Who cares about integrability?

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  • Physica D 51 (1991) 343-359 North-Holland

    Who cares about integrability?

    Harvey Segur Program in Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, USA

    Problems that admit solitons and are completely integrable have been analyzed in detail since their discovery by Zabusky and Kruskal 25 years ago. Their study has influenced the development of both mathematics and physics during that time. The questions addressed here are:

    (i) Has this subject run its natural course? (ii) Will it significantly influence the nonlinear science that develops over the next decade?

    Supporting examples will be chosen from several disciplines, but especially from the study of water waves.

    I. Introduction (b) The nonlinear Schr6dinger equation,

    Nonlinear partial differential equations that admit solitons and are completely integrable are of interest mathematically because they possess a rich mathematical structure. Rarely do nonlinear equations admit so much structure as is present in all "soliton equations". Some of these equa- tions are also of interest because they model various physical phenomena. This paper reviews briefly the current status of integrable equations, and attempts to identify those aspects of the subject that might be important over the next decade. There is still no adequate definition of what a "soliton equation" is, and I will base my remarks on results from three well-known exam- ples.

    (a) Solitons were first discovered by Zabusky and Kruskal [32] in the Korteweg-de Vries equa- tion:

    u, + 6uux + uxx ~ = 0. (KdV)

    Many of the special properties of integrable equations were discovered first for this equation, including the method of inverse scattering [13], and complete integrability for an infinite dimen- sional Hamiltonian system [6, 34].

    i4,t + qJxx + 210120 = 0, (NLS)

    was the second important equation to be solved by inverse scattering methods, by Zakharov and Shabat [37]. Their discovery showed that the KdV equation was not just an isolated curiosity. (There are actually two NLS equations, which differ in the sign of their nonlinear terms. Both are com- pletely integrable, but only this version admits solitons that vanish as Ix l ---' ~.)

    (c) Kadomtsev and Petviashvili [20] introduced their equation,

    (u, + 6uu x + Uxxx) x + 3Uyy = O, (KP)

    to study the stability of KdV solitons to trans- verse perturbations, but it also happens to be integrable; it was solved on x2+y2< oo by Ablowitz, Bar Yaacov and Fokas [1]. (There are also two KP equations, which differ in the relative signs of the last two terms. In the terminology of Zakharov [33], this is KP2.)

    An important ingredient in the development of the theory of solitons and of complete integrabil- ity has been the interplay between mathematics and physics. The three examples listed above were chosen not only because they are integrable,

    0167-2789/91/$03.50 1991- Elsevier Science Publishers B.V. (North-Holland)

  • 344 H. Segur / Who cares about integrabilio,?

    but also because each has been derived in several physical contexts as a model of physical phenom- ena. In fact, each equation describes approxi- mately the evolution of ordinary water waves under the right conditions. To emphasize this interplay between mathematics and physics, I now describe briefly the sense in which each equation models water waves. The interested reader can find more details in ch. 4 of ref. [2].

    lmagine a long, straight channel, with a flat, horizontal bottom and vertical sides, containing water. At one end of the channel is a mechanical device to create waves on the surface of the water; these waves propagate to the other end of the channel, where they are absorbed. The waves that one commonly sees at the beach are caused primarily by gravity, with much weaker effects from surface tension and viscosity; this will be the situation of interest here.

    According to the standard theory of waves of infinitesimal amplitude (e.g. ref. [24]), gravity- induced water waves are dispersi~'e, and waves with longer wavelengths travel faster than shorter waves. The result is that water waves emanating from a spatially and temporally confined source tend to sort themselves out, with longer waves moving ahead of the shorter waves. (One can observe this phenomenon by throwing a rock into a quiet pond: in the wave pattern that radiates outward, the long waves lead, followed by waves of decreasing wavelength.) This suggests the fea- sibility of studying experimentally the evolution of water waves with only a narrow range of wavenumbers, because waves with different wavenumbers will separate from each other spa- tially.

    The KdV equation was first derived by Korte- weg and de Vries [22] to describe approximately the evolution of long waves of moderate ampli- tude propagating in one direction. The KdV equation applies in a coordinate system moving with the speed of long waves of infinitesimal amplitude; x measures horizontal distance, t rep- resents time, and u represents the displacement of the water surface above its mean level. All

    three variables have been scaled appropriately. The equation implicitly assumes that there are no variations across the (narrow) channel, and it ignores any dissipation due to viscosity.

