear Schrdinger equatin. Physical Review Letters 47, Janssen, P. A. E. M., 98. Mdulatinal instability and the Fermi Pasta Ulam recurrence. Physics f Fluids 4, 3 6. Ma, Y. C., 98. he resnant interactin amng lng and shrt waves. Wave Mtin 3, Ma, Y. C., Redek, L. G., 979. Sme slutins ertaining t the resnant interactin f lng and shrt waves. Physics f Fluids, Sharma R. P., Batra, K., Verga, A. D., 005. Nnlinear evlutin f the mdulatinal instability and chas using ne dimensinal Zakharv equatins and a simlified mdel. Physics f Plasma, 03 (7 ages).

13 3 January 7, 006 Simaeys, G. V., Emlit, P., Haelterman, M., 00. Exerimental study f reversible behavir f mdulatinal instability in tical fibers. Jurnal f the Otical Sciety f America B 9, aha,. R., Ablwitz, M. J., 984. Analytical and numerical asects f certain nnlinear evlutin equatins. II Numerical nnlinear Schrdinger equatin. Jurnal f Cmutatinal Physics 55, sang, S. C., Chw, K. W., 004. he evlutin f eridic waves f the culed nnlinear Schrdinger equatins. Mathematics and Cmuters in Simulatins 66, sang, S. C., Nakkeeran, K., Malmed, B. A., Chw, K. W., 005. Culed eridic waves with site disersins in nnlinear tical fiber. Otics Cmmunicatins 49, 7 8. Xu, Y., Wei, R., 99. Fermi Pasta Ulam recurrence f bund slitary waves in a rectangular water trugh. Jurnal f the Acustical Sciety f America 9, Yshinaga,., Wakamiya, M. Kakutani,., 99. Recurrence and chatic behavir frm nnlinear interactin between lng and shrt waves. Physics f Fluids A 3, Yuen, H. C., Lake, B. M., 98. Nnlinear dynamics f dee water gravity waves. Advances in Alied Mechanics, 67 9.

14 4 January 7, 006 Figures 0.5 Ω I Figure a: he grwth rate fr mdulatinal instabilities with gru velcity mismatch V = 0 and initial amlitudes A=0., 0 B=0. 0 L X Figure b: Lng-time evlutin f the lng wave L with gru velcity mismatch V = 0, wave number f erturbatin = 0.5, and initial amlitudes A 0 =0., B=0. 0

15 5 January 7, 006 L X Figure c Lng-time evlutin f the lng wave L with gru velcity mismatch V = 0, wave number f erturbatin =, and initial amlitudes A 0 =0., B 0 =0.

16 6 January 7, Ω I Figure a: he grwth rate fr mdulatinal instabilities with gru velcity mismatch V = and initial amlitudes A 0 =0., B 0 =0. L X Figure b Lng-time evlutin f the lng wave L with gru velcity mismatch V =, wave number f erturbatin =, and initial amlitudes A 0 =0., B 0 =0.

17 7 January 7, Ω I Figure 3a: he grwth rate fr mdulatinal instabilities with gru velcity mismatch V = and initial amlitudes A 0 =0.0, B 0 =0. Figure 3b: Lng-time evlutin f the lng wave L with gru velcity mismatch V =, wave number f erturbatin = 0., and initial amlitudes A 0 =0.0, B 0 =0.

18 8 January 7, Ω I Figure 4a: he grwth rate fr mdulatinal instabilities with gru velcity mismatch V = and initial amlitudes A 0 =0., B 0 =0.0 L X Figure 4b: Lng-time evlutin f the lng wave L with gru velcity mismatch V =, wave number f erturbatin =, and initial amlitudes A 0 =0., B 0 =0.0

19 9 January 7, 006 Ω I Figure 5a: he grwth rate fr mdulatinal instabilities with gru velcity mismatch V = and initial amlitudes A 0 =0., B 0 =0. L X Figure 5b: Lng-time evlutin f the lng wave L with gru velcity mismatch V =, wave number f erturbatin =, and initial amlitudes A 0 =0., B 0 =0.

20 0 January 7, 006 L X Figure 5c: Lng time evlutin f the lng wave L with gru velcity mismatch V =, wave number f erturbatin = 0.5, and initial amlitudes A=0., 0 B=0. 0

21 January 7, 006 Ω I Figure 6a: he grwth rate fr mdulatinal instabilities with gru velcity mismatch V = 3 and initial amlitudes A 0 =0., B 0 =0. L X Figure 6b: Lng-time evlutin f the lng wave L with gru velcity mismatch V = 3, wave number f erturbatin = 3, and initial amlitudes A 0 =0., B 0 =0.

22 January 7, 006 Ω I Figure 7a: he grwth rate fr mdulatinal instabilities fr site disersins, with gru velcity mismatch V = and initial amlitudes A 0 =0., B 0 =0. L X Figure 7b: Lng-time evlutin f the lng wave L fr site disersins, with gru velcity mismatch V =, wave number f erturbatin =, and initial amlitudes A 0 =0., B 0 =0.

23 3 January 7, 006 A X B X Figure 8: Lng time evlutin f the shrt wave enveles A (t) and B (bttm) with arameters = λ = µ =, =, λ = 0.5, µ =, and gru velcity mismatch V =, wave number f erturbatin = 0.5, and amlitude A 0 =0., B=