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1 Ma-Planck-Institt für Mathematik in den Natrwissenschaften Leipzig Nmerical simlation of generalized KP type eqations with small dispersion by Christian Klein, and Christof Sparber Preprint no.: 2 26

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3 NUMERICAL SIMULATION OF GENERALIZED KP TYPE EQUATIONS WITH SMALL DISPERSION CHRISTIAN KLEIN AND CHRISTOF SPARBER Abstract. We nmerically stdy nonlinear dispersive wave eqations of generalized Kadomtsev-Petviashvili type in the regime of small dispersion. To this end we inclde general power-law nonlinearities with different signs. A particlar focs is on the Korteweg-de Vries sector of the corresponding soltions. version: Febrary 3, 26. Introdction One of the most important models for nonlinear dispersive waves is the + dimensional Korteweg-de Vries eqation (KdV) (.) t +6 + ɛ 2 =, describing shallow water waves, which essentially propagate in only one dimension, cf. [, 22] for frther details. In (.) we already introdced a scaling sitable for the so-called small dispersion limit where ɛ. To this end one starts from the nscaled (dimensionless) KdV model (.2) T +6 X + XXX =, where (T,X) are now considered as the natral length scales in this problem. Introdcing slowly varying soltions of the form = (ɛt, ɛx), <ɛ, plgging them into (.2) and denoting (t, ) =(ɛt, ɛx), we obtain the scaled eqation (.). Ths, in all what follows the notion of the coordinates refers to the slow scales t,. In order to take into accont also (weak) transverse effects in the propagation of shallow water waves, a natral generalization of the KdV eqation is the 2 + dimensional Kadomtsev-Petviashvili eqation (KP) ( (.3) t +6 + ɛ 2 ) + λ yy =, λ = ±, see, e.g., [, ] for a physical derivation. The case λ = corresponds to the so-called KP-I model, valid for strong srface tension, whereas λ = + is sally referred to as KP-II eqation, sed in the case of weak srface tension. Clearly, any y-independent soltion of (.3) is a soltion of (.), forming the so-called KdV sector. Both, the KdV and the KP eqation, are known to be completely integrable, ths comprising an infinite nmber of conservation laws []. Moreover, eplicit soltions for both of them can be obtained via the inverse scattering method [2, 9]. 2 Mathematics Sbject Classification. 35Q53, 74S25, 34E5. Key words and phrases. (modified) Korteweg-de Vries eqation, Kadomtsev-Petviashvili eqation, nonlinear dispersive models, modlation theory. This work has been spported by the APART research grant of C.S. fnded by the Astrian Academy of Sciences.

4 2 C. KLEIN AND C. SPARBER In this work we shall consider the following generalization of the KP eqation (gkp) ( (.4) t +6σ n + ɛ 2 ) + λ yy =, σ = ±, where n N allows for higher order nonlinearities as stdied in, e.g., [6]. Ths,if we consider y-independent soltions of (.4) we get a generalized KdV type model (gkdv) of the following form, (.5) t +6σ n + ɛ 2 =. For n = 2andσ = ±, eqation (.5) frnishes the focssing resp. defocssing modified KdV eqation [2]. Note however, that the case n = 2 for (.4) does not yield the modified KP eqation, which carries a frther nonlinear term and can be obtained as a higher order eqation within the so-called KP hierarchy [4]. In what follows we shall stdy (.4) in its evoltionary form, i.e. { t +6σ n + ɛ 2 + λ (.6) yy =, t= = φ(, y), where the anti-derivative is defined as a Forier mltiplier with the singlar symbol i/k (see [4, 7, 8] for several mathematical reslts on (.6) with n =). In this work we shall be interested in a nmerical stdy of soltions to (.4), and their corresponding KdV sector, in the regime ɛ. In particlar we are interested in the inflence of the considered higher order nonlinearities and their respective signs. For the classical KdV eqation (.), the limit ɛ has attracted lots of interest in varios branches of mathematics. Obviosly this is a highly singlar limiting regime which reqires particlar care. Nowadays, a rigoros asymptotic treatment of this problem has been established for the cases n = (classical KdV) and n =2(modifiedKdV),cf. [9, 6, 7, 2] and the references given therein. The qoted reslts heavily rely on inverse scattering techniqes and complete integrability, which in particlar implies that the case n>2 is still open. Let s also refer to [8, 2] for nmerical stdies of sch problems. On the other hand, an analogos comprehensive treatment for KP type models is an nsolved mathematical problem so far. To or knowledge, the only rigoros reslts in this direction are crrently available in [3, 5]. To shed some light on these qestions a nmerical stdy of the small dispersion KP eqation has been recently given by the athors in [3]. The present paper can be seen as a contination of this earlier work, taking into accont higher order nonlinearities. We hope that or stdies allow for frther insight in the role of dispersion in nonlinear wave eqations. 2. Remarks on the dispersionless limits The formal limit, as ɛ, of (.6) yields the dispersionless gkp eqation { t +6σ n + λ (2.) yy =, t= = φ(, y). However we do not epect this eqation to be a correct description of the dispersionless limit for all times. In particlar for y-independent soltions we find the following (generalized) Hopf type eqation (2.2) t +6σ n =, which is known to provide a reliable approimation of the limiting KdV soltion only p to some finite time t c. This can be seen by integrating eqation (2.2)

