# Korteweg–de Vries limit for the Fermi–Pasta–Ulam system

###### Abstract.

In this paper, we develop dispersive PDE techniques for the Fermi–Pasta–Ulam (FPU) system with infinitely many oscillators, and we show that general solutions to the infinite FPU system can be approximated by counter-propagating waves governed by the Korteweg–de Vries (KdV) equation as the lattice spacing approaches zero. Our result not only simplifies the hypotheses but also reduces the regularity requirement in the previous study [45].

## 1. Introduction

The Fermi–Pasta–Ulam (FPU) system is a simple nonlinear dynamical lattice model describing a long one-dimensional chain of vibrating strings with nearest neighbor interactions. This model was first introduced by Fermi, Pasta, and Ulam in the original Los Alamos report [12] in 1955 with regard to their numerical studies on nonlinear dynamics. At that time, it was anticipated that the energy initially given to only the lowest frequency mode would be shared by chaotic nonlinear interactions and it would be eventually thermalized to equilibrium. However, numerical simulations showed the opposite behavior. The energy is shared among only a few low-frequency modes and it exhibits quasi-periodic behavior. This phenomenon is known as the FPU paradox. Since then, the FPU paradox has emerged as one of the central topics in various fields, and it has stimulated extensive studies on nonlinear chaos.

Among the various important studies in this regard, the most remarkable one is the fundamental work of Zabusky and Kruskal [49], in which the problem was solved for the first time by discovering a connection to the Korteweg–de Vries (KdV) equation. The authors showed that the FPU system is formally approximated by the KdV equation and its quasi-periodic dynamics is thus explained in connection with solitary waves for the KdV equation. From an analysis perspective, Friesecke and Wattis proved that the FPU system has solitary waves [17], confirming the numerical observation [11], whereas Friesecke and Pego established their convergence to the soliton solutions to the KdV equation [13]. Moreover, various qualitative properties have been proved for the FPU solitary waves [14, 15, 16, 38].

The KdV approximation problem has also been investigated for general states without restriction to solitary waves. For an infinite chain, Schneider and Wayne showed that the FPU flow can be approximated by counter-propagating KdV flows (see (1.9) below) via the multi-scale method [45]. This approach has been applied to a periodic setting [40] as well as to generalized discrete models [37, 6, 19]. Furthermore, with a different scaling, the cubic nonlinear Schrödinger equation is derived from the FPU system [44] (see also [21, 22]).

In contrast, the FPU paradox can be explained in a completely different manner, i.e., by the approach of Izrailev and Chirikov [27], which involves the Kolmogorov–Arnold–Moser (KAM) theory: quasi-periodicity occurs because the FPU system can be approximated by a finite-dimensional integrable system (see [39, 41, 42] for this direction). We also note that the quasi-periodic dynamics vanishes after a sufficiently long time-scale as predicted originally [18]. This phenomenon is called metastability, and it has been investigated rigorously (e.g., [2, 1]). Overall, the dynamics problem for the FPU system has garnered considerable research attention and it has been explored from various perspectives. We refer to the surveys -[48, 20] and the references therein for a more detailed history and an overview of the problem.

In this article, we follow the approach of Zabusky and Kruskal [49]. Our objective is to provide a rigorous justification of the KdV approximation for general solutions, including solitary waves, to infinite FPU chains. Let us begin with introducing the setup of the problem. Consider the FPU Hamiltonian

(1.1) |

for a function . Here, denotes for the position and momentum of the -th string, and the potential function determines the potential energy from nearest-neighbor interactions. We assume that

(1.2) |

Such potentials include the cubic FPU potential , the Lennard-Jones potential . and the Toda potential , a more general polynomial potential

The above-mentioned Hamiltonian generates the FPU system

(1.3) |

where . By combining the two equations in the system and then rewriting them for the relative displacement between two adjacent points, , we can simplify the system as

(1.4) |

where . Next, by rescaling with

(1.5) |

for small , we obtain

(1.6) |

where is a discrete Laplacian on , i.e.,

Finally, by extracting the linear term from the right-hand side of (1.6), we derive a discrete nonlinear wave equation, which we refer to hereafter as the FPU system

(1.7) |

where . This reformulated equation is still a Hamiltonian equation with the Hamiltonian^{1}^{1}1It is derived from (1.1).

(1.8) |

Through the formal analysis described in Section 2, one would expect that the solutions to FPU (1.7) are approximated by counter-propagating waves

(1.9) |

where each is a solution to the KdV equation

(1.10) |

This method of deriving the two KdV flows can be regarded as an infinite-lattice version of the method of Zabusky and Kruskal [49].

In this study, we revisit the KdV limit problem for general solutions, albeit through a rather different approach. Indeed, in a broad sense, a dynamical system approach was adopted in all the aforementioned studies [45, 21, 40, 37, 22, 44, 6, 19]. By regarding the FPU system (1.7) as a nonlinear dispersive equation, we exploit its dispersive and smoothing properties, and we then employ them to justify the KdV approximation. This approach enables us to not only simplify the assumptions on the initial data in the previous study but also reduce the regularity requirement.

For the statement of the main theorem, we introduce the basic definitions of function spaces, the Fourier transform and differentials on a lattice domain, and the linear interpolation operator. For , the Lebesgue space is defined by the collection of real-valued functions on a lattice domain equipped with the -norm

For , we define its (discrete) Fourier transform by

Meanwhile, for a periodic function , its inverse Fourier transform is given by

Then, Parseval’s identity,

(1.11) |

extends the discrete Fourier transform (resp., its inversion) to (resp., ).

There are several ways to define differentials on a lattice domain . Throughout the paper, we use the following different types of differentials, all of which are consistent with differentiation on the real line as the Fourier multiplier of the symbol as .

###### Definition 1.1 (Differentials on ).

(resp., , ) denotes the discrete Fourier multiplier of the symbol

For , we define the Sobolev space (resp., ) by the Banach space equipped with the norm

(1.12) |

In particular, when , we denote

To compare functions on different domains, we introduce the linear interpolation

(1.13) | ||||

for all with . Note that the linear interpolation converts a function on a lattice into a continuous function on the real line.

Now, we are ready to state our main result.

###### Theorem 1.2 (KdV limit for FPU).

If satisfies (1.2), then for any , there exists such that the following holds. Suppose that for some ,

(1.14) |

Let be the solution to FPU (1.7) with initial data , and let be the solution to KdV (1.10) with interpolated initial data .

(Continuum limit)

(1.15) |

(Small amplitude limit) Scaling back,

is a solution to FPU (1.4). Moreover, it satisfies

(1.16) |

We remark that the assumption on the initial data is simplified compared to the previous work [45]. We assume only a uniform bound on the size of the initial data (see (1.14)) in a natural Sobolev norm (without any weight), and the mean-zero momentum condition is not imposed. Furthermore, the regularity requirement is reduced to .

As for the regularity issue, we emphasize that reducing the regularity is not only a matter of mathematical curiosity but it may also lead to a significant improvement in the continuum limit (1.15). As stated in our main theorems, the KdV approximation is stated in the form of either a continuum limit or a small amplitude limit. Mathematically, they are equivalent; however, the continuum limit (1.15) seems rather weaker because it holds only in a short time interval , whereas the small amplitude limit (1.16) is valid almost globally in time . Thus, it would be desirable to extend the time interval arbitrarily for the continuum limit. For comparison, we state that for discrete nonlinear Schrödinger equations (DNLS), the continuum limit is established in a compact time interval of any size [24], and an exponential-in-time bound is obtained. In the proof, conservation laws obviously play a crucial role. However, unlike DNLS, the FPU system does not have a conservation law controlling a higher regularity norm, say the norm. Only an -type quantity is controlled by its Hamiltonian (1.8). Therefore, it would be desirable to establish the continuum limit for -data. If such a low regularity convergence is achieved, then one may try to employ the conservation law to extend the size of the interval to be arbitrarily large. Although the regularity is significantly reduced in this study, our assumption that is still far from the desired case of . At the end of this section, we mention the technical obstacle that prevents us from going below . Instead of sharpening the estimates, a new idea seems necessary to reduce the regularity. as

The main contribution of this article is to present a new approach to the KdV limit problem from the perspective of the theory of nonlinear dispersive PDEs. In spite of the dispersive nature of the FPU system, which is clear from its connection to the KdV equation, to the best of authors’ knowledge, there has been no attempt to tackle the problem using dispersive PDE techniques thus far.

Our approach is achieved on the basis of the following two observations. First, as outlined in Section 2, we reformulate the FPU system (1.7) by separating its Duhamel formula into two coupled equations (2.3), which we refer to as the coupled FPU. Indeed, this is a standard method to deal with inhomogeneous wave equations; however after implementing it, we realized that it is much easier to understand the limit procedure by analyzing the symbols of the linear propagators and their asymptotics (see Remark 2.1). By this refomulation, we introduce a different convergence scheme to the KdV equation via the decoupled FPU (2.12). It makes the problem more suitable and clearer for analysis by dispersive PDE techniques.

Second, we discover that the linear propagators for the coupled and decoupled FPUs exhibit properties similar to those of the Airy flows in many aspects. A technical but crucial feature of our analysis is that the phase functions of the linear FPU propagators are comparable with those of the Airy propagators at different derivative levels. Indeed, direct calculations show that

on the frequency domain 1.2). This allows us to recover the Strichartz estimates, the local smoothing and maximal function estimates (Proposition 5.1), and the bilinear estimates (Lemma 6.1) for the linear FPU flows owing to the “magical” property of Zabusky and Kruskal’s transformation of the FPU system in their original study [49]. Indeed, dispersive equations on a lattice domain do not enjoy smoothing in general. For instance, the phase function for the linear Schrödinger flow is comparable with that for the linear Schrödinger flow on , i.e., on is far from near the high frequency edge (see Figure 1.1). Therefore, the discrete linear Schrödinger flow does not enjoy local smoothing at all (see [26]). With various dispersive and smoothing estimates for the linear FPU flows, we follow a general strategy (see [24] for instance) to prove the convergence from the coupled to the decoupled FPU and the convergence from the decoupled FPU to the KdV equation. First, we employ the linear and bilinear estimates to obtain -uniform bounds for solutions to the coupled and decoupled FPUs. Then, using the uniform bounds, we directly measure the differences to prove the convergences. ; however, its derivative for the discrete Fourier transform (