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Some open problems
Physica 18D (1986)1-12 North-Holland, Amsterdam
SOME OPEN PROBLEMS Harvey SEGUR Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA and A.R.A.P., P.O. Box 2229, Princeton, NJ 08540, USA
This conference proceedings covers a variety of topics, related by a common theme: solitons and coherent structures. An objective of this paper is to exhibit some of the relations between topics, by identifying some unsolved but well-defined problems that either were or might have been discussed during the conference, and by suggesting how they might be solved using methods that were discussed here. The problems were chosen to identify unifying themes, rather than to identify the most important problems in each area. For example I will not discuss any of the important work of Baxter, despite its fundamental role in the development of solvable lattice models. The problems discussed come from quantum field theory, transonic flow, water waves, and the two-dimensional motion of a perfect fluid.
1. Integrable quantum field theories An outstanding problem in this subject has been to understand and to make precise the apparent connection between integrable quantum field theories and their classical counterparts. The full scope of this connection has been described in detail in well-written articles by Kulish and Sklyanin [1] and by Thacker [2]. Despite the important advances that are reviewed in these articles, however, the analogy between the classical and quantum problems is still incomplete, questions that can be answered classically remain open in the analogous quantum problem, and vice versa. The incompleteness of this analogy is most apparent in the two models that may be called "the nonlinear SchfiSdinger equation," which I will use for illustration. The quantum-mechanical model is a one-dimensional nonrelativistic field theory. It is defined by a Hamiltonian, Hq=
f dx { 0xff* Ox~ + cff*q~*q~),
(1)
and by equal-time commutation relations, [ + ( x ) , q~*(y)] =
8(x-y),
[q~(x), ~ ( y ) ] = 0 =
[q,*(x),q~*(y)].
(2)
Here q~(x) is a bosonic field operator, and c is a real number. The bosons interact repulsively if c > 0, and attractively if c < 0. Virtually all of the published work on this subject treats the repulsive case. The Hamiltonian in (1) is normally ordered if we interpret ~ ( x ) and q~*(x) as annihilation and creation operators, respectively. Any Hamiltonian, of course, implies a dynamical (i.e., time-dependent) model, but the questions that have received the most attention to date are time-independent. Two questions of fundamental interest for these integrable quantum field models are: (i) how to diagonalize the Hamiltonian; and (ii) how to compute correlation functions and Green's functions. The classical version of the one-dimensional nonlinear SchrSdinger equation also can be represented by a Hamiltonian,
He= ½ f dx (I ax, i2 + cl,l'},
(3)
where q~(x, t) is a complex-valued scalar function, and c is a real number. To use H c as a Hamiltonian, we choose q~ and q~* as conjugate variables, use the usual Poisson bracket [analogous to (2)],
0167-2789/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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H. Segur/Some open problems
and derive from (3) the evolution equation, iq,, = '/'xx - 2clq'12q ',
(4)
where subscripts denote partial derivatives. This is known to be a well-posed initial-value problem on - ~ < x < ~ if we prescribe initial data, ~ ( x ) , such that
Jcp 2f dxf (x)l
Iclf dxl 12< , (5)
and seek a solution of (4) for t >_ 0 satisfying
~,(x,O)=~(x), fdxl~f 2 < ~ .
(6)
In contrast to the quantum model, the time-dependent nature of the solution is of central interest in the classical model. Thus we have two similar Hamiltonians, representing models with quite different interpretations. Even so, it is desirable to develop theories that are as parallel as possible, and the review articles mentioned above describe the quantum field theories in terms of such a parallel development. The first step in solving the quantum problem, is to diagonalize the Hamiltonian. This was done, using Bethe's ansatz, by Lieb and Lininger [3], Berezin, Pokhil and Finkelberg [4] and McGuire [5], before there was any systematic study of integrable problems. The practical value of diagonalizing the Hamiltonian is that one can calculate its S-matrix (and its mass spectrum, for a relativistic theory) both of which are observables. Some people feel that a quantum field model is "solved'once its Hamiltonian has been diagonalized. In the classical problem the corresponding step was accomplished by Zakharov and Shabat [6]. They found two linear equations (i.e., a Lax pair) whose compatibility implies (4). Then they used one of the auxiliary equations to map the solution of (4) into another set of variables (scattering data), and showed that while evolves according to (4), the scattering data evolve in a very simple way. The reason for this simple evolution, of course, is that the (canonical) transforma-
q,(x,t)
tion to scattering da~, diagonalizes the Hamiltonian. No one would assert that the classical problem is solved once it has been diagonalized in this way. It is also necessary to construct the inverse mapping which recovers ~(x, t), the solution of (4), from the (time-evolved) scattering data. This mapping was also given by Zakharov and Shabat [6] using a generalization of the Gel'fandLevitan-Marchenko linear integral equation. Their results aply for - ~ < c < ~ , with some differences if c > 0 or c < 0. In the quantum problem, the analogue of constructing the inverse mapping is to compute correlations functions. We need not worry here about defining "correlation function" precisely, because no correlation functions have been computed for almost any integrable quantum model, as Kulish and Sklyanin [1] showed clearly in their table I. The single exception has been the nonlinear SchrtSdinger equation (1), where progress has been made in the strong coupling (c --* ~ ) limit. Thacker [2] reviews the work of Jimbo, Miwa, Mfri and Sato [7] and of Creamer, Thacker and Wilkinson [8] in obtaining the first few terms of a (1/c)expansion of two-point correlation functions in terms of Painlev6 transcendents. There is little reason to believe that information at finite values of the coupling constant (c) can be obtained from any reasonable extension of this series. Computing correlation functions at finite coupling strength for (1), or for any of the other integrable quantum problems, is a major open question in the quantum theory. In principle, an advantage of having developed the classical and quantum theories in a parallel fashion is that the (solved) classical theory might provide insight into the (still unsolved) quantum theory. Unfortunately this approach has not proven successful, so we are left in an unsatisfactory state: in the classical problem (3) we can reconstruct 4,(x, t) for any value of the coupling constant (c), but in the analogous quantum problem (1), we can compute correlation functions for no finite values of c.
H. Segur / Some open problems
Conversely, we can compute quantum correlation functions in the limit c ~ oo, but this is precisely the limit in which the classical theory fails, because of (5). Many people have tried unsuccessfully to extend the quantum theory to finite coupling strengths, but the idea of extending the classical theory to the stong coupling limit seems to be new. One might hope that if the classical theory were available to both for [cl < oe and Icl = ~ , then these two versions of the classical theory might suggest how to extend the quantum theory to finite coupling strengths.
Problem # 1 . Solve the classical nonlinear SchrSdinger equation, (4), in the (singular) limit c---, oo. Relate the solutions of the classical and quantum problems in this limit. A possible method of solution of this problem is described below.
2.
lntegrable problems in the small dispersion limit
In their original work, Zabusky and Kruskal [9] were interested in the Korteweg-de Vries equation in the small dispersion limit:
ut+UUx+82Uxxx=O, 8 2 << 1,
--o0 < x <
u(x,O) = U ( x ) , given.
oo,
(7)
Their pioneering work led to the discovery that the K o r t e w e g - d e Vries equation could be solved exactly, for any 8, and the question of finding the limiting form of the solution as 8 ~ 0 lost some of its urgency. Moreover, the usual theory [e.g., Delft and Trubowitz [10]] requires that the initial data for (7) satisfy
8-2/~ (1 + x2)lV(x)[ d x
< oo,
(8)
a_oo
so it is clear that the small dispersion limit (8 ---, 0) is singular. Let us recall briefly the difficulties of letting 8 ~ 0 in (7). Naively, one expects that as 8 --, 0 the
3
solution of (7) would approach the solution of the limiting problem obtained by actually setting 8 = 0,
v,+vv~=O,
v(x,O)=U(x).
(9)
This nonlinear, hyperbolic equation has characteristics given by dx
-d-? = v(x, t).
