Non-Analytic Solutions of Nonlinear Wave Models

Transcription

1 Non-Analtic Solutions of Nonlinear Wave Models Yi A. Li School of Mathematics Universit of Minnesota Minneapolis, MN Peter J. Olver School of Mathematics Universit of Minnesota Minneapolis, MN U.S.A. Philip Rosenau School of Mathematical Sciences Tel Aviv Universit Tel Aviv ISRAEL Abstract. This paper surves recent work of the coauthors on nonanaltic solutions to nonlinear wave models. We demonstrate the connection between nonlinear dispersion and the eistence of a remarkable variet of nonclassical solutions, including peakons, compactons, cuspons, and others. Nonanalticit can onl occur at points of genuine nonlinearit, where the smbol of the partial differential equation degenerates, and thereb provide singularities in the associated dnamical sstem for traveling waves. We propose the term pseudo-classical to characterize such solutions, and indicate how the are recovered as limits of classical analtic solutions. Supported in part b NSF Grant DMS and BSF Grant Supported in part b BSF Grant Jul 8, 1998

2 Historicall, the mathematical stud of phsical phenomena has proceeded in three phases. The introductor phase relies on linear models, and is a necessar prerequisite to the deeper understanding of the actual nonlinear regime. Thus, for instance, the unidirectional linear wave equation u t + u = provides the prototpe of linear hperbolic waves. An initial data propagates completel unchanged with speed c = 1. (For simplicit, we adopt units in which all phsical parameters are 1 throughout this surve.) Restricting our attention to conservative sstems, the net interesting case is the dispersive wave equation u t + u + u =. Since different wave numbers propagate at different speeds, localized initial data will disperse in time, albeit with conservation of energ. In certain phsical regimes, these simple linear models provide the satisfactor eplanations of the phsicall observed wave propagation. Inspired b the dramatic observations of J. Scott Russell, [26], Boussinesq initiated a sstematic derivation of nonlinear wave models that could describe the weakl nonlinear regime, which is the second of the three historical phases. The most celebrated of Boussinesq s models is the equation u t + u + (u 2 ) + u =, (1.1) which initiall appears in both [4; eq. (), p. 77] and [5; eqs. (28, 291)]. This well-studied equation, now known as Korteweg-de Vries (KdV) equation in honor of its rediscover, [14], over two decades later, models the weakl nonlinear regime in which nonlinear effects are balanced b dispersion. As recognized b Boussinesq, its solitar wave solutions are described b analtic sech 2 functions that have eponentiall decaing tails. The discover of the inverse scattering granted to the KdV a status of a paradigm, for it was the first equation shown to be completel integrable, and so a complete analsis can be effected, cf. [9]. Its solitar waves are nowadas referred to as solitons since the interact cleanl, ecept for a phase shift. It admits two distinct, compatible Hamiltonian structures, and hence an infinite hierarch of local higher order smmetries and associated conservation laws, cf. [2]. Moreover, it is now understood that the KdV and its integrable cousins are not just esoteric equations describing particular phenomena, but ubiquitous descriptions of a weakl nonlinear regime wherein linear dispersion counterbalances the governing nonlinearit to form steadil propagating patterns. Numerical studies indicate that much of the basic weakl nonlinear phenomenolog associated with the KdV and other integrable models is shared with man nonintegrable models in the same phsical regime. A good eample is the BBM or regularized long wave model, [2], u t + u + uu u t =. (1.2) Like the KdV, (1.2) also admits analtic solitar wave solutions, and localized initial data is observed to break up into a finite collection of solitar waves, [12]. Unlike the KdV, (1.2) has solitar waves of both positive and negative elevation; the positive waves interact almost cleanl there is a ver small but noticeable dispersive effect, [, 19]. (On the other hand, collisions between positive and negative waves are quite dramaticall inelastic, [25].) We are still a long wa from being able to establish rigorousl these numerical observations et, one might venture that the basic phenomenolog of wave mechanics in the weakl nonlinear regime is fairl well understood. 2

