Cnoidal Waves on Fermi–Pasta–Ulam Lattices
We study a chain of infinitely many particles coupled by nonlinear springs, obeying the equations of motion
with generic nearest-neighbour potential . We show that this chain carries exact spatially periodic travelling waves whose profile is asymptotic, in a small-amlitude long-wave regime, to the KdV cnoidal waves. The discrete waves have three interesting features: (1) being exact travelling waves they keep their shape for infinite time, rather than just up to a timescale of order wavelength suggested by formal asymptotic analysis, (2) unlike solitary waves they carry a nonzero amount of energy per particle, (3) analogous behaviour of their KdV continuum counterparts suggests long-time stability properties under nonlinear interaction with each other. Connections with the Fermi-Pasta-Ulam recurrence phenomena are indicated. Proofs involve an adaptation of the renormalization approach of  to a periodic setting and the spectral theory of the periodic Schrödinger operator with KdV cnoidal wave potential.
ams:70F, 70H12, 35Q51, 35Q53, 82B28
Keywords: Fermi–Pasta–Ulam problem, Korteweg–de Vries equation, cnoidal wave solutions, solitons.
The Fermi-Pasta-Ulam model consists of a one-dimensional chain of particles coupled by nonlinear springs, obeying the equations of motion
Here is the displacement of the n particle out of equilibrium at time , and is an anharmonic potential such as with (the FPU- chain) or with (the FPU- chain). This model provides a fascinating paradigm of nonlinear Hamiltonian many-particle dynamics. On the one hand, it is simple enough to allow insight by rigorous mathematical analysis. On the other hand, it already exhibits a rich spectrum of phenomena of wider importance: coherent signal- and energy transport (as seen in biomolecules such as DNA); near-integrable behaviour (as documented by the fact that in certains regimes, the FPU model is well approximated by the Korteweg-de Vries equation, of which more below); dispersive shocks (as seen in molecularly resolved gas dynamics); and statistical irreversibility and thermalization effects despite microscopic reversibility (as described by statistical mechanics).
A central numerical phenomenon in the system (1) is a crossover from energy trapping in a few long-wave modes at low initial energy per particle to ergodic-like spreading of energy to short-wave modes at high initial energy per particle. See Fermi, Pasta, Ulam and Tsingou111Mary Tsingou was involved in the numerical work as acknowledged in the original report.  for first observations at low energy, Israilev and Chirikov  for first observations at high energy, Dreyer and Herrmann  for energy transfer to short-wave modes via dispersive shocks, and  for a nice review.
A significant amount of understanding of the recurrent and non-statistical behaviour at low energy has emerged via approximation of the FPU model by completely integrable systems. In this paper
– we argue that this level of understanding, reviewed below, is not completely satisfactory because it fails to cover the physically most interesting regime, fixed nonlinearity, nonzero energy per particle, and larger and larger system size
– we hope to convince readers that a deeper understanding could come via establishing existence, and long-time stability under interaction with each other, of spatially periodic and quasi-periodic waves in the infinite FPU chain
– and we rigorously carry out a first step, establishing existence of certain spatially periodic travelling waves in FPU which are good candidates for such special interaction properties, namely waves which are, in a small-amplitude long-wave limit, asymptotic to the celebrated KdV cnoidal waves.
In the remainder of this Introduction we review current theoretical understanding of non-statistical behaviour of FPU at low energy, and informally describe our results.
1. The invariant tori explanation. One line of thought, going back to , is to consider a fixed and finite number of particles only, and linearize, i.e. approximate the anharmonic interaction potential by a purely harmonic one. The resulting system is, of course, completely integrable, with the phase space being foliated by the invariant tori given by the set of states with a given fixed amount of energy in each normal mode. By Kolmogorov-Arnol’d-Moser (KAM) theory, when switching on the anharmonic terms, many invariant tori survive. This prevents typical solutions from spreading energy to the entire phase space. This argument can be made rigorous by using Birkhoff normal forms .