(10)
Eq. (9) states that v is constant along these characteristics. It follows that the characteristics are straight lines, each of which carries its own initial value of v(x,O). If dU/dx < 0 over a finite interval, then two or more of these characteristics will cross at some finite time. Denote the earliest such time (which depends on U(x)) by t*. Because the solution of (9) becomes multi-valued at t* (and at some particular x*), (9) has no solution for t > t* unless we reinterpret the problem to allow discontinuous solutions, or "shocks". Certainly there is no reason to expect the solutions of (7) and (9) to be close after t*. Moreover, IVxxxl--" ~ as t ~ t* (near x = x*), so the neglected term in (7) becomes important for any 8 2 > 0 even for t < t*. A method to solve (7) in the small dispersion limit was invented by Lax and Levermore [11] and extended by Venakides [12]. Their work is reviewed by Lax in these proceedings. The accomplishment of Lax, Levermore and Venakides was to give precise meaning to lim
u(x, t; 8),
8 ---, 0
where u(x, t; 8) solves (7). They showed that in the context of square integrable functions, this limit exists as a strong limit for t < t* and as a weak limit for all t. They also showed how to compute both this limit and lim lim t~oo
u(x, t; 8).
840
The method of Lax, Levermore and Venakides probably has wider application than just the
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H. Segur/Some open problems
K o r t e w e g - d e Vries equation. Here are two other problems where their method might be valuable. The first problem is to solve the classical nonlinear SchrSdinger equation (4) in the strong coupling (Icl ~ oo) limit. To recast this problem as a small dispersion limit, define
8 = c -1,
~= - c t .
(11)
lems in this small dispersion (or strong coupling) limit.
Problem # 2 . Solve (12) for 0 < - 6 << 1. Problem # 3 . D o e s the L a x - L e v e r m o r e Venakides method have a direct quantum analogue?
Then the problem becomes iq,~ = 21q,12~- 8eOxx, ,t,(x,
= o;
181 << 1, o
=
asx
----~ + 0 0 ,
(12)
find :
[
A second set of problems that can be viewed as small dispersion limits arise in the study of tran sonic flow in two dimensions. Krupp and Cole [13] show that an equation equivalent to
(u, + UUx) ~ + Uw = 0,
lim t x, ,r; 8 ).
(15)
8~0
As with (7), we obtain the limiting problem by setting 6 = 0 in (12),
i~b, = 21~1%,
~(x,0) = ~(x).
(13)
The solution of (13) is + ( x , r ) = ~ ( x ) exp (-2i1~(x)12~-).
(14)
In this case, the solution exists for all time, so the small dispersion limit of (12) should be less singular than that in (7). Even so q~x, the omitted term for (12), grows like ~.2 as z ~ oo, so one expects a naive perturbation expansion of the solution of (12) to break down when 8T 2 = 0(1), if not earlier. Thus (12) can be solved exactly for any finite 8, as can (7). The limit 6 ~ 0 is singular in both cases, because (5) and (8) break down. In both cases, the limiting (8 = 0) problem can be solved explicitly, but the omitted dispersive term ( 8 ~ x or 6 2 u ~ ) grows without bound for t > 0. Lax and Levermore [11] treated initial data which evolves primarily into solitons whereas Venakides [12] solved a problem with no solitons. The analogy in (12) is that its solution has no solitons if 6 > 0 (the repulsive case), and solitons if 8 < 0.
Problem # 1A. Solve (12) for 0 < 8 << 1. Relate the solutions of the classical and quantum prob-
plus appropriate boundary and initial conditions, approximately describe the unsteady, nearly sonic flow of an inviscid fluid over a thin two-dimensional airfoil. Roughly speaking, x represents distance along the wing, y measure distance normal to it, t is time, and u represents the streamwise component of the perturbation from the velocity far upstream. The model seems to be due to Timman [14] originally. A well-known model of steady transonic flow in two dimensions, q~vv= (k + ~bx)~bxx,
(16)
follows from (15) if we seek steady traveling waves (0, = - k a x ) and set u = -fix- [See Jameson [15] for a discussion of computational aspects of (16).] Both (15) and (16) are hyperbolic, like (9), and both admit shocks. It is known experimentally that these shocks contribute significantly to the drag of the airfoil, and a major objective in the design of airfoils to be used in transonic flight is to minimize the strength of these shocks. A challenging problem in computational fluid dynamics is to design a numerical scheme that can determine accurately the location and strength of these shocks. One method to compute shocks numerically is to smooth them with a little viscosity. That is, instead of starting with the equations for an in-
H. Segur/ Some open problems
viscid, compressible gas, start with the (compressible) Navier-Stokes equations, in which the viscosity is small but finite. Then virtually the same derivation that led to (15) leads instead to (17)
( u t + uu~) x + u w = ~,u . . . .
where ~ represents the fluid viscosity (0 < ~, << 1). This equation has the same physical significance as (15), its inviscid limit, but for 1, > 0 the shocks of (17) have finite thickness. If we omit (Uvv) from (17), it reduces to a derivative of the famous equation of Burgers [16]. Just as Burgers' equation describes weak onedimensional shocks, so (17) describes weak shocks that are also weakly two-dimensional. That is, (17) represents a physical generalization of Burgers' equation to two spatial dimensions. Burgers' equation is known to be integrable, and can be linearized by the transformation of Cole [17] and Hopf [18]. It is reasonable to ask, therefore, whether (17) might also be integrable. Applying the Painlev6 analysis of Weiss, Tabor and Carnevale [19] yields a negative result; fully twodimensional solutions of (17), i.e., those which cannot be transformed into solutions of Burgers' equation, do not have the Painlev6 property. An alternate method to smooth the shocks in (15) is to add a little dispersion, so (15) is replaced by (ut+uU~)x+U,,='u
.....
Icl <<1.
(18)
This replacement may be regarded simply as a computational convenience: the solutions of (18) remain continuous for c ~: 0, and they should approach the solutions of (15) as c --, 0. For E ~ 0 , (18) is the equation studied by Kadomtsev and Petviashvili [20], and it is known to be integrable. The method of solution of (18) is discussed both by Ablowitz and by Beals in these proceedings. For our purposes it is sufficient to note the following facts: (i) For c > 0, (18) can be solved exactly by a method due to Manakov [21]. For c < 0, (18) also
5
can be solved exactly, but Manakov's method fails and must be replaced by a method of Ablowitz, BarYaacov and Fokas [22]. (ii) Independent of the method, the qualitative nature of the solution of (18) depends of the sign of ~, because Kadomtsev and Petviashvili [20] showed that KdV-type (plane-wave) solitons are unstable for c < 0, but not for ~ > 0. (iii) For either sign of c, the limit c --, 0 in (18) is a small dispersion limit, completely analogous to the limit carried out by Lax, Levermore and Venakides for (7). (iv) For steady traveling waves, ( 0 t = - k O x ) , and (18) reduces to the Boussinesq equation Uyy=kuxx-(½u2)~
+cu .....
(19)
which also can be solved exactly. The small dispersion (~ -~ 0) limit of (19) is equivalent to (16). P r o b l e m # 4 . For c > 0, solve (18) in the small
dispersion limit, to find lim u ( x , y , t; E). ¢---~ 0
P r o b l e m # 5 . For ~ <0, solve (18) in the small
dispersion limit (~ ~ 0). P r o b l e m # 6 . One has mathematical justification
to define the solution of (15) to be the solution of problem # 4, or to be the solution of problem # 5, or to be the small viscosity (v ~ 0) limit of the solution of (17). A priori, there is no reason to believe that the three limits coincide. If they fail to coincide, then (15) would not have a unique solution, and a more careful reformulation would be required to make (15) well-posed. This (hypothetical) nonuniqueness of (15) raises serious practical questions, because a given numerical scheme might be approximating any one of the three solutions. P r o b l e m # 7 . Solve (19) in the small dispersion
(~ ~ 0) limit. Compare the solution of (16) obtained in this way with the results obtained by more conventional numerical algorithms. Does (16) have a unique solution?