3 The final, and most difficult, phase in the modeling process is the transition to full nonlinear models. Linear and weakl nonlinear models will onl predict the phsical behavior of weak disturbances indeed, the are tpicall derived b rescaling the full phsical sstem to include one or more small parameters, epanding in a power series, then truncating to eliminate some or all of the higher order terms. However, in a genuinel nonlinear regime, nonlinearit plas a dominant role rather than being a higher order correction, and so will introduce new complications, new phenomena, and require new analtical techniques. Weakl nonlinear models cannot describe the full nonlinear, real world phenomena such as wave breaking, shocks, waves of maimal height, [27], large amplitude disturbances, and so on. The problem of how one derives realistic nonlinear models for such highl nonlinear effects is of fundamental importance but has not as et received the attention it deserves. One tpe of full nonlinear phenomena shock waves in gas dnamics and similar first order hperbolic sstems alread has a long histor. The simplest case is the inviscid Burgers or dispersionless KdV or Riemann equation u t + (u 2 ) =, obtained b replacing the linear convective term in the unidirectional linear model b a full nonlinear version. The hodograph transformation (, t, u) (u, t, ) will linearize the equation, but the analtic solutions to the transformed equation revert to multipl-valued solutions of the Riemann equation. For the gas dnamics and continuum mechanics applications, a phsical wave cannot be represented b a multipl-valued solution, and one must replace it with a discontinuous, nonanaltic shock wave, the placement of the shock governed b a suitable entrop condition, cf. [29]. On the other hand, if one uses this equation to model surface waves, then multipl-valued solutions are phsicall acceptable the nonphsical solutions are when the curve crosses itself and the weak characterization is no longer applicable. Therefore, the analtical characterization of acceptable solutions depends on the phsics underling the model. Just as linear dispersion dramaticall alters the hperbolic traveling wave, so one epects nonlinear dispersion to be capable of causing deep qualitative changes in the nature of dispersive phenomena. Mimicking the passage from the linear hperbolic equation to the nonlinear Riemann equation, one is naturall led to introduce a simple nonlinearl dispersive model of the form u t ± (u 2 ) + (u 2 ) =, (1.±) known as the K(2, 2) equation. The ± sign plas an important role in the structure, and we shall refer to (1.+) as the focusing model, while (1. ) is the defocusing version. Unlike the KdV model, (1.1), the K(2, 2) equation will model dispersive waves in the full nonlinear regime. It is one of a large class of simple nonlinearl dispersive wave models, incorporating different powers of u in the nonlinear terms, first proposed in [22, 24]. We shall use (1.) as our initial paradigm for the structures that ma appear in the full nonlinear regime, and then indicate additional, stranger denizens, of the nonlinear terra incognita. At this point, it is important to distinguish between a full nonlinear model, in which the highest order terms in the partial differential equation are nonlinear (tpicall quasilinear), and the points in phase space where the nonlinearit is actuall manifested. These