2. The soliton explanation. Another approach, going back to Zabusky and Kruskal , is to keep the number of particles infinite, consider a suitable small-amplitude long-wave regime, and observe (see Remark 2.1 below for their heuristic argument) that (1) is then well-approximated by a nonlinear, infinite-dimensional integrable equation, the KdV equation
(More precisely, the latter approximates the FPU- chain; in case of the FPU- chain one obtains the mKdV equation.) The explanation of non-statistical behaviour advocated in  then went as follows: the latter equation carries ’solitons’, i.e. solitary waves which propagate exactly under the nonlinear KdV dynamics; numerically these solitons also show persistent shapes and velocities after nonlinearly interacting with each other; and general spatially localized initial data can be well approximated by superposition of finitely many such solitons. Hence, again, energy is prevented from spreading to the entire phase space. Significant aspects of this picture has nowadays become rigorous mathematics. In particular, the inverse scattering transform introduced in 1974 by Gardner et al.  allows to prove the radiationless interaction of solitons, as well as show that all spatially localized initial data asymptotically split up into a superposition of solitons (see e.g.  for more information and further references). And the approximation of (1) by (2) can be made rigorous up to timescales of order (wavelength) for general localized solutions , and globally in time for special solutions of soliton type [12, 13, 14, 15] and multi-soliton type [38, 37].
From a physical point of view, neither the invariant tori explanation nor the soliton explanation are completely satisfactory. This is because they are limited to regimes satisfying certain undesirable restrictions:
The invariant tori explantion is based on finite-dimensional KAM theory, and hence the allowed size of the anharmonicity tends rapidly to zero as the particle number gets large . It thus does not apply to the natural situation of many particles interacting via a fixed nonlinear potential. Very interesting results are available on KAM theory for perturbations of infinite-dimensional systems , but perturbing the harmonic lattice (or indeed other infinite-dimensional integrable systems like the Toda lattice or KdV) into FPU lies well beyond the scope of these results.
The soliton explanation does apply to infinitely many particles interacting via a fixed nonlinear potential, but it is only valid for spatially localized states; but spatial localization forces the energy per particle to be zero.222Curiously, the numerical KdV simulations in initial paper  on the soliton explanation were done at finite energy per particle, via imposing periodic boundary conditions, but the subsequent theoretical analysis of soliton interactions was done at zero energy per particle, via just considering a fixed number of localized solitons on the whole real line.
How, then, could we access the behaviour of infinite FPU chains with fixed nonlinear potential and nonzero energy per particle? Our own, admittedly rather modest, contribution towards this challenge is the following. We show that (1) carries exact spatially periodic travelling waves
with small-amplitude long-wave profile
and near-sonic velocity
where is the relative displacement between neighbouring particles and is a KdV cnoidal wave profile of speed 1, i.e. a spatially periodic function such that solves the KdV equation (2). Such solutions to KdV exist and are known explicitly via algebraic and geometric methods, see Figure 1 and the end of this Introduction.
While interesting non-explicit existence results on periodic FPU travelling waves have been obtained previously [46, 11, 39, 21], the key point which makes the new result (3)–(5) promising in the context of energy trapping in FPU at low energy per particle is now the following. The discrete cnoidal waves (3)–(5) are not arbitrary FPU travelling waves with finite energy per particle, but are good candidates for exhibiting special stability properties under nonlinearly interacting with each other, on account of analogous remarkable properties of their continous KdV counterparts. As made more precise below, the KdV cnoidal waves are the spatially periodic analoga of the KdV solitons, and appropriate nonlinear superpositions, the so-called finite-gap KdV solutions, are (spatially quasi-periodic) analoga of the KdV multi-solitons. A long-term goal, then, which lies well beyond the scope and tools of this article, would be to investigate existence and long-time stability of discrete FPU-analoga of general finite-gap KdV solutions. Perhaps infinite-time stability of such waves in the spirit of the deep recent results of Mizumachi [38, 37] on discrete FPU-analoga of multi-solitons no longer holds, but it is conceivable that long finite-time results analogous to those of Hoffman and Wayne  on FPU solitary wave interactions can be transferred to a quasi-periodic setting.