5 GENERALIZED KP EQUATION WITH SMALL DISPERSION 3 (implicitly) via the method of characteristics, i.e. (2.3) (t, ) =f(ξ), = ξ +6σtf n (ξ), for some initial condition (,) = f(). This constrction however only works p to the time [ ] σ (2.4) t c =min ξ 6 ξ (f n, (ξ)) where the minimm is determined for ξ given by (t, ) =f(ξ), i.e. as the maimm of f (ξ) for negative f (ξ). For times t>t c,thesoltion(t, ), as given by (2.3), becomes mlti-valed. Also, at t = t c the gradient of (t, ) w.r.t. R diverges, marking the onset of a shock wave in the corresponding weak soltion to (2.2). For small bt still nonzero ɛ, it is well known, that soltions of the gkdv eqation for t<t c will behave qalitatively as the corresponding soltion to the generalized Hopf eqation, the latter being a strong limit of the former as ɛ. For t t c, however, the soltions to the gkdv eqation will develop a zone of rapid modlated oscillations with wavelength of the order /ɛ. These oscillations conseqently smear ot the shock fronts visible in the weak soltion of (2.2). Moreover the oscillatory zone within the gkdv eqation roghly coincides with the region where the corresponding Hopf soltion is mlti-valed. As already pointed ot in [3] the analogos sitation in the KP case is less clear, althogh or nmerical stdies show several qalitative similarities. Essentially this is de to the fact that the dispersionless KP eqation cannot be integrated by the method of characteristics. Moreover it is not clear how to solve, in an appropriate way, the corresponding Whitham modlation eqations, whicharesedinthekdv case to identify the oscillatory zones and to constrct the asymptotic soltion there. In [3], we established nmerically that for t<t c the KP eqation converges to the corresponding dispersionless model with the rate O(ɛ 3/2 ). From the nmerical point of view we circmvented the problem of shock soltions within the limiting dispersionless model in [3] by sing a dissipative reglarization. More precisely we add to (2.) a small dissipative term of the form (2.5) ( t +6 n µ )+λ yy =. where <µ being some small real-valed parameter, sch that µ O(), as ɛ. Althogh (2.5) does no longer conserve the mass, i.e. (2.6) M[(t)] := d dy 2 (t,, y), R R we still epect that, as µ, the corresponding reglarized soltion tends to some kind of entropy soltion for the dispersionless eqation. More details on the nmerical treatment of (.4) of the considered eqations can be fond in [3] and in Appendi A below. 3. Nmerical stdy of small dispersion regimes As a concrete eample we shall consider the initial data given by (3.) φ δ = sech 2 (r), r := 2 + δy 2, where δ [, ] is a small deformation parameter. This choice of φ conseqently allows for a continos deformation of the KdV sector within the soltion of the