4 points will be determined b the complete degeneration of the smbol of the partial differential equation. For eample, the full nonlinear K(2, 2) equation has degenerate smbol when u =, and these will be the points of genuine nonlinearit in this case. A crucial difference between the linear or weakl nonlinear models, versus the full nonlinear models like (1.) is that the latter admit nonanaltic solutions. However, unlike the abrupt discontinuities of shock waves in the first order hperbolic regime, these discontinuities are more subtle, tpicall occurring at higher order derivatives; moreover, standard eistence theorems, e.g., Cauch Kovalevskaa, indicate that one can onl epect such singularities at the points of genuine nonlinearit in the phase space. This is in contrast to the nonanaltic shock solutions of nonlinear hperbolic models, which can occur at an point in the phase space, but whose implementation requires additional phsical assumptions, e.g., entrop conditions. The first observed manifestations of nonanalticit were compactons solitar wave solutions of compact support, [24], and peakons solitar wave solutions having peaks or corners, [6]. Both tpes of solutions appear in the basic K(2, 2) equation, as well as other nonlinear models of this tpe. How does one characterize such nonclassical solutions? Motivated b the entrop characterization of shock waves, the first idea that might come to mind is to view them as weak solutions in the traditional sense, [29]. Although this is a valid formulation, and tpicall does serve to distinguish them, both compactons and peakons are solutions in a considerabl stronger (although not quite classical) sense. In fact, the are piecewise analtic, satisfing the equation in a classical sense awa from the points of genuine nonlinearit; moreover, each term (or, possibl, certain combinations of terms) in the equation is well-defined at the singularities. The remain solutions even when subjected to nonlinear changes of variables a process that has disastrous consequences for the weak formulation. Consequentl, one does not need to introduce an artificial (or phsical) entrop condition to prescribe the tpe of singularit. In [17, 18], we proposed the term pseudo-classical solution to distinguish such non-analtic solutions. Yet another characterization is based on the observation that these solutions can be realized as the limits of classical, analtic solutions. Note that (1.) is genuinel nonlinear when the undisturbed (asmptotic) height of the solution is lim u =, but becomes weakl nonlinear at nonzero amplitudes. Therefore, the solutions with lim u = α are analtic, and, in favorable situations, will converge to a solution as α. Alternativel, one can replace u b u+α in (1.) the parameter α then measures linear dispersion. The convergence of analtic solitar wave solutions under vanishing linear dispersion or asmptotic height distinguishes those non-analtic solitar wave solutions which are genuine pseudo-classical solutions. Since the analsis of general solutions is much too difficult, we shall now concentrate on the traveling wave solutions to the equations under consideration. Substituting the basic ansatz u = φ( ct) for a traveling wave of speed c reduces the partial differential equation in, t to an ordinar differential equation, which can then be analzed b dnamical sstems methods. The classical, analtic traveling wave solutions are given b classical analtic orbits to the dnamical sstem: solitar wave solutions correspond to homoclinic orbits at a single equilibrium point; periodic waves come from periodic orbits; while heteroclinic orbits connecting two equilibrium points ield analtic kink solutions. 4

5 Awa from singular points of the dnamical sstem one can appl standard eistence and analtic dependence results to prove that the corresponding solutions are analtic. Therefore, nonanaltic behavior will onl arise at the singular points, which are precisel the points of genuine nonlinearit of the original partial differential equation. The admission of nonanaltic traveling wave solutions indicates that classical dnamical sstems analsis must be significantl broadened to eplain how a classical analtic orbit can give rise to a nonanaltic solution! In the case of the K(2, 2) model (1.), the traveling wave solutions satisf a threedimensional dnamical sstem cφ ± (φ 2 ) + (φ 2 ) =. (1.4±) Each point (α,, ) ling on the φ-ais in the (φ, φ, φ ) phase space provides a fied point for the dnamical sstem (1.4) representing a constant (undisturbed) solution of height α. The equation (1.4) can be integrated twice b standard methods, leading to φ 2 (φ ) 2 = 1 4 φ4 + 1 cφ + aφ 2 + b or (φ ) 2 = b φ 2 + a + c φ 1 4 φ2. (1.5) where a, b are the integration constants. If b then the potential on the right hand side of (1.5) has a pole of order 2 at φ = ; on the other hand, if b = then the pole disappears nevertheless, φ = remains a singular point of the original form of the equation, and one still anticipates that nonanaltic behavior can occur when an orbit crosses the singular ais in the phase plane. Thus, even when the singular pole is no longer present owing to a cancellation of factors, its nonanaltic legac will remain. Note particularl that the singular points of the dnamical sstem arise from the points where the smbol of the original partial differential equation (1.) degenerates. For simplicit, we onl consider the case c > of positive wave speed. In the focusing case (1.4+), for each < α < c 2, there is a homoclinic orbit at the fied point (α,, ) that represents an analtic solitar wave solution with asmptotic height α. Since each homoclinic orbit can be obtained as the limit of periodic orbits, the solitar wave solution is the limit of periodic traveling wave solutions as the period becomes infinite. When α =, the origin is a singular point for the dnamical sstem (1.4), which implies that the associated homoclinic orbit is traversed in finite time. Therefore, the corresponding traveling wave solution is no longer analtic, but has compact support. In the present case, the dnamical sstem can be integrated eplicitl, leading to the compacton 4c ct cos2, ct 2π, φ c ( ct) = 4 (1.6), ct > 2π, which can be realized as the limit of solitar waves solutions as their asmptotic height α. (See [17, 18] for the basic method of proof of such results.) In the defocusing case (1.4 ), the fied point (α,, ) has a homoclinic orbit and corresponding analtic solitar wave solution onl when α > 2c. At the limiting point α = 2c the homoclinic orbit passes through the singularit, and the result is a nonanaltic traveling wave that has a 5