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H. Segur/Some openproblems
3. Nonlinear stability in Hamiltonian systems Nonlinear stability is of fundamental importance to this conference, because the physical significance of a "coherent state" depends on how stable it is. Typically, the more stable a state is the more easily it can be observed experimentally. In these proceedings, Marsden and Holm discuss an algorithm that they and their collaborators have developed to prove the nonlinear stability of certain equilibrium solutions of Hamiltonian systems. The basic ideas in this algorithm had been used earlier in isolated problems [cf. Kruskal and Oberman [23], Arnol'd [24], but Holm, Marsden, Ratiu and Weinstein [25] refined and organized these ideas into a clear, systematic algorithm]. In his lecture, Marsden used their algorithm to show that the single soliton is a stable solution of the Korteweg-de Vries equation,
u, + 6uu~ + uxx x = 0.
(20)
More generally, a nearly identical analysis shows that the simplest periodic solution of (20), known as a cnoidal wave, is also stable. An example of a cnoidal wave is shown in fig. 1. The soliton and cnoidal wave are equilibrium solutions of (20) in the sense that each is stationary (i.e., time-independent) in an appropriately translating coordinate system. In fact, they are the only bounded real solutions of (20) that are stationary.
A cruder, but far simpler, approach to stability can be summarized as follows: (i) Korteweg and de Vries (1895) originally derived their equation to model the one-dimensional propagation of waves of moderate amplitude in shallow water. (ii) Mathematical stability corresponds roughly to physical observability. (iii) Therefore, if cnoidal waves are stable, we should observe them in shallow water. Fig. 2, taken from an ancient National Geographic (1933), shows a very regular train of periodic, one-dimensional waves off the coast of Panama. Certainly the quahtative features of these waves (periodic waves with steep, localized crests and flat, broad troughs) correspond to those of the cnoidal waves in fig. 1. Thus, fig. 2 can be viewed as a qualitative corroboration of the nonlinear stability of cnoidal waves. Quantitative measurements of periodic waves in shallow water also support this positon, and ocean engineers have regarded cnoidal waves as "typical" (i.e., stable) one-dimensional, periodic waves in shallow water ever since Wiegel [27] brought the waves to their attention [e.g., see Sarpkaya and Isaacson [28]]. As a model of periodic waves in shallow water, cnoidal waves have the obvious limitation that they are one-dimensional, whereas the water surface is two-dimensional. This limitation raises two questions: a) What are the appropriate two-dimensional generalizations of the cnoidal wave and the soliton?
Fig. 1. A representative cnoidal wave solution of the KdV equation. The KP equation also admits one-dimensional, cnoidal wave solution.
H. Segur/ Some open problems
7
Fig. 2. Periodic waves in shallow water. The original caption read " A s they near shallow water close to the coast of Panama, huge, deep-sea waves, relics of a recent storm, are transformed into waves that have crests, but little or no troughs. A light breeze is blowing diagonally across the larger waves to produce a cross-chop. Three A r m y bombers, escorted by a training ship, are proceeding from Albrook Field, Canal Zone, to David, Panama."
b) Are the two-dimensional generalizations nonlinearly stable? Kadomtsev and Petviashvili [20] showed that their equation (i.e., (18), with c a fixed, non-zero constant) arises as a two-dimensional generalization of (20) when one relaxes slightly the requirement that the waves be strictly one-dimensional. In the context of water waves, this means that the fluid depth should be shallow in comparison with a typical wavelength in the direction of propagation, which in turn should be small in comparison with a transverse wavelength (see Ablowitz and Segur [29], for the full derivation). For gravity-induced water waves, the final result can be written in a standard form as
(u, + 6uu:, + u:,.~x) x + 3ur~v = O.
(21)
Obviously, every solution of (20) is a y-independent solution of (21). More generally, Satsuma [30] and Krichever [31] showed that (21) admits exact
solutions in the form
u(x, y, t) = 2 0 2 1 n 0 ,
(22)
just as does the KdV equation. Both N-soliton and N-phase quasi-periodic solutions can be obtained in this way. Fig. 3 shows a 2-soliton solution of the KP equation. Away from the region of interaction, each wave is essentially a KdV-type soliton. As expected, there is a phase shift because of the interaction. This solution has two phases, and it moves on a two-dimensional surface, so there is a uniformly translating coordinate system in which the entire wave pattern is stationary. The 2-soliton solutions of KP that a r e stationary in this sense can be viewed a two-dimensional generalizations of a single (plane-wave) soliton. The 2-soliton solution has a periodic analogue, called a solution of genus 2. An example is shown in fig. 4. Locally, the solution looks like two soli-
8
H. Segur/Some open problems
Fig. 3. Typical 2-soliton solution of the KP equation.
Fig. 4. Genus 2 solution of the KP equation. This family of exact solutions contains as limiting cases both the 2-soliton solution (fig. 3) and the cnoidal wave (fig. 1).
tons, but it is periodic in two directions. This solution also is stationary in an appropriately translating coordinate system. It represents a two-dimensional generalization of the cnoidal wave. (See Segur and Finkel [32] for a detailed discussion of these KP solutions of genus 2 as models of two-dimensional periodic waves in shallow water.) Given these two-dimensional generalizations of cnoidal waves and solitons, we can ask about their stability. The experimental approach is: Are these waves ever observed? Fig. 5 shows two interacting, soliton-type waves in shallow water, photographed by Terry Toedtemeier off the coast of Oregon. The qualitative agreement between the wave patterns in figs. 3 and 5 is remarkably good, but we have so little quantitative information about the ocean
waves that the apparent agreement can only be considered suggestive. Similarly, an observation of two-dimensional waves in shallow water that is in qualitative agreement with fig. 4 was made by Joe H a m m a c k , [cf. fig. 1.9 of Segur, Finkel and Philander [33]. Neither of these observations provides enough quantitative information to justify a conclusive statement about the validity of the stationary K P solutions of genus 2 as models of actual water waves. In particular, the stability of these K P solutions is open. Problem # 8. Use the stability algorithm of Holm,
Marsden, Ratiu and Weinstein [25] to determine the nonlinear stability of the KP solutions of genus 2 that are stationary in an appropriate coordinate system.
H. Segur/ Some open problems
9
Fig. 5. Oblique interaction of two nearly solitarywaves in shallow water. (Photographcourtesy of T. Toedtemeier.)
4. Two-dimensional motion of a perfect fluid The last problem also involves questions of stability: to determine the long-time behavior of an inviscid, incompressible fluid as it moves in a two-dimensional plane. The problem arises as a simplified model with applications in geophysics [e.g., Salmon [34]] and plasmas [e.g., Morrison [35]]. It was first considered by Euler, with more recent contributions by several of the speakers at this conference. Here is a statement of the problem. Let x = (x, y ) define a position in a two-dimensional plane, u = (u, v) represent the velocity vector of an inviscid, incompressible fluid moving in that plane, p (x, t) denote the fluid pressure, and t represent time. The dynamical equations of the fluid are Euler's equations in two dimensions, Du
Dt
= ( ~t + u" W)u
W.u=0.
= - WP'
(23)
Initial and boundary conditions are also needed to make (23) into a well-posed problem. For definiteness, we impose periodic boundary conditions in (x, y ) and we require that u(x, t = 0) be smooth and periodic. It should be emphasized that the long-time behavior of the solution of (23) seems to be rather delicate, and the result may depend sensitively on the initial and boundary conditions.
Problem # 9. Given smooth initial data, does the solution of (23) with periodic boundary conditions tend to any limit as t ---, oo? Does the existence of such a limit depend on the initial data? Conversely, does (23) admit solutions that are recurrent in time? Is the system ergodic? Is it mixing (in the sense of statistical mechanics)? The governing equation become particularly simple in terms of the vorticity. Define the (scalar) vorticity by 6o = v x -
Uy
= (~TX u ) . k .
(24)
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H. Segur / Some open problems
Then (23) reduces to the very simple form D~
Dt
=0.