    The nonlinear Schr6dinger equation (NLS) de- scribes the evolution of short waves, propagating in the same channel. Suppose one creates (per- haps with an oscillatory paddle at one end of the channel) a packet of waves, all with nearly the same wavelength ("nearly monochromatic"), which is much shorter than the mean fluid depth. One follows this packet by travelling along the channel with the group velocity of the waves, and NLS describes approximately the evolution of the complex envelope of the wavepacket as it propa- gates. In NLS, x measures spatial distance in this moving coordinate system, t represents time, and

    represents the complex amplitude of the packet. As with KdV, all three variables have been scaled properly, no variation across the channel is per- mitted, and the effects of viscosity are ignored.

    The KP equation describes almost the same situation as KdV, except that waves are no longer required to be strictly one-dimensional; the KP equation also allows weak transverse variations. In the channel discussed above, one could imag- ine simply moving the sidewalls outward to per- mit transverse variations.

    2. Extra structure of integrable equations

    Every integrablc equation has a long list of special properties that hold for integrable equa- tions, and only for them. We now review this list briefly; more details can be found in any book on soliton theory, including that of Ablowitz and Segur [2].

    2.1. Solitons

    The first surprise is that these equations admit exact, N-soliton solutions. (This statement holds for the examples given, but there are integrable equations that admit no bounded, localized solu-

  • H. Segur / Who cares about integrability? 345

    \

    \

    jx , \ I \

    , Fig. 1. A typical interaction of two KdV solitons at four successive times.

    tions, and no solitons.) A single soliton is simply a solitary wave: a spatially localized wave of perma- nent form. For KdV, we have

    U(x,t)=2K2sech2[K(X--4KZt+x,~)], (1)

    where K and x 0 are constants. Solitary waves are not unusual; what is unusual is that the KdV equation also admits exact solutions in closed

    form for 2 solitons, and for N solitons, for N > 1. Fig. 1 shows a 2-soliton solution for the KdV equation, at four different times, showing how 2 solitons interact. The two isolated waves that emerge as t ~ + ~ are precisely those that were present as t ~ -~, except that each has been phase-shifted from where it would have been had there been no interaction.

    For KP, each soliton is a plane wave, an one- dimensional, KdV-type soliton, but travelling in an arbitrary direction in the x-y plane:

    u(x,y,t) =2K 2

    sech2(K[x + .Y - (302 + 4K2), + x,,]} (2)

    Fig. 2 shows a 2-soliton solution for the KP equation. The figure is a snapshot taken at a particular time, but as time changes this entire wave pattern simply translates.

    Recall that the KP equation describes waves in shallow water; i.e. waves with wavelengths much longer than the mean fluid depth. Fig. 3 shows a photograph, taken by Terry Toedtemeier on the coast of Oregon, showing the interaction of two nearly plane, nearly solitary waves in shallow water. Note that each wave experiences a phase shift as a result of its interaction with the other wave. No quantitative information about these waves is available, but the qualitative resem- blance of the wave patterns in figs. 2 and 3 is striking.

    Fig. 2. A 2-soliton solution of the KP equation, for one choice of the parameters.

  • 346 H. Segur / Who cares about integrability?

    Fig. 3. Oblique interaction of two nearly solitary waves in shallow water. (Photograph courtesy of T. Toedtemeier.)

    For NLS, a soliton is a localized, complex wave

    packet:

    0 (x , t) = aexp[ i (b 2 - a2) t - ibx + c~,,]

    sech[a(x - 2bt +/3,,)]. (3)

    At a fixed time (or at a fixed location), Re ) looks like the solid curve in fig. 4, while 101 corre- sponds to the dashed curve. However, the solid line in fig. 4 is not simply the graph of Re 0. Fig. 4 actually shows an experimental wave record for a packet of waves in deep water, measured by Joe Hammack in a narrow channel like that described above. The solid line is the time-history of the water surface at a fixed gauge-site. In this experi- ment, he created a packet of water waves corre- sponding to a NLS soliton with an oscillating

    paddle at one end of the channel. According to NLS theory, this wave packet should propagate down the channel without change of form. Fig. 4a shows the wave that passed a gage 6 m down- stream of the paddle, while fig. 4b shows the same wave packet 30 m downstream. In each case, the dashed line is the shape of the packet with the measured peak amplitude, as predicted by NLS theory, (3). The close agreement between the (dashed) envelope shape and the (solid) mea- sured wave amplitude shows how well NLS pre- dicts actual waves in deep water.