6 corner at its crest φ =, known as a peakon, which can also be realized as a limit of analtic solitar wave solutions. There is also a closed-form formula for the peakon: φ p ( ct) = 2c ( e ct 2 1 ). (1.7) The peakon solution corresponds to the particular case of (1.5) where the integration constant b =, and so the pole is canceled out, but, as discussed above, the φ = singularit remains and permits the non-smooth corner. φ -2c/ φ -2c/ φ α α φ φ 4c/ 4c/ φ Fig. 1. Trajectories and graphs of the peakon, loopon and compacton 6

7 solutions of the K(2, 2) equation. 7

8 When α >, the solutions that pass through the singular ais φ = take the form of multipl-valued looped solitar waves loopons. Loopons first appeared in an integrable model of [28], which, in contrast to the K(2, 2) equation, required a changeable sign in the equation. One can justif such solutions either b skewing the coordinate aes, replacing b = + kφ, or b parametrizing the graph b arc length instead of the horizontal coordinate. As with the Riemann equation, their admissibilit as phsical solutions depends on the phsical sstem being modeled; unlike breaking surface waves, it seems unlikel that these could emerge from standard initial data. We shall defer the detailed analsis of these loopons until a subsequent publication. In Figure 1, we displa the phase-plane portrait and the corresponding solution for the peakon, compacton and loopon solutions. Note that the phase plane aes are labeled b φ, φ and the curves represent the integral curves of the dnamical sstem (1.5) with appropriate integration constants a, b. The reader should compare these with the more standard phase portrait for the KdV equation in that case, changing the integration constants merel translates the picture; in contrast, here the change of integration constants fundamentall alters the potential, allowing new families of solutions to emerge. The K(2, 2) eample alread indicates the possibilit of significantl more interesting phenomena in the full nonlinear regime. An even richer eample is provided b the K(, ) equation u t ± (u ) + (u ) =. (1.8±) The resulting dnamical sstem for its traveling wave solutions u = φ( ct) is twice integrated, with constants a, b, ielding (φ ) 2 = b φ 4 + a φ + c φ2. (1.9) The occurrence of two different tpes of poles at the singularit φ = alread demonstrates that the K(, ) equation has, potentiall, a much richer structure. Let us indicate the tpes of solutions associated with different fied points (φ, φ ) = (α, ) in the phase space. Consider first the focusing case (1.8+). For positive wave speeds c >, if < α < c, there is an analtic solitar wave of elevation having asmptotic height α. As α these solitar waves converge to a compacton c ct cos, ct π φ c ( ct) = 2 2,, ct > π. (1.1) 2 Even though (1.1) has discontinuous first derivatives, its cube is C 2, and so φ c is a pseudo-classical (and hence weak) solution to (1.8+). In addition, when < α < c, the homoclinic tails have the singular φ ais as an asmptote, and hence there is a corresponding solution which attains an infinite slope in a finite time. But, as indicated in the phase portrait in figure 2, the two points of infinite slope can be connected through an analtic solution. The resulting nonanaltic solution forms a smmetric wave of depression that is analtic ecept for two points of infinite slope, called a tipon, [2]. In Figure 2, 8