(25)
Each fluid particle has some initial vorticity and (25) says that the fluid particle carries that vorticity with it as it moves around. At each instant, therefore, the velocity field is obtained by integrating over the vorticity, and then the vorticity is transported by the velocity field so generated. Conceptually, the motion is extremely simple: the vorticity field simply convects itself around. Given this conceptual simplicity, it is surprising how difficult it has been to characterize (correctly) the nature of the resulting flow. Several properties of this system are known (e.g., see Holm, et al. [25] for more details). (i) Arnol'd showed that the system is Hamiltonian. The Hamiltonian function is the kinetic energy,
H =ff½u.udA,
(26)
where the integral is taken over a (two-dimensional) period. The exact form of the appropriate Poisson bracket is not needed for our purposes. (ii) Another positive-definite, conserved quantity is called the enstrophy,
E = ff½~e dA.
(27)
(iii) Because the vorticity is conserved for each fluid particle, and because the fluid motion simply rearranges these particles, it follows that the integral of any smooth function of the vorticity is conserved. In particular, (23) conserves the moments of vorticity,
I,=ffoa'dA,
n=1,2,3 ....
(28)
Therefore (23) has infinitely many constants of the motion.
(iv) In these proceedings, Marsden shows that the moments of vorticity are examples of "Casimirs"; i.e., they annihilate the Poisson bracket appropriate for (23). It follows that this infinite set of constants of the motion cannot be used to prove that (23) is completely integrable, contrary to a conjecture of Ebin [36]. This point was first made by Marsden and Weinstein [37], who conjectured that (23) is not completely integrable. Certainly it is widely believed in the geophysical community that (23) is not completely integrable, but rather that its solutions exhibit an irreversible cascade of energy to low wavenumbers and of enstrophy to high wavenumbers [cf. Rhines [38] or Salmon [34]. Evidence to support the plausibility of these cascades can be found in the numerical simulations of Zabusky (in these proceedings). The energy cascade appears there as "vortex merging", while the enstrophy cascade corresponds to "illamentation." The argument that such a cascade necessarily occurs seems to rest on a very simple argument given by Salmon [34], among others. That argument shows that if the solution of (23) has a limit as t-~ oo, then it is very likely that energy will tend to lower wavenumbers as t -~ o0. On the other hand, if a solution of (23) is recurrent, with any reasonable recurrence time, then there is no limit as t -~ oo, so the usual argument is irrelevant. I am aware of no strong argument which precludes the possibility of recurrent motion or even gives a lower bound on a possible recurrence time. At this time, the ubiquity of the e n e r g y / e n s t r o p h y cascades probably should be regarded as open. For numerical purposes, it is sometimes convenient to represent the solution of (23) with a time-dependent Fourier series, and to replace (23) with an infinite set of ordinary differential equations for the evolution of the Fourier coefficients. In this formulation, different numerical schemes correspond to different truncations of this infinite set of equations. It is reasonable to consider only those truncations that conserve at least energy (26) and enstrophy (27). Hald [39] found several lowdimensional truncations that conserved energy, en-
H. Segur / Some open problems
s t r o p h y , a n d e n o u g h o t h e r c o n s t a n t s that the t r u n c a t e d s y s t e m s were c o m p l e t e l y integrable. Cert a i n l y H a l d ' s t r u n c a t e d systems exhibit no irreversible cascades. O n the o t h e r hand, Lee [40] a n d Kells and O r s z a g [41] s h o w e d that H a l d ' s extra integrals d e p e n d o n H a l d ' s p a r t i c u l a r truncations, a n d disa p p e a r if the t r u n c a t i o n is d o n e " i s o t r o p i c a l l y . " Lee [40] c o n c l u d e d that (23) with p e r i o d i c b o u n d a r y c o n d i t i o n s " h a s no e x t r a n e o u s c o n s t a n t s of m o t i o n [i.e., b e y o n d energy a n d enstrophy], unless the p r e s e n c e o f some other constants of m o t i o n is d i s c o v e r e d . " Because of its p r a c t i c a l i m p o r t a n c e , (23) has b e e n the subject o f a great deal of n u m e r i cal work. G l a z [42] reviews m u c h of this work. H e c o n c l u d e s t h a t all of the " n u m e r i c a l studies of the e r g o d i c i t y / m i x i n g questions for the t r u n c a t e d E u l e r e q u a t i o n s (including this one) have been inconclusive." T o s u m m a r i z e , (23) with p e r i o d i c b o u n d a r y cond i t i o n s is the subject of strongly held opinions, b u t t h e r e seems to be little conclusive evidence to s u p p o r t these opinions. T h e n a t u r e of the l o n g - t i m e b e h a v i o r o f the solutions (23) r e m a i n s an o p e n question.
Acknowledgements T h i s w o r k was s u p p o r t e d in p a r t b y the U.S. A r m y R e s e a r c h Office, a n d b y the N a t i o n a l Scie n c e F o u n d a t i o n u n d e r G r a n t No. PHY82-17853, s u p p l e m e n t e d b y f u n d s f r o m the N a t i o n a l A e r o n a u t i c s a n d Space A d m i n i s t r a t i o n .
References [1] P.P. Kulish and E.K. Sklyanin, in: Integ. Quantum Field Theories, Springer Lecture Notes in Physics, vol. 151, J. Hietarinta and C. Mononen, eds. (Springer, Berlin, 1982) pp. 61-119. [2] H.B. Thacker, in Integ. Quantum Field Theories, Springer Lecture Notes in Physics vol. 151, J. Hietarinta and C. Montanen, eds. (Springer, Berlin, 1982) pp. 1-60.
11
[3] E.H. Lieb and W. Lininger, Phys. Rev. 130 (1963) 1605-1616. [4] F.A. Berezin, G.P. Pokhil and V.M. Finkel'berg, Vestn. MGU, Ser. 1. (1964) 21-28. [5] J.B. McGuire, J. Math. Phys. 5 (1964) 622-636. [6] V.E. Zakharov and A.B. Shabat, Soy. Phys. JETP 34 (1972) 62-69. [7] M. Jimbo, T. Miwa, Y. Mrri and M. Sato Physica 1D (1980) 80-158. [8] D.B. Creamer, J.B. Thacker and D. Wilkenson, J. Math. Phys. 22 (1981) 1084; Phys. Rev. D23. (1982) 3081-3084. [9] N.J. Zabusky and M.D. Kruskal, Phys. Rev. Lett. 15 (1965) 240-243. [10] P. Deift and E. Trubowitz, Comm. Pure. Appl. Math 32 (1979) 121-251. [11] P.D. Lax and D.D. Levermore, Comm. Pure. Appl. Math 36 (1983) I, 253-290; II, pp. 571-593; III, pp. 809-830. [12] S. Venakides, Comm. Pure Appl. Math., to appear. [13] J.A. Krupp and J.P. Cole, (1976) UCLA Rept. UCLAEng-76/04. [14] R. Timman, in: Symposium Transsonicum Aachen, K. Oswatitsch, ed. (Springer, Berlin, 1962) pp. 394-401. [15] A. Jameson, J. Appl. Mech., Trans. ASME 50 (1983) 1053-1070. [16] J.M. Burgers, Adv. Appl. Mech 1 (1948) 171-199 [17] J.D. Cole, Quart. Appl. Math 9 (1951) 225-236. [18] E. Hopf, Comm. Pure Appl. Math 3 (1950) 201-230. [19] J. Weiss, M. Tabor and G. Carnevale, J, Math. Phys. 24 (1983) 522-526. [20] B.B. Kadomtsev and V.I. Petviashvili, Sov. Phys. Doklady 15 (1970) 539-541. [21] S.V. Manakov, Physica 3D (1981) 420-427; see also H. Segur in Math. Meth. Hydrodyn. and Integ. Dyn. Syst., AIP Conf. Proc. #88. M. Tabor and Y.M. Treve, eds. (1982) pp. 211-228; A.S. Fokas and M.J. Ablowitz, Stud. Appl. Math 69 (1983) 211-228. [22] M.J. Ablowitz, D. Bar Yaacov and A.S. Fokas, Stud. Appl. Math. 69 (1983) 135-143. [23] M.D. Kruskal and C. Oberman Phys. Fluids 1 (1958) 275-280. [24] V.I. Arnol'd, Usp. Mat. Nauk 24 (1969) 225-226. [25] D.D. Holm, J.E. Marsden, T. Ratiu and A. Weinstein, Phys. Rep., to appear. [26] National Geographic 63 (1933) 611. [27] R.L. Wiegel, J. Fluid Mech 7 (1960) 273-286. [28] T. Sarpkaya and M. Isaacson, Mech. of Wave Forces on Offshore Structures (Van Nostrand Reinhold, New York, 1981). [29] M.J. Ablowitz and H. Segur J. Fluid Mech. 92 (1979) 691-715. [30] J. Satsuma, J. Phys. Soc. Japan 40 (1976) 286-290. [31] I.M. Krichever, Russ. Math. Surveys 32 (1977) 185-213. [32] H. Segur and A. Finkel, Stud. App. Math, to appear. [33] H. Segur. A. Finkel and H. Philander, in: Nonlinear Phenomena, Springer Lecture Notes Physics, vol 189, K.B. Wolf. ed. (Springer, Berlin, 1983) pp. 212-232. [34] R. Salmon, in: Proc. Int. School Phys. "Enrico Fermi", vol. 80, A.R. Osborne and P. Malanotte Rizzoli, eds.