The above line of thought also suggests, as to the best of our knowledge has not been pointed out previously, that the linear normal modes which were nonlinearly evolved in the original FPU simulations  and underlie the ”KAM explanation”  enjoy special interaction properties under nonlinear FPU dynamics. This is because, in the limit of low amplitude, the KdV cnoidal waves become close to linear waves plus a constant; see the dotted curve in Figure 1 and (41) in Section 4.
Next, let us compare our result (3)–(5) to previous work on localized waves in the FPU model. Existence of travelling waves with constant asymptotic values at infinity has been established via different approaches: variational methods [16, 43, 11, 39, 21], see also [42, 22] for a generalization to nonconvex potentials, center manifold arguments , and comparison to KdV . The latter two approaches are limited to small amplitude, but also deliver the waveform.
We close this Introduction by describing briefly the above-mentioned finite-gap KdV solutions, and explaining how they encode and reveal remarkable nonlinear interaction properties of the KdV cnoidal waves in Figure 1. The finite-gap solutions can be constructed via algebraic and geometric methods going back to Its, Matveev, Dubrovin, McKean, and van Moerbeke [26, 27, 9, 8, 35], and are given by
where denotes the scalar product in and is a constant. Moreover, is a symmetric matrix and denotes the vector of the generalized phases , , which are determined by an underlying prescribable Riemann surface (see B). In general, the ’s and ’s are incommensurable quantities and thus KdV solutions of the form (6) are quasi-periodic in and . In the one-gap case () the solutions (6) reduce to the, spatially periodic, KdV cnoidal waves given by
Physically, the general finite-gap solutions (6) can be interpreted as a linear superposition of cnoidal waves of the form (7) plus a nonlinear interaction term. Here the diagonal elements of the symmetric matrix determine the periods of these cnoidal waves, while the off-diagonal elements give their nonlinear interaction. In this precise sense, these solutions are finite-energy-per-unit-volume analoga of the KdV multi-solitons.
For a more detailed description of quasi-periodic KdV solutions we refer to B.
2 Main result
We now describe precisely or result on existence and shape of cnoidal-type waves in Fermi-Pasta-Ulam chains.
The governing equations are
where denotes the displacement out of equilibrium of the th particle. Formally, the associated Hamiltonian energy
(with denoting the particle momenta) is conserved along solutions, but our interest is in infinite-energy solutions. In this paper we make the following assumptions on the nearest-neighbour potential:
Well-known examples are given by the potential of the cubic FPU chain , the Lennard-Jones-(12,6) potential , and the Toda potential . The latter gives rise to a completely integrable Hamiltonian system. Let us denote by
the distortion of the th bond length out of equilibrium. Moreover, we introduce the shift operators for the backward and forward shifts along the lattice, respectively. With these notations the equations of motion (8) become
One can write this equation as a first-order Hamiltonian system of the form
where , , and denotes the functional gradient of . By making the travelling wave ansatz equation (12) becomes
Since we aim to consider spatially periodic solutions, the function spaces appropriate for our setting will be the periodic Sobolev spaces
where is the set of -periodic distributions, i.e. the set of all continuous linear functionals from (the set of all smooth -periodic functions from into ) into . The Fourier series of and its Fourier coefficients are given by
The space is a Hilbert space with the inner product
which describes waves travelling at approximately the speed of sound while showing a negligible temporal and spatial change in their shape. Then by Taylor expanding differences and neglecting terms of order in equation (13) one obtains a KdV equation - arising as the coefficients of the -terms - of the form
From comparison of the -terms one obtains the relation for the speed of sound.