9 the solid curve in the phase portrait is the homoclinic orbit that gives rise to the analtic solitar wave, while the dashed curve indicates the homoclinic tails and connecting orbit that provide the nonanaltic tipon. φ α α φ Fig. 2. Trajectories and graphs of the analtic and tipon solitar wave solutions of the K(, ) equation. Further analsis leads to the following sequence of solutions as we move through the fied points (α,, ): (a) α = a compacton. (b) < α < c analtic solitar waves of elevation and solitar tipons of depression. As α the solitar waves converge to the compacton, while the amplitudes of the solitar tipons decrease to zero. On the other hand, when α c, the analtic solitar waves shrink to zero while solitar tipons produce a solitar tipon. (c) α = c there is no analtic solitar wave, but a solitar tipon of depression remains. (d) c < α < c a famil of analtic periodic waves and a second famil of periodic tipons, which are even, periodic pseudo-classical solutions having isolated points of infinite slope. As the period of the waves increases to, the converge, respectivel, to the solitar wave and the solitar tipon at the complementar fied point α = 1 2 ( α + 4c α 2 ). As α c from the right, the analtic periodic waves converge to zero while the periodic tipons converge to the solitar tipon. As α c, the analtic periodic waves converge to the compacton at α =, whereas the periodic tipons shrink to zero. (e) α = c analtic periodic waves, which converge to the compacton at α = as the period becomes infinite. (f ) α > c a famil of periodic waves that begin as analtic, then, at a critical period converge to a periodic cuspon (also called a billiard solution because the cusps are all upside down touching the real ais) and then become periodic tipons. The defocusing case (1.8 ) is also interesting. For c > there are no periodic or solitar traveling waves at all. For c <, the following sequence of solutions successivel 9

10 emerges as we move through the fied points (α,, ): (a) < α < c analtic periodic waves. (b) α = c no solitar or periodic waves. (c) c < α < c 2 a unique analtic solitar wave of depression for each α. As α c 2, the solitar waves converge to a cuspon, which is a wave of depression having an infinite slope at its crest. (d) α = c 2 the homoclinic orbit reaches the singularit φ = and becomes a cuspon solution. (e) c 2 < α < c the cuspon mutates into a tipon (of depression) whose width becomes larger and larger. (In the K(, ) equation, we obtain tipons instead of the K(2, 2) loops because the associated potential has a pole of order 4 here.) (f ) α = c two tipon kinks, each analtic ecept for a single point of infinite slope. (g) α > c no solitar or periodic waves. Unlike the KdV equation, the general K(m, n) equations are not integrable. For eample, when m = n the admit onl four local conservation laws. Moreover, a zeromass ripple is left after a collision of, sa, two compactons, although their shape remains the same after the collide, [24]. This robustness cannot be eplained in terms known to us from the conventional soliton theor. There are, however, a wide variet of integrable nonlinearl dispersive sstems, and admit analtic multi-soliton solutions. Historicall, the first work in this direction was b Wadati et. al., [28], on cusped and looped solitons. Presentl, the most well-studied is the Camassa-Holm equation u t + u νu t + (u 2 ) ν(u 2 ) + ν(u 2 ) =, (1.11) which was derived as a model for ocean dnamics, [6, 7], and shown to be integrable b inverse scattering; see also [1, 8]. The bihamiltonian structure of (1.11) had, in fact, been first written down (albeit with an error in one of the coefficients) b Fuchssteiner, [11; (5.)]. The equation for traveling waves u = φ( ct) becomes, upon two integrations, ( c ) ν 2 φ (φ ) 2 = ( c 2 φ ) [ (φ ) ] 2 + a + b, (1.12) where a, b are the integration constants. In contrast to the dnamical sstem (1.5), the location of the singularit is not universal, but depends on the wave speed; this is reflected in the fact that the characteristic equation ξ 2 τ + 2uξ = for the original partial differential equation (1.11) has solution-dependent degeneracies, and so the points of genuine nonlinearit are solution-dependent. It is worth re-emphasizing that both cases are fundamentall different from what occurs in the formation of shock waves or more general weak solutions, whose discontinuities are not at all associated with points of genuine nonlinearit. In the focusing case ν = +1, the solitons are peakons, φ p ( ct) = ( c ) e ct 1 4, (1.1) This has not, in fact, been rigorousl established, but seems almost certain. 1