12
H. Segur/ Some open problems
(1982) pp. 30-78. [35] P.J. Morrison, in: Math. Methods Hydrodyn. and Integ. Dyn. Syst., AIP Conf. Proc. #88, M. Tabor and Y.M. Treve, eds. (1982) pp. 13-46. [36] D.G. Ebin, Comm. Pure Appl. Math. 36 (1983) 37-56. [37] J.E. Marsden and A. Weinstein, Physica 7D (1983) 305-323.
[38] [39] [40] [41]
P.B. Rhines, Ann. Rev. Fluid. Mech. 11 (1979) 401-442. O. Hald, Phys. Fluids 19 (1976) 914-915. J. Lee Phys. Fluids 20 (1977) 1250-1254. L.C. Kells and S.A. Orszag, Phys. Fluids 21 (1978) 162-168. [42] H. Glaz, SIAM J. Appl. Math. 41 (1981) 459-479.
SOME OPEN PROBLEMS Harvey SEGUR Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA and A.R.A.P., P.O. Box 2229, Princeton, NJ 08540, USA
This conference proceedings covers a variety of topics, related by a common theme: solitons and coherent structures. An objective of this paper is to exhibit some of the relations between topics, by identifying some unsolved but well-defined problems that either were or might have been discussed during the conference, and by suggesting how they might be solved using methods that were discussed here. The problems were chosen to identify unifying themes, rather than to identify the most important problems in each area. For example I will not discuss any of the important work of Baxter, despite its fundamental role in the development of solvable lattice models. The problems discussed come from quantum field theory, transonic flow, water waves, and the two-dimensional motion of a perfect fluid.
1. Integrable quantum field theories An outstanding problem in this subject has been to understand and to make precise the apparent connection between integrable quantum field theories and their classical counterparts. The full scope of this connection has been described in detail in well-written articles by Kulish and Sklyanin [1] and by Thacker [2]. Despite the important advances that are reviewed in these articles, however, the analogy between the classical and quantum problems is still incomplete, questions that can be answered classically remain open in the analogous quantum problem, and vice versa. The incompleteness of this analogy is most apparent in the two models that may be called "the nonlinear SchfiSdinger equation," which I will use for illustration. The quantum-mechanical model is a one-dimensional nonrelativistic field theory. It is defined by a Hamiltonian, Hq=
f dx { 0xff* Ox~ + cff*q~*q~),
(1)
and by equal-time commutation relations, [ + ( x ) , q~*(y)] =
8(x-y),
[q~(x), ~ ( y ) ] = 0 =
[q,*(x),q~*(y)].
(2)
Here q~(x) is a bosonic field operator, and c is a real number. The bosons interact repulsively if c > 0, and attractively if c < 0. Virtually all of the published work on this subject treats the repulsive case. The Hamiltonian in (1) is normally ordered if we interpret ~ ( x ) and q~*(x) as annihilation and creation operators, respectively. Any Hamiltonian, of course, implies a dynamical (i.e., time-dependent) model, but the questions that have received the most attention to date are time-independent. Two questions of fundamental interest for these integrable quantum field models are: (i) how to diagonalize the Hamiltonian; and (ii) how to compute correlation functions and Green's functions. The classical version of the one-dimensional nonlinear SchrSdinger equation also can be represented by a Hamiltonian,
He= ½ f dx (I ax, i2 + cl,l'},
(3)
where q~(x, t) is a complex-valued scalar function, and c is a real number. To use H c as a Hamiltonian, we choose q~ and q~* as conjugate variables, use the usual Poisson bracket [analogous to (2)],
0167-2789/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
2
H. Segur/Some open problems
and derive from (3) the evolution equation, iq,, = '/'xx - 2clq'12q ',
(4)
where subscripts denote partial derivatives. This is known to be a well-posed initial-value problem on - ~ < x < ~ if we prescribe initial data, ~ ( x ) , such that
Jcp 2f dxf (x)l
Iclf dxl 12< , (5)
and seek a solution of (4) for t >_ 0 satisfying
~,(x,O)=~(x), fdxl~f 2 < ~ .
(6)
In contrast to the quantum model, the time-dependent nature of the solution is of central interest in the classical model. Thus we have two similar Hamiltonians, representing models with quite different interpretations. Even so, it is desirable to develop theories that are as parallel as possible, and the review articles mentioned above describe the quantum field theories in terms of such a parallel development. The first step in solving the quantum problem, is to diagonalize the Hamiltonian. This was done, using Bethe's ansatz, by Lieb and Lininger [3], Berezin, Pokhil and Finkelberg [4] and McGuire [5], before there was any systematic study of integrable problems. The practical value of diagonalizing the Hamiltonian is that one can calculate its S-matrix (and its mass spectrum, for a relativistic theory) both of which are observables. Some people feel that a quantum field model is "solved'once its Hamiltonian has been diagonalized. In the classical problem the corresponding step was accomplished by Zakharov and Shabat [6]. They found two linear equations (i.e., a Lax pair) whose compatibility implies (4). Then they used one of the auxiliary equations to map the solution of (4) into another set of variables (scattering data), and showed that while evolves according to (4), the scattering data evolve in a very simple way. The reason for this simple evolution, of course, is that the (canonical) transforma-
q,(x,t)
tion to scattering da~, diagonalizes the Hamiltonian. No one would assert that the classical problem is solved once it has been diagonalized in this way. It is also necessary to construct the inverse mapping which recovers ~(x, t), the solution of (4), from the (time-evolved) scattering data. This mapping was also given by Zakharov and Shabat [6] using a generalization of the Gel'fandLevitan-Marchenko linear integral equation. Their results aply for - ~ < c < ~ , with some differences if c > 0 or c < 0. In the quantum problem, the analogue of constructing the inverse mapping is to compute correlations functions. We need not worry here about defining "correlation function" precisely, because no correlation functions have been computed for almost any integrable quantum model, as Kulish and Sklyanin [1] showed clearly in their table I. The single exception has been the nonlinear SchrtSdinger equation (1), where progress has been made in the strong coupling (c --* ~ ) limit. Thacker [2] reviews the work of Jimbo, Miwa, Mfri and Sato [7] and of Creamer, Thacker and Wilkinson [8] in obtaining the first few terms of a (1/c)expansion of two-point correlation functions in terms of Painlev6 transcendents. There is little reason to believe that information at finite values of the coupling constant (c) can be obtained from any reasonable extension of this series. Computing correlation functions at finite coupling strength for (1), or for any of the other integrable quantum problems, is a major open question in the quantum theory. In principle, an advantage of having developed the classical and quantum theories in a parallel fashion is that the (solved) classical theory might provide insight into the (still unsolved) quantum theory. Unfortunately this approach has not proven successful, so we are left in an unsatisfactory state: in the classical problem (3) we can reconstruct 4,(x, t) for any value of the coupling constant (c), but in the analogous quantum problem (1), we can compute correlation functions for no finite values of c.