    Note that the peak amplitude of the wave packet in fig. 4b is smaller than that in fig. 4a. This decrease is due to the viscosity of the water in the experiment, an effect ignored in the NLS approximation. NLS theory describes the evolu- tion of nearly monochromatic wave packets on a

  • H. Segur / Who cares about integrability? 347

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    time-scale much longer than the period of a wave, but much shorter than the time of viscous decay. Similar statements apply to KdV and KP.

    2.2. Quasi-periodic solutions

    Equations that admit solitons also admit exact, N-phase, quasiperiodic solutions that can be writ- ten in terms of Riemann theta functions. These solutions are generalizations of N-soliton solu- tions, which they approach in the appropriate limit. For both KdV and KP, they take the form

    u = 2a~ In 0 (6 , ,62 . . . . . 6N)- (4 )

    00. . /

    00

    Fig. 4. Measured displacement of water surface, showing the evolution of envelope soliton at two downstream locations. Water depth, h = 1 m; wave frequency, ~o = 1 Hz; wavenum- ber, kh = 4.0. Solid line: measured history of surface displace- ment; dashed line: envelope shape predicted by NLS. (a) 6 m downstream of wavemaker; (b) 30 m downstream of wave- maker. (Courtesy of Joe Hammack.)

    Here O is a Riemann theta function of genus N, and "quasiperiodic" means that O is periodic in each of its N phase variables if the other N- 1 variables are held fixed. (Theta functions are de- fined explicitly in section 4.) For KdV,

    dpj = I.tjx + ogjt + qSjo , (5a)

    while for KP

    (5b)

    In the simplest case, N = 1, (4) is equivalent to the "cnoidal wave" found by Korteweg and de Vries [22]. Fig. 5 shows a KP solution of genus 1 for one choice of the parameters, while fig. 6 shows a KP solution of genus 2. The solution in

    Fig. 5. A typical KP solution of genus 1. The solution is one-dimensional, and it is a periodic generalization of one soliton.

  • 348 tt. Segur / l~7to cares about integrahilio'?

    Fig. 6. A KP solution of genus 2, near the 2-soliton limit.

    fig. 6 is near the 2-soliton limit, and the sense in which this solution is a periodic generalization of that in fig. 2 is apparent. KP solutions of genus 2 reduce to 2 solitons in one limit, to genus-1 solutions in another limit, and to 2 Fourier modes in a third limit. However, there is no need to be in any of these limits, and fig. 7 shows a KP solution of genus 2 away from any of these limits.

    2.3. Solution of the initial-t'alue problem

    A third surprising feature of integrable prob- lems, separate from their large families of special solutions, is that one can actually soh'e these

    equations as initial-value problems, with arbitrary initial data in a specified class. (From this stand- point, one could define "integrablc" to mean simply that one can integrate the equation for- ward in time.) The method to accomplish this is called the "method of inverse scattering", or the "inverse scattering transform". Many nonlinear equations can be solved for short times, but the method of inverse scattering provides solutions to integrable equations for arbitrarily long times. Moreover, the solutions that evolve from two "nearby" sets of initial data (within the appropri- ate class) can diverge linearly, but not exponen- tially; i.e. there is no chaotic behaviour in these solutions.

    Fig. 7. A KPsolut ion o fgenus2 , awayfrom any particular limit.

  • H. Segur / Who cares about integrability? 349

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    0 20 40 60 80 100 120 140 160 180

    t(glh)t- (x-b)/h = %-r

    Fig. 8. Evolution of a long, nearly rectangular, water wave of elevation into 3 KdV solitons plus radiation. Water depth, h=5 cm. (a) wave profile at x/h=O. (b) x/h=20. (c) x/h = 180. (d) x/h = 400. Solid line: measured wave profile; dotted line: soliton profile computed from (1) [19].

    The solutions that evolve from these initial data often have simple characterizations. For ex- ample, if the initial data for KdV are smooth, if u(x, 0) and all of its derivatives vanish rapidly as x ~ _+~, then the solution evolves into N soli- tons, which separate from each other as t--* ~, plus "radiation", which decays in amplitude as t ~ ~. (Evolution into solitons plus radiation also occurs for more general initial data. The delicate question of exactly what initial data are allowed has been examined by Cohen [8], Delft and Trubowitz [9] and others.)

    This evolution into solitons plus radiation can be seen in the water wave records shown in fig. 8. In this experiment, a long-wave disturbance was created at one end of a long channel, then the

    evolving shape of the disturbance was measured as it passed four successive gages down the chan- nel. The front of the disturbance is to the...