11 which can be realized as limits of analtic solitar wave solutions associated with fied points (α, ) in the phase plane as α 1 4 from the right. When α < 1 4 or α > 1 c, the 2 solutions are cuspons. As α 1 4 from the left, the cusps suddenl open up into a peak in the limit, so the solitar waves, peakon and cuspons form a single famil of solutions. In the defocusing case ν = 1, the singular point α = c 2 gives rise to a compacton 1 φ c ( ct) = 2 c ( ) c cos 2 ct, ct π, 2, ct π, (1.14) which is a limit of analtic solitar wave solutions as the asmptotic height goes to zero. However, in this case, (1.11) is not well-posed. See [17, 18] for details. The Camassa-Holm equation (1.11) can be associated to the Korteweg-deVries equation (1.1) b a process called trihamiltonian dualit, which was applied in [1, 21] to sstematicall additional nonlinearl dispersive, integrable sstems associated with man other classical weakl nonlinear soliton equations. A particularl interesting eample is the generalized Boussinesq sstem v t + 2w t 2v t = w v + ( 1 2 v2 + vw) ( 1 2 v2 ), w t + 2w t = w + (vw w2 + vw ), (1.15) that forms the integrable dual to a Boussinesq sstem arising in shallow water theor that was studied b Whitham, [29], and solved b inverse scattering methods in [1, 15]. Let us summarize the tpes of solutions found in [16] for the case c < 1, focusing on novel features not seen in the preceding eamples. The dnamical sstem for traveling waves v = φ( ct), w = ψ( ct) is cφ 2cψ + 2cφ = ψ φ + ( 1 2 φ2 + φψ φφ ), cψ 2cψ = ψ + ( φψ ψ2 + φψ ). (1.16) The fied points of (1.16) all lie in the (φ, ψ) plane; moreover, the line φ = 2c 1 consists of singular points. Let Γ denote the triangle formed b the three lines φ = 2c 1, ψ = and ψ = 1 φ. Let Ω be the inscribed ellipse defined b the equation φ 2 + φψ + ψ 2 + 2cφ ψ + c 2 =. The curves Γ and Ω divide the (φ, ψ) plane into regions containing fied points with different properties: (a) inside the ellipse Ω periodic traveling wave solutions, (b) between Ω and Γ solitar wave solutions, (c) on the line segments ψ = or ψ = 1 φ of Γ two kink solutions. 11

12 ψ (1,) (-c, ) (-2c-1,) φ Γ Ω (-2c-1,2(c+1)) Fig.. The phase plane for the Boussinesq sstem. The most interesting point is the verte V = (1, ) of Γ, which corresponds to an infinite famil of different nonanaltic solutions! Each of these solutions can be realized as a limit of analtic solitar wave solutions associated with points on the interior of Γ, but the limiting solution depends on how the interior points converge to V. More specificall, let P be a curve inside Γ (but outside Ω) which terminates at V. The solitar wave solutions (φ a, ψ a ) corresponding to a P will converge to a nonanaltic traveling wave (φ, ψ ) as a V, but this solution depends on the asmptotics of the curve P at V. For eample, if P is not tangent to Γ as it approaches V, then the solitar wave solutions converge to a peakon of the form φ p (ξ) = 1 2(c + 1)e ξ /2, ψ p (ξ) = (c + 1)e ξ /2 (sign(ξ) + 1), where ξ = ct. (1.17) On the other hand, for the path defined b the equation ψ(ψ + φ 1) + e 2λ/(φ 1) =, which tangent to the sides of Γ, the solitar wave solutions converge to a solution that we call a mesaon owing to its flat top; the width 2µ = 2λ/(c + 1) of the top is governed b the parameter λ; in the zero-width limit µ, the mesaon becomes a peakon. There are two different versions, depending on the branch of the path chosen b the quadratic equation (1.17); one of these is ( ) 1 2(c + 1)e (µ+ξ)/2,, ξ < µ, (φ m, ψ m )(ξ) = ( 2c 1, ) ξ < µ, (1 2(c + 1)e (µ ξ)/2, 2(c + 1)e (µ ξ)/2 ), ξ > µ. (1.18) Note that the mesaon (1.18) is constant at the fied point ( 2c 1, ) of the dnamical sstem, which lies on the line of singularities. (The other mesaon is constant at the second 12