6 4 C. KLEIN AND C. SPARBER gkp model. Note that the initial data are rapidly decreasing as, y. Another reason for or initial condition is that φ δ is sbject to the constraint (3.2) dφ δ (, y) =, y R, R which trns ot to be a necessary condition for admissible KP initial data, cf. [3] for more details on this. In the present paper we will only consider the cases δ = (KdV) and δ =. For the discssion of other vales of δ, see[3]. 3.. Generalized Hopf eqation with dissipative reglarization. The stdy of a dissipative reglarization of the gkp eqation as in (2.5) allows on the one hand to identify regions where shocks form, and on the other hand, to nmerically approimate the corresponding entropy soltion. We illstrate this concept in the gkdv case. For eamples in the KP setting, see again [3]. First, consider the classical qadratic case n =andσ = with the initial data (3.). As can be seen from Fig., there is only a single shock front which forms roghly for t =.8. This is different from the sitation encontered in, e.g., the Figre. Soltion to the reglarized Hopf eqation (n = )for initial data (3.), µ =. and σ = attimet =.9. cbic case where n =2andσ =, as it is shown in Fig. 2. A first (reglarized) shock appears for t. in the region of negative initial data. The width of the shock front is of the order /µ wherewehavechosenµ =.. However, for t.22, a second shock front appears in the region of positive initial data. De to the dispersive reglarization for small bt non-zero ɛ, rapid oscillations are epected to smooth ot these shocks. As was shown in [8], these oscillations always appear for finite ɛ shortly before the time t c is reached. Conseqently we epect only a single oscillatory zone in the classical (qadratic) KdV setting to be discssed below gkdv eqation with small dispersion. LetsnowfocsontheKdV sector, i.e. δ = in (3.), choosing ɛ =.. We show the corresponding soltion to the classical KdV eqation, i.e. n = σ =, for several vales of t in Fig. 3. It can be seen that the first oscillations form at t.8. In this eample the oscillations are more prononced in the region of negative initial data. The analogos sitation is shown for n = bt σ = in Fig. 4. In this case the oscillations start mch earlier, roghly at time t.6. As epected there is only one oscillatory zone

7 GENERALIZED KP EQUATION WITH SMALL DISPERSION 5 t= t= t=.22 t= Figre 2. Soltion to the reglarized generalized Hopf eqation for several vales of t, as obtained from the initial data (3.), with µ =., n =2,andσ =. which is located between the etrema of the initial data. The oscillations are similar to those for negative initial data in the case σ =, bt mch more prononced. In the case n =2andσ = (again ɛ =.) the corresponding soltion can be seen in Fig. 5. Here, the oscillations appear mch earlier than in the classical case above. Indeed, the first oscillation is clearly visible at t.8. Again the oscillations are more prononced in the region of negative initial data. For σ =, the corresponding sitation is given in Fig. 6. This time however, the oscillations start later, roghly at time t.2. The oscillations in the region of negative initial data are similar to those in the case σ =. In contrast to the case n =, σ =, there are two oscillatory regions in this eample. The plots for the above soltions at time t =.4 are presented in one frame in Fig. 7. For higher nonlinearities, the sitation is qalitatively the same. For eample, in the case n =3,σ =, as shown in Fig. 8 for t =.4, the oscillations are similar to the ones in the case n =,σ =,cf. Fig. 3, bt they are smaller in amplitde. Note that the almost linear part of the soltion, appearing between the oscillatory regions in the KdV case, is strongly deformed in the present eample. Finally, the soltion for n =4andσ =attimet =.4 is shown in Fig. 9. Again, the sitation is qalitatively similar to the case n =2,σ = in Fig. 5, bt there a fewer oscillations. Ths the higher nonlinearity seems to sppress oscillations in both eamples.

8 6 C. KLEIN AND C. SPARBER.5 t=.5 t= t=.28.5 t= Figre 3. Soltion to the KdV eqation for initial data (3.) and ɛ =., σ = for several vales of t gkp eqation with small dispersion. In this section we will stdy soltions to the generalized KP eqations for the initial data (3.). We will restrict orselves to the case λ = andɛ =., i.e. the generalized KP-I eqation in the limit of small dispersion. The soltion for the classical KP-I eqation (i.e. for n =andσ =)attime t =.4 was already presented in [3]. It it shown Fig.. Note that tails towards have formed. The oscillations close to these tails are enhanced in comparison to the KdV case. On the other hand, the oscillations for negative are weaker than in the KdV case. In [3], this led s to the conclsion that the mass which is transferred to the KP tails leads to stronger gradients within the corresponding dkp eqation and ths to stronger oscillations in the KP-I eqation. The same sitation for σ = can be seen in Fig.. As in the KdV case, there are only oscillations in the region corresponding to negative initial data. In the case n = 2 and σ =, the corresponding soltion can be seen in Fig. 2. Again the oscillations are enhanced on the side, where the tails appear, as is more visible from the soltion on the -ais in Fig. 4. As can be seen in Fig. 3, there are no oscillations on the side of the tails in the case n =2andσ =, bt the tails are very strong. The relation to the gkdv case is more obvios from Fig. 4 from which it can be inferred that the oscillations on the opposing side of the tails are weakened.