The ansatz (15) can be justified rigorously via a renormalization appraoch adapted from  (see Section 3), spectral theory for the Schrödinger operator with KdV cnoidal wave potential (see Section 4 and A), and an implicit function theorem argument (see Section 5). This leads to the following rigorous persistence result for cnoidal waves when deforming the limiting KdV equation back into the lattice equation.
Assume that the nearest neighbour interaction potential satisfies the assumptions (10), and let denote the sonic wave velocity. Fix the constants and such that , where denotes the complete elliptic integral of the first kind, i.e. . Then the following statements hold:
(existence and local uniqueness) Given sufficiently small, there exists such that, for and , the governing equation (13) for the periodic travelling wave profile admits a unique solution in the set
where denotes the KdV cnoidal wave profile with speed that solves the integrated KdV travelling wave equation
Explicitly, is given by
Here denotes one of the Jacobian elliptic functions (see Section 4 for its precise definition).
(asymptotic shape) The solution from (a) satisfies the estimate
where is independent of .
(smoothness) The mapping from into is .
3 Renormalization and the continuum limit
The formal argument in Remark 2.1 which relates the lattice equation to the KdV equation is not mathematically satisfactory. This is because it involves uncontrolled truncation of a Taylor expansion of a difference operator into (more and more unbounded) difference operators. To overcome this problem, Friesecke and Pego  introduced – in the context of spatially localized waves – a renormalization approach which recasts both the lattice travelling wave equation (13) and the limiting KdV travelling wave equation in a form involving only bounded operators (see (20) below). The ensuing lattice Fourier multiplication operator governing lattice waves and its limiting continuum counterpart can then be shown, via a careful analysis of the location of their complex poles, to be rigorously close in an appropriate operator norm. As we show in this section, this approach can be adapted to the periodic setting. However, as we will see in Section 4, the limiting KdV equation (17) becomes more subtle in the periodic setting, admitting a two-parameter family of solutions in place of the one-parameter soliton family.
Let us make the assumptions (10) on the nearest-neighbour potential . We isolate the nonlinear from the linear part of the restoring force by writing
Since for and , this equation can be transformed into the fixed point equation
where denotes the pseudo-differential operator with symbol
Here denotes the usual sinc function.
Let us introduce a small parameter by setting
where and are fixed real constants. By rescaling variables via
the renormalization equation (20) turns into an (-dependent) fixed point equation for :
where and the operator has the symbol
From this and the Taylor expansion one obtains for the pointwise limit of of the symbol as :
The nonlinearity satisfies . Hence as the fixed point equation (24) converges to the equation
By applying the operator to both sides, we see that this equation is equivalent to the integrated KdV equation
which stems from integrating the usual KdV travelling wave equation
with vanishing integration constant.
A key advantage of the above analysis over the formal asymptotics in Section 2 is the fact, shown in , that the convergence in (27) is in fact uniform. This will allow us to show that the multipliers and are close in operator norm on the periodic Sobolev space introduced in the previous section. It is useful to introduce the notation
the -independent functions and coincide with those considered in .
a) was proved in [12, Lemma 3.1], by careful analysis of the location of the poles of in the complex plane. To show b), note that, for any operator on of form ,
Applying this with , yields
Estimating the above supremum by that over and applying a) yields the assertion. ∎
4 Periodic KdV travelling wave solutions
In this section we explicitly describe the periodic travelling wave solutions of the (integrated) KdV equation (29), see also [6, 40]. Here a main difference from the solitary wave case is present. Mathematically, there arises a non-zero integration constant after multiplying the KdV equation (17) by and integrating. This forces us to deal with a 2-parameter system (for instance represented physically by the speed and period of the wavetrain) in contrast to the soliton case, which is fully described by one parameter corresponding to the velocity (or equivalently the amplitude or width) of the single soliton. The class of periodic solutions contains soliton solutions as a degenerate limit, namely infinite period.