13 fied point ( 2c 1, 2c+2) on the singular line.) The occurrence of fied points which are also singularities is the ke propert that allows the peakon to open up into a flat topped mesaon of arbitrar width. Indeed, in contrast, the line of singularities of the Camassa- Holm dnamical sstem (1.12) contains no embedded fied points, and this prevents the Camassa-Holm peakons from opening up. If our path P coincides with a side of the triangle, the associated analtic kink solutions converge to et another nonanaltic solution at V, called a rampon or semi-compact kink, of which an eample is (φ r, ψ r )(ξ) = { ( 2c 1, ), ξ < 2 log( 2(c + 1)), ( e ξ/2 + 1, e ξ/2), ξ > 2 log( 2(c + 1)). (1.19) Finall, points on the singular line φ = 2c 1 admit compactons which are found as limits of solitar waves; an eample is 2 b(b 2c 2) cosh ξ + 1, ξ < µ, φ c (ξ) = 2 (1.2) 2c 1, ξ > µ, where µ = 2 cosh 1 c 1 determines the support of the compacton. b(b 2c 2) c= b -2 Fig. 4. Graphs of the peakon, mesaon, rampon and compacton solutions of the dual Boussinesq sstem. Figure 4 shows the different tpes of nonanaltic solitar wave solutions admitted b (1.15). From left to right, the show a peakon, a mesaon, a rampon and a compacton. The solid line is v = φ, while the dashed line shows w = ψ. Note particularl that w is not 1

14 continuous it jumps back to zero at a certain point in the first three cases. Nevertheless, the solution is still quasi-classical and does not require an sort of entrop condition to fi the position of the discontinuit. It is clearl beond the capacit of the classical dnamical sstems theor to eplain this phenomenon, and so passage to the full nonlinear regime introduces new and unepected phenomena even in such well plowed fields as ordinar dnamical sstems! These eamples provide concrete evidence that both the integrable and nonintegrable, nonlinearl dispersive eamples treated to date are merel the tip of the proverbial iceberg. Even though, the nature of the nonanalticit changes in each case, these differences are technical, showing merel the different facets of nonlinearit. All in all, we have not even begun to understand the nature of the nonlinear interactions that govern these equations. References [1] Beals, R., Sattinger, D.H., and Szmigielski, J., Acoustic scattering and the etended Korteweg devries hierarch, preprint, Universit of Minnesota, [2] Benjamin, T.B., Bona, J.L., and Mahone, J.J., Model equations for long waves in nonlinear dispersive sstems, Phil. Trans. Ro. Soc. London A 272 (1972), [] Bona, J.L., Pritchard, W.G., and Scott, L.R., Solitar wave interaction, Phs. Fluids 2 (198), [4] Boussinesq, J., Théorie des ondes et des remous qui se propagent le long d un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl. 17 (2) (1872), [5] Boussinesq, J., Essai sur la théorie des eau courants, Mém. Acad. Sci. Inst. Nat. France 2 (1) (1877), [6] Camassa, R., and Holm, D.D., An integrable shallow water equation with peaked solitons, Phs. Rev. Lett. 71 (199), [7] Camassa, R., Holm, D.D., and Hman, J.M., A new integrable shallow water equation, Adv. Appl. Mech. 1 (1994), 1. [8] Constantin, A., and McKean, H., The integrabilit of a shallow water equation with periodic boundar conditions, preprint, Courant Institute, [9] Drazin, P.G., and Johnson, R.S., Solitons: An Introduction, Cambridge Universit Press, Cambridge, [1] Fokas, A.S., Olver, P.J., and Rosenau, P., A plethora of integrable bi-hamiltonian equations, in: Algebraic Aspects of Integrable Sstems: In Memor of Irene Dorfman, A.S. Fokas and I.M. Gel fand, eds., Progress in Nonlinear Differential Equations, vol. 26, Birkhäuser, Boston, 1996, pp

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