9 GENERALIZED KP EQUATION WITH SMALL DISPERSION 7 t= t= t=.28 t= Figre 4. Soltion to the gkdv eqation with n =andσ = for initial data (3.) and ɛ =. for several vales of t. We show the above presented soltions on the -ais in Fig. 4 where it can be compared directly with Fig. 7 for the corresponding KdV soltions. A qalitative similarity is obvios, as are the differences which mainly show that the gradients and ths the oscillations are enhanced in the vicinity of the tails. The same tendency can also be observed for higher nonlinearities. In Fig. 5 we show the case n =3,σ =att =.4. With respect to the corresponding gkdv case in Fig. 8, the oscillations near the tails are enhanced, whereas those on the opposing side are weakened. This is even more obvios from Fig. 6 where the potential is shown on the -ais. The same holds tre for the case n =4,σ =as can be seen from Fig. 7 and Fig. 9. The corresponding soltion on the -ais is shown in Fig. 8. Appendi A. The nmerical method In this paper we discss nmerical soltions to initial vale problems to the generalized KP eqation (.6). We are interested in initial data which decay rapidly as, y (for eample Schwartz fnctions). This allows for the se of Forier spectral methods for the spatial coordinates. Ths we treat the problem in a periodic setting w.r.t. and y with periods 2π/L and 2π/L y, respectively. It is known that for sch initial data the KP soltion will develop tails which decay only algebraically in both spatial coordinates, as can be already seen from the Green s fnction of the linearized model, cf. [3]. These tails conseqently lead

10 8 C. KLEIN AND C. SPARBER t= t= t= t= Figre 5. Soltion to the gkdv eqation with n =2andσ = for initial data (3.) and ɛ =. for several vales of t. to a discontinity of the soltion at the bondaries of the domain which yields a Gibbs phenomenon. In addition, the periodicity condition implies the eistence of echoes for these tails. However, as discssed in more detail in [3], we can choose the domain via L and L y large enogh sch that neither the Gibbs phenomenon de to the tails nor the echoes inflence the oscillatory regions we are interested in. The accracy of the code will be controlled by the conservation of the mass (2.6), a featre which is not implemented directly in the code. Ths mass conservation presents a strong test for the accracy of or method. We denote by û(t, k,k y ):= (t,, y)e i(k+kyy) d dy, R 2 the Forier transform of (t) w.r.t. and y, andbyûn+ the Forier transform of n+. Then we get from eqation (.6) (A.) t û + i6k σ ɛ 2 ik 3 n +ûn+ û +iλ k2 y û =. k This is eqivalent to ( ( (A.2) t ep i(k 2 y /k ɛ 2 ik 3 )t) û ) +ep ( i(ky 2 /k ɛ 2 ik 3 )t) i6k σ =. n +ûn+ The se of an integrating factor elliminates a stiff part in the eqations and allows for the se of eplicit schemes with larger time steps. In the present work, we

11 GENERALIZED KP EQUATION WITH SMALL DISPERSION 9 t= t= t=.28 t= Figre 6. Soltion to the gkdv eqation with n =2andσ = for initial data (3.) and ɛ =. for several vales of t. se a forth order Rnge Ktta method. The code is stable for time steps of the order t = 4 with 248 modes in the gkdv eamples, and t =8 5 with N = 248, N y = 28 modes (L = L y = 5). With these vales we achieve relative mass conservation better than 4 for the gkp comptations and better than 8 in the gkdv cases. References. M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evoltion eqations and inverse scattering, Cambridge Univ. Press, Cambridge (99). 2. J. C. Aleander, R. L. Pego, and R. L. Sachs, On the transverse instability of solitary waves in the Kadomtsev-Petviashvili eqation, Phys. Lett. A 226 (997), V. N. Bogaevskiĭ, On Korteweg-de Vries, Kadomtsev-Petviashvili, and Bossinesq eqations in the theory of modlations, Zh. Vychisl. Mat. i Mat. Fiz. 3 (99), no., 487 5; translation in U.S.S.R. Compt. Math. and Math. Phys. 3 (99), no. 5, M. Boiti, F. Pempinelli, and A. Pogrebkov, Properties of soltions of the Kadomtsev- Petviashvili I eqation, J. Math. Phys. 35 (994), isse 9, A. de Board and Y. Martel, Noneistence of L 2 -compact soltions of the Kadomtsev- Petviashvili II eqation, Math. Annalen. 328 (24), A. de Board and J. C. Sat, Symmetries and decay of the generalized Kadomtsev- Petviashvili solitary waves, SIAM J. Math. Anal. 28 (997), no. 5, J. Borgain, On the Cachy problem for the Kadomtsev-Petviashvili eqation, Geom. Fnct. Anal. 3 (993), no.4, T. Grava and C. Klein, Nmerical soltion of the small dispersion limit of Korteweg de Vries and Whitham eqations, preprint,arxiv:math-ph/5.