H. Segur / Some open problems
Conversely, we can compute quantum correlation functions in the limit c ~ oo, but this is precisely the limit in which the classical theory fails, because of (5). Many people have tried unsuccessfully to extend the quantum theory to finite coupling strengths, but the idea of extending the classical theory to the stong coupling limit seems to be new. One might hope that if the classical theory were available to both for [cl < oe and Icl = ~ , then these two versions of the classical theory might suggest how to extend the quantum theory to finite coupling strengths.
Problem # 1 . Solve the classical nonlinear SchrSdinger equation, (4), in the (singular) limit c---, oo. Relate the solutions of the classical and quantum problems in this limit. A possible method of solution of this problem is described below.
2.
lntegrable problems in the small dispersion limit
In their original work, Zabusky and Kruskal [9] were interested in the Korteweg-de Vries equation in the small dispersion limit:
ut+UUx+82Uxxx=O, 8 2 << 1,
--o0 < x <
u(x,O) = U ( x ) , given.
oo,
(7)
Their pioneering work led to the discovery that the K o r t e w e g - d e Vries equation could be solved exactly, for any 8, and the question of finding the limiting form of the solution as 8 ~ 0 lost some of its urgency. Moreover, the usual theory [e.g., Delft and Trubowitz [10]] requires that the initial data for (7) satisfy
8-2/~ (1 + x2)lV(x)[ d x
< oo,
(8)
a_oo
so it is clear that the small dispersion limit (8 ---, 0) is singular. Let us recall briefly the difficulties of letting 8 ~ 0 in (7). Naively, one expects that as 8 --, 0 the
3
solution of (7) would approach the solution of the limiting problem obtained by actually setting 8 = 0,
v,+vv~=O,
v(x,O)=U(x).
(9)
This nonlinear, hyperbolic equation has characteristics given by dx
-d-? = v(x, t).
(10)
Eq. (9) states that v is constant along these characteristics. It follows that the characteristics are straight lines, each of which carries its own initial value of v(x,O). If dU/dx < 0 over a finite interval, then two or more of these characteristics will cross at some finite time. Denote the earliest such time (which depends on U(x)) by t*. Because the solution of (9) becomes multi-valued at t* (and at some particular x*), (9) has no solution for t > t* unless we reinterpret the problem to allow discontinuous solutions, or "shocks". Certainly there is no reason to expect the solutions of (7) and (9) to be close after t*. Moreover, IVxxxl--" ~ as t ~ t* (near x = x*), so the neglected term in (7) becomes important for any 8 2 > 0 even for t < t*. A method to solve (7) in the small dispersion limit was invented by Lax and Levermore [11] and extended by Venakides [12]. Their work is reviewed by Lax in these proceedings. The accomplishment of Lax, Levermore and Venakides was to give precise meaning to lim
u(x, t; 8),
8 ---, 0
where u(x, t; 8) solves (7). They showed that in the context of square integrable functions, this limit exists as a strong limit for t < t* and as a weak limit for all t. They also showed how to compute both this limit and lim lim t~oo
u(x, t; 8).
840
The method of Lax, Levermore and Venakides probably has wider application than just the
4
H. Segur/Some open problems
K o r t e w e g - d e Vries equation. Here are two other problems where their method might be valuable. The first problem is to solve the classical nonlinear SchrSdinger equation (4) in the strong coupling (Icl ~ oo) limit. To recast this problem as a small dispersion limit, define
8 = c -1,
~= - c t .
(11)
lems in this small dispersion (or strong coupling) limit.
Problem # 2 . Solve (12) for 0 < - 6 << 1. Problem # 3 . D o e s the L a x - L e v e r m o r e Venakides method have a direct quantum analogue?
Then the problem becomes iq,~ = 21q,12~- 8eOxx, ,t,(x,
= o;
181 << 1, o
=
asx
----~ + 0 0 ,
(12)
find :
[
A second set of problems that can be viewed as small dispersion limits arise in the study of tran sonic flow in two dimensions. Krupp and Cole [13] show that an equation equivalent to
(u, + UUx) ~ + Uw = 0,
lim t x, ,r; 8 ).
(15)
8~0
As with (7), we obtain the limiting problem by setting 6 = 0 in (12),
i~b, = 21~1%,
~(x,0) = ~(x).
(13)
The solution of (13) is + ( x , r ) = ~ ( x ) exp (-2i1~(x)12~-).
(14)
In this case, the solution exists for all time, so the small dispersion limit of (12) should be less singular than that in (7). Even so q~x, the omitted term for (12), grows like ~.2 as z ~ oo, so one expects a naive perturbation expansion of the solution of (12) to break down when 8T 2 = 0(1), if not earlier. Thus (12) can be solved exactly for any finite 8, as can (7). The limit 6 ~ 0 is singular in both cases, because (5) and (8) break down. In both cases, the limiting (8 = 0) problem can be solved explicitly, but the omitted dispersive term ( 8 ~ x or 6 2 u ~ ) grows without bound for t > 0. Lax and Levermore [11] treated initial data which evolves primarily into solitons whereas Venakides [12] solved a problem with no solitons. The analogy in (12) is that its solution has no solitons if 6 > 0 (the repulsive case), and solitons if 8 < 0.
Problem # 1A. Solve (12) for 0 < 8 << 1. Relate the solutions of the classical and quantum prob-
plus appropriate boundary and initial conditions, approximately describe the unsteady, nearly sonic flow of an inviscid fluid over a thin two-dimensional airfoil. Roughly speaking, x represents distance along the wing, y measure distance normal to it, t is time, and u represents the streamwise component of the perturbation from the velocity far upstream. The model seems to be due to Timman [14] originally. A well-known model of steady transonic flow in two dimensions, q~vv= (k + ~bx)~bxx,
(16)
follows from (15) if we seek steady traveling waves (0, = - k a x ) and set u = -fix- [See Jameson [15] for a discussion of computational aspects of (16).] Both (15) and (16) are hyperbolic, like (9), and both admit shocks. It is known experimentally that these shocks contribute significantly to the drag of the airfoil, and a major objective in the design of airfoils to be used in transonic flight is to minimize the strength of these shocks. A challenging problem in computational fluid dynamics is to design a numerical scheme that can determine accurately the location and strength of these shocks. One method to compute shocks numerically is to smooth them with a little viscosity. That is, instead of starting with the equations for an in-
H. Segur/ Some open problems
viscid, compressible gas, start with the (compressible) Navier-Stokes equations, in which the viscosity is small but finite. Then virtually the same derivation that led to (15) leads instead to (17)
( u t + uu~) x + u w = ~,u . . . .
where ~ represents the fluid viscosity (0 < ~, << 1). This equation has the same physical significance as (15), its inviscid limit, but for 1, > 0 the shocks of (17) have finite thickness. If we omit (Uvv) from (17), it reduces to a derivative of the famous equation of Burgers [16]. Just as Burgers' equation describes weak onedimensional shocks, so (17) describes weak shocks that are also weakly two-dimensional. That is, (17) represents a physical generalization of Burgers' equation to two spatial dimensions. Burgers' equation is known to be integrable, and can be linearized by the transformation of Cole [17] and Hopf [18]. It is reasonable to ask, therefore, whether (17) might also be integrable. Applying the Painlev6 analysis of Weiss, Tabor and Carnevale [19] yields a negative result; fully twodimensional solutions of (17), i.e., those which cannot be transformed into solutions of Burgers' equation, do not have the Painlev6 property. An alternate method to smooth the shocks in (15) is to add a little dispersion, so (15) is replaced by (ut+uU~)x+U,,='u
.....
Icl <<1.
(18)
This replacement may be regarded simply as a computational convenience: the solutions of (18) remain continuous for c ~: 0, and they should approach the solutions of (15) as c --, 0. For E ~ 0 , (18) is the equation studied by Kadomtsev and Petviashvili [20], and it is known to be integrable. The method of solution of (18) is discussed both by Ablowitz and by Beals in these proceedings. For our purposes it is sufficient to note the following facts: (i) For c > 0, (18) can be solved exactly by a method due to Manakov [21]. For c < 0, (18) also
5
can be solved exactly, but Manakov's method fails and must be replaced by a method of Ablowitz, BarYaacov and Fokas [22]. (ii) Independent of the method, the qualitative nature of the solution of (18) depends of the sign of ~, because Kadomtsev and Petviashvili [20] showed that KdV-type (plane-wave) solitons are unstable for c < 0, but not for ~ > 0. (iii) For either sign of c, the limit c --, 0 in (18) is a small dispersion limit, completely analogous to the limit carried out by Lax, Levermore and Venakides for (7). (iv) For steady traveling waves, ( 0 t = - k O x ) , and (18) reduces to the Boussinesq equation Uyy=kuxx-(½u2)~
+cu .....
(19)
which also can be solved exactly. The small dispersion (~ -~ 0) limit of (19) is equivalent to (16). P r o b l e m # 4 . For c > 0, solve (18) in the small
dispersion limit, to find lim u ( x , y , t; E). ¢---~ 0
P r o b l e m # 5 . For ~ <0, solve (18) in the small
dispersion limit (~ ~ 0). P r o b l e m # 6 . One has mathematical justification
to define the solution of (15) to be the solution of problem # 4, or to be the solution of problem # 5, or to be the small viscosity (v ~ 0) limit of the solution of (17). A priori, there is no reason to believe that the three limits coincide. If they fail to coincide, then (15) would not have a unique solution, and a more careful reformulation would be required to make (15) well-posed. This (hypothetical) nonuniqueness of (15) raises serious practical questions, because a given numerical scheme might be approximating any one of the three solutions. P r o b l e m # 7 . Solve (19) in the small dispersion
(~ ~ 0) limit. Compare the solution of (16) obtained in this way with the results obtained by more conventional numerical algorithms. Does (16) have a unique solution?
6
H. Segur/Some openproblems
3. Nonlinear stability in Hamiltonian systems Nonlinear stability is of fundamental importance to this conference, because the physical significance of a "coherent state" depends on how stable it is. Typically, the more stable a state is the more easily it can be observed experimentally. In these proceedings, Marsden and Holm discuss an algorithm that they and their collaborators have developed to prove the nonlinear stability of certain equilibrium solutions of Hamiltonian systems. The basic ideas in this algorithm had been used earlier in isolated problems [cf. Kruskal and Oberman [23], Arnol'd [24], but Holm, Marsden, Ratiu and Weinstein [25] refined and organized these ideas into a clear, systematic algorithm]. In his lecture, Marsden used their algorithm to show that the single soliton is a stable solution of the Korteweg-de Vries equation,
u, + 6uu~ + uxx x = 0.
(20)
More generally, a nearly identical analysis shows that the simplest periodic solution of (20), known as a cnoidal wave, is also stable. An example of a cnoidal wave is shown in fig. 1. The soliton and cnoidal wave are equilibrium solutions of (20) in the sense that each is stationary (i.e., time-independent) in an appropriately translating coordinate system. In fact, they are the only bounded real solutions of (20) that are stationary.
A cruder, but far simpler, approach to stability can be summarized as follows: (i) Korteweg and de Vries (1895) originally derived their equation to model the one-dimensional propagation of waves of moderate amplitude in shallow water. (ii) Mathematical stability corresponds roughly to physical observability. (iii) Therefore, if cnoidal waves are stable, we should observe them in shallow water. Fig. 2, taken from an ancient National Geographic (1933), shows a very regular train of periodic, one-dimensional waves off the coast of Panama. Certainly the quahtative features of these waves (periodic waves with steep, localized crests and flat, broad troughs) correspond to those of the cnoidal waves in fig. 1. Thus, fig. 2 can be viewed as a qualitative corroboration of the nonlinear stability of cnoidal waves. Quantitative measurements of periodic waves in shallow water also support this positon, and ocean engineers have regarded cnoidal waves as "typical" (i.e., stable) one-dimensional, periodic waves in shallow water ever since Wiegel [27] brought the waves to their attention [e.g., see Sarpkaya and Isaacson [28]]. As a model of periodic waves in shallow water, cnoidal waves have the obvious limitation that they are one-dimensional, whereas the water surface is two-dimensional. This limitation raises two questions: a) What are the appropriate two-dimensional generalizations of the cnoidal wave and the soliton?
Fig. 1. A representative cnoidal wave solution of the KdV equation. The KP equation also admits one-dimensional, cnoidal wave solution.
H. Segur/ Some open problems
7
Fig. 2. Periodic waves in shallow water. The original caption read " A s they near shallow water close to the coast of Panama, huge, deep-sea waves, relics of a recent storm, are transformed into waves that have crests, but little or no troughs. A light breeze is blowing diagonally across the larger waves to produce a cross-chop. Three A r m y bombers, escorted by a training ship, are proceeding from Albrook Field, Canal Zone, to David, Panama."
b) Are the two-dimensional generalizations nonlinearly stable? Kadomtsev and Petviashvili [20] showed that their equation (i.e., (18), with c a fixed, non-zero constant) arises as a two-dimensional generalization of (20) when one relaxes slightly the requirement that the waves be strictly one-dimensional. In the context of water waves, this means that the fluid depth should be shallow in comparison with a typical wavelength in the direction of propagation, which in turn should be small in comparison with a transverse wavelength (see Ablowitz and Segur [29], for the full derivation). For gravity-induced water waves, the final result can be written in a standard form as
(u, + 6uu:, + u:,.~x) x + 3ur~v = O.
(21)
Obviously, every solution of (20) is a y-independent solution of (21). More generally, Satsuma [30] and Krichever [31] showed that (21) admits exact
solutions in the form
u(x, y, t) = 2 0 2 1 n 0 ,
(22)
just as does the KdV equation. Both N-soliton and N-phase quasi-periodic solutions can be obtained in this way. Fig. 3 shows a 2-soliton solution of the KP equation. Away from the region of interaction, each wave is essentially a KdV-type soliton. As expected, there is a phase shift because of the interaction. This solution has two phases, and it moves on a two-dimensional surface, so there is a uniformly translating coordinate system in which the entire wave pattern is stationary. The 2-soliton solutions of KP that a r e stationary in this sense can be viewed a two-dimensional generalizations of a single (plane-wave) soliton. The 2-soliton solution has a periodic analogue, called a solution of genus 2. An example is shown in fig. 4. Locally, the solution looks like two soli-
8
H. Segur/Some open problems
Fig. 3. Typical 2-soliton solution of the KP equation.
Fig. 4. Genus 2 solution of the KP equation. This family of exact solutions contains as limiting cases both the 2-soliton solution (fig. 3) and the cnoidal wave (fig. 1).
tons, but it is periodic in two directions. This solution also is stationary in an appropriately translating coordinate system. It represents a two-dimensional generalization of the cnoidal wave. (See Segur and Finkel [32] for a detailed discussion of these KP solutions of genus 2 as models of two-dimensional periodic waves in shallow water.) Given these two-dimensional generalizations of cnoidal waves and solitons, we can ask about their stability. The experimental approach is: Are these waves ever observed? Fig. 5 shows two interacting, soliton-type waves in shallow water, photographed by Terry Toedtemeier off the coast of Oregon. The qualitative agreement between the wave patterns in figs. 3 and 5 is remarkably good, but we have so little quantitative information about the ocean
waves that the apparent agreement can only be considered suggestive. Similarly, an observation of two-dimensional waves in shallow water that is in qualitative agreement with fig. 4 was made by Joe H a m m a c k , [cf. fig. 1.9 of Segur, Finkel and Philander [33]. Neither of these observations provides enough quantitative information to justify a conclusive statement about the validity of the stationary K P solutions of genus 2 as models of actual water waves. In particular, the stability of these K P solutions is open. Problem # 8. Use the stability algorithm of Holm,
Marsden, Ratiu and Weinstein [25] to determine the nonlinear stability of the KP solutions of genus 2 that are stationary in an appropriate coordinate system.