By multiplying the (integrated) KdV equation (29) with and integrating once more, we get
Let us denote the polynomial in the bracket on the right hand side by . For finding periodic solutions of (32) it is useful to consider the roots of . The function can by factorized by
where , , and denoted the three roots. It turns out that real periodic solutions occur if the three zeros of the polynomial are real and distinct, i.e. . The dependence of the zeros on the parameters and is thus given by
The real and bounded solutions are contained in the region , whereas in the region the solutions are unbounded. Introduce the normalized variable , then (32) becomes
where we have set for . The new variable takes values in the interval . Assume that a maximum value of is at which can always be achieved by a coordinate translation. Since the critical points of are at the boundary of the interval we conclude . Moreover, we have and . Hence a bounded solution oscillates between the values and .
In order to construct such a solution let us again make a change of variables by defining implicitly via
where is continuous with . With this definition (36) becomes
We want to use the standard notation in this context and therefore define the parameters
which fulfill and . Using this notation (37) can be written in the form
which can be solved implicitly by integration
where is a final arbitrary constant of integration. Here is called the standard elliptic integral of the first kind and the elliptic modulus. The Jacobian elliptic function is defined as the inverse of the function via . Another basic elliptic function, the cnoidal function , is defined in terms of via . Thus
Hence by recasting variables appropriately the solution of (29) is finally given by the so-called cnoidal wave
where the travelling wave coordinate is
and the elliptic modulus is given by
The solution (39) is periodic if with period
is called the complete elliptic integral of the first kind.
It is worthwhile to consider two limiting cases of these KdV cnoidal wave solutions.
The ”most nonlinear” limit: In the case (corresponding to ) we have and . Thus in this case the solution of (16) with corresponds to a single soliton solution
where . Due to the invariance of the KdV equation under Galilean transformations, i.e. transformations of the form
this solution is equivalent to the well-known standard form
The ”linear” limit: In the case which corresponds to the limit (amplitude tending to zero) we have and . Therefore the limiting behaviour of (16) with as is
where . Here we made use of . Thus in leading orders of this small amplitude limit the cnoidal wave describes a linear wave.
From the considerations above we deduce that the periodic solutions of the type (39) can be considered an intermediate form between linear waves and highly nonlinear solitons.
By solving the KdV travelling wave equation (30) via integration one has in general to deal with a 3-parameter system, stemming from the cubic polynomial . Thus the KdV solutions constructed in this way depend on 3 parameters, reflected by the roots , the integration constants and velocity , or elliptic modulus, period and velocity . Due to the choice of the renormalization ansatz (23) the governing equations (13) for the string are related to an integrated KdV equation (29) with zero integration constant, i.e. . Hence the 3-parameter reduces to a 2-parameter system. We aim to admit periodic KdV travelling waves to our considerations; for that purpose it is essential that the second integration constant is non-zero. Otherwise the system would include only one free parameter giving rise to soliton solutions.
The cnoidal wave solution (39) is given in terms of the roots , , and . In the following we derive an equivalent expression in the parameters , , and . Since the latter model directly the ”physical” quantities like the shape, period, and velocity of the wavetrain this will help us to get a less abstract picture of the solutions we are dealing with. Besides it enables us to directly relate our spectral analysis results derived in Section 5.1 to studies on the Lamé equation, cf. A.
Let us start by making the ansatz with . Insert this expression into (29) and use the relations , , and . Then comparing coefficients yields the following system of equations
Since we assume to be periodic with period and is a -periodic function we have . Hence from (44) we get
Moreover, from (43) one obtains
Finally, inserting these formulas into (44) gives the following quadratic equation
which has the positive solution
Thus we can write the cnoidal wave solution (39) in the convenient form
where is the travelling wave coordinate with the velocity
Note that an equivalent renormalization approach would be the following: replace the renormalization ansatz (23) by , where denotes an arbitrary constant. In the limit this leads to an integrated KdV equation for given by
where . Indeed, just set in (29). When integrating this equation once more assume the second integration constant to vanish. Then one of the zeros