12 C. KLEIN AND C. SPARBER.5 n=, σ=.5 n=, σ= n=2, σ=.5 n=2, σ= Figre 7. Soltion to the gkdv eqations for initial data (3.) and ɛ =. attimet =.4 for several vales of n and σ Figre 8. Soltion to the gkdv eqation for initial data (3.) and ɛ =., n =3andσ =attimet = T. Grava and Fei-Ran Tian, The generation, propagation, and etinction of mltiphases in the KdV zero-dispersion limit, Comm. Pre Appl. Math. 55 (22), no. 2, R. S. Johnson, The classical problem of water waves: a reservoir of integrable and nearly integrable eqations, J. Nonl. Math. Phys. (23)

13 GENERALIZED KP EQUATION WITH SMALL DISPERSION Figre 9. Soltion to the gkdv eqation for initial data (3.) and ɛ =., n =4andσ =attimet =.4. Figre. Soltion to the gkp-i eqation with n =andσ = at t =.4, for initial data (3.) and ɛ =... B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl. 5 (97), A. M. Kamchatnov, A. Spire, and V. V. Konotop, On dissipationless shock waves in a discrete nonlinear Schrödinger eqation, J. Phys. A37 (24), no. 2, C. Klein, C. Sparber, and P. Markowich, Nmerical stdy of oscillatory regimes in the Kadomtsev-Petviashvili eqation, preprint,arxiv:math-ph/ B. G. Konopechenko and V. G. Dbrovsky, Inverse spectral transform for the modified Kadomtsev-Petviashvili eqation, Std. Appl. Math. 86 (992), no. 3, I. M. Krichever, The averaging method for two-dimensional integrable eqations, Fnktsional. Anal. i Prilozhen. 22 (988), no. 3, 37 52; translation in Fnct. Anal. Appl. 22 (989), no. 3, P. D. La and C. D. Levermore, The small dispersion limit of the Korteweg de Vries eqation I,II,III, Comm. Pre Appl. Math. 36 (983), , ,

14 2 C. KLEIN AND C. SPARBER Figre. Soltion to the gkp-i eqation with n =andσ = att =.4, for initial data (3.) and ɛ =.. Figre 2. Soltion to the gkp-i eqation with n =2andσ = at time t =.4, for initial data (3.) and ɛ =.. 7. C. D. Levermore, The hyperbolic natre of the zero dispersion KdV limit, Comm. Part. Diff. Eq. 3 (988), no. 4, L. Molinet, J. C. Sat, and N. Tzvetkov, Well-posedness and ill-posedness reslts for the Kadomtsev-Petviashvili-I eqation Dke Math. J. 5 (22), no. 2, S.Novikov,S.V.Manakov,L.P.Pitaevskii,andV.E.Zakharov,Theory of Solitons: The Inverse Scattering Method Springer Monographs in Contemporary Mathematics, Springer F.R.Tian,Oscillations of the zero dispersion limit of the Korteweg de Vries eqations, Comm. Pre App. Math. 46 (993),

15 GENERALIZED KP EQUATION WITH SMALL DISPERSION 3 Figre 3. Soltion to the gkp-i eqation with n =2andσ = att =.4, for initial data (3.) and ɛ =.. n=, σ= n=, σ= n=2, σ= n=2, σ= Figre 4. Soltion to the gkp eqations for initial data (3.) and ɛ =. attimet =.4 for several vales of n and σ.

16 4 C. KLEIN AND C. SPARBER Figre 5. Soltion to the gkp-i eqation with n =3andσ = at t =.4, for initial data (3.) and ɛ = Figre 6. Soltion to the gkp-i eqation with n =3andσ = at t =.4 onthe-ais, for initial data (3.) and ɛ =.. 2. M. Wadati, The modified Korteweg-de Vries eqation, J. Phys. Soc. Japan 34 (973), G. B. Whitham, Linear and nonlinear waves, Wiley (974). Ma Planck Institte for Mathematics in the Sciences address: klein@mis.mpg.de Wolfgang Pali Institte Vienna & Faclty of Mathematics, Vienna University, Nordbergstraße 5, A-9 Vienna, Astria address: christof.sparber@nivie.ac.at

17 GENERALIZED KP EQUATION WITH SMALL DISPERSION 5 Figre 7. Soltion to the gkp-i eqation with n =4andσ = at t =.4, for initial data (3.) and ɛ = Figre 8. Soltion to the gkp-i eqation with n =4andσ = at t =.4 onthe-ais, for initial data (3.) and ɛ =..