H. Segur/ Some open problems
9
Fig. 5. Oblique interaction of two nearly solitarywaves in shallow water. (Photographcourtesy of T. Toedtemeier.)
4. Two-dimensional motion of a perfect fluid The last problem also involves questions of stability: to determine the long-time behavior of an inviscid, incompressible fluid as it moves in a two-dimensional plane. The problem arises as a simplified model with applications in geophysics [e.g., Salmon [34]] and plasmas [e.g., Morrison [35]]. It was first considered by Euler, with more recent contributions by several of the speakers at this conference. Here is a statement of the problem. Let x = (x, y ) define a position in a two-dimensional plane, u = (u, v) represent the velocity vector of an inviscid, incompressible fluid moving in that plane, p (x, t) denote the fluid pressure, and t represent time. The dynamical equations of the fluid are Euler's equations in two dimensions, Du
Dt
= ( ~t + u" W)u
W.u=0.
= - WP'
(23)
Initial and boundary conditions are also needed to make (23) into a well-posed problem. For definiteness, we impose periodic boundary conditions in (x, y ) and we require that u(x, t = 0) be smooth and periodic. It should be emphasized that the long-time behavior of the solution of (23) seems to be rather delicate, and the result may depend sensitively on the initial and boundary conditions.
Problem # 9. Given smooth initial data, does the solution of (23) with periodic boundary conditions tend to any limit as t ---, oo? Does the existence of such a limit depend on the initial data? Conversely, does (23) admit solutions that are recurrent in time? Is the system ergodic? Is it mixing (in the sense of statistical mechanics)? The governing equation become particularly simple in terms of the vorticity. Define the (scalar) vorticity by 6o = v x -
Uy
= (~TX u ) . k .
(24)
10
H. Segur / Some open problems
Then (23) reduces to the very simple form D~
Dt
=0.
(25)
Each fluid particle has some initial vorticity and (25) says that the fluid particle carries that vorticity with it as it moves around. At each instant, therefore, the velocity field is obtained by integrating over the vorticity, and then the vorticity is transported by the velocity field so generated. Conceptually, the motion is extremely simple: the vorticity field simply convects itself around. Given this conceptual simplicity, it is surprising how difficult it has been to characterize (correctly) the nature of the resulting flow. Several properties of this system are known (e.g., see Holm, et al. [25] for more details). (i) Arnol'd showed that the system is Hamiltonian. The Hamiltonian function is the kinetic energy,
H =ff½u.udA,
(26)
where the integral is taken over a (two-dimensional) period. The exact form of the appropriate Poisson bracket is not needed for our purposes. (ii) Another positive-definite, conserved quantity is called the enstrophy,
E = ff½~e dA.
(27)
(iii) Because the vorticity is conserved for each fluid particle, and because the fluid motion simply rearranges these particles, it follows that the integral of any smooth function of the vorticity is conserved. In particular, (23) conserves the moments of vorticity,
I,=ffoa'dA,
n=1,2,3 ....
(28)
Therefore (23) has infinitely many constants of the motion.
(iv) In these proceedings, Marsden shows that the moments of vorticity are examples of "Casimirs"; i.e., they annihilate the Poisson bracket appropriate for (23). It follows that this infinite set of constants of the motion cannot be used to prove that (23) is completely integrable, contrary to a conjecture of Ebin [36]. This point was first made by Marsden and Weinstein [37], who conjectured that (23) is not completely integrable. Certainly it is widely believed in the geophysical community that (23) is not completely integrable, but rather that its solutions exhibit an irreversible cascade of energy to low wavenumbers and of enstrophy to high wavenumbers [cf. Rhines [38] or Salmon [34]. Evidence to support the plausibility of these cascades can be found in the numerical simulations of Zabusky (in these proceedings). The energy cascade appears there as "vortex merging", while the enstrophy cascade corresponds to "illamentation." The argument that such a cascade necessarily occurs seems to rest on a very simple argument given by Salmon [34], among others. That argument shows that if the solution of (23) has a limit as t-~ oo, then it is very likely that energy will tend to lower wavenumbers as t -~ o0. On the other hand, if a solution of (23) is recurrent, with any reasonable recurrence time, then there is no limit as t -~ oo, so the usual argument is irrelevant. I am aware of no strong argument which precludes the possibility of recurrent motion or even gives a lower bound on a possible recurrence time. At this time, the ubiquity of the e n e r g y / e n s t r o p h y cascades probably should be regarded as open. For numerical purposes, it is sometimes convenient to represent the solution of (23) with a time-dependent Fourier series, and to replace (23) with an infinite set of ordinary differential equations for the evolution of the Fourier coefficients. In this formulation, different numerical schemes correspond to different truncations of this infinite set of equations. It is reasonable to consider only those truncations that conserve at least energy (26) and enstrophy (27). Hald [39] found several lowdimensional truncations that conserved energy, en-
H. Segur / Some open problems
s t r o p h y , a n d e n o u g h o t h e r c o n s t a n t s that the t r u n c a t e d s y s t e m s were c o m p l e t e l y integrable. Cert a i n l y H a l d ' s t r u n c a t e d systems exhibit no irreversible cascades. O n the o t h e r hand, Lee [40] a n d Kells and O r s z a g [41] s h o w e d that H a l d ' s extra integrals d e p e n d o n H a l d ' s p a r t i c u l a r truncations, a n d disa p p e a r if the t r u n c a t i o n is d o n e " i s o t r o p i c a l l y . " Lee [40] c o n c l u d e d that (23) with p e r i o d i c b o u n d a r y c o n d i t i o n s " h a s no e x t r a n e o u s c o n s t a n t s of m o t i o n [i.e., b e y o n d energy a n d enstrophy], unless the p r e s e n c e o f some other constants of m o t i o n is d i s c o v e r e d . " Because of its p r a c t i c a l i m p o r t a n c e , (23) has b e e n the subject o f a great deal of n u m e r i cal work. G l a z [42] reviews m u c h of this work. H e c o n c l u d e s t h a t all of the " n u m e r i c a l studies of the e r g o d i c i t y / m i x i n g questions for the t r u n c a t e d E u l e r e q u a t i o n s (including this one) have been inconclusive." T o s u m m a r i z e , (23) with p e r i o d i c b o u n d a r y cond i t i o n s is the subject of strongly held opinions, b u t t h e r e seems to be little conclusive evidence to s u p p o r t these opinions. T h e n a t u r e of the l o n g - t i m e b e h a v i o r o f the solutions (23) r e m a i n s an o p e n question.
Acknowledgements T h i s w o r k was s u p p o r t e d in p a r t b y the U.S. A r m y R e s e a r c h Office, a n d b y the N a t i o n a l Scie n c e F o u n d a t i o n u n d e r G r a n t No. PHY82-17853, s u p p l e m e n t e d b y f u n d s f r o m the N a t i o n a l A e r o n a u t i c s a n d Space A d m i n i s t r a t i o n .
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11
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H. Segur/ Some open problems
(1982) pp. 30-78. [35] P.J. Morrison, in: Math. Methods Hydrodyn. and Integ. Dyn. Syst., AIP Conf. Proc. #88, M. Tabor and Y.M. Treve, eds. (1982) pp. 13-46. [36] D.G. Ebin, Comm. Pure Appl. Math. 36 (1983) 37-56. [37] J.E. Marsden and A. Weinstein, Physica 7D (1983) 305-323.
[38] [39] [40] [41]
P.B. Rhines, Ann. Rev. Fluid. Mech. 11 (1979) 401-442. O. Hald, Phys. Fluids 19 (1976) 914-915. J. Lee Phys. Fluids 20 (1977) 1250-1254. L.C. Kells and S.A. Orszag, Phys. Fluids 21 (1978) 162-168. [42] H. Glaz, SIAM J. Appl. Math. 41 (1981) 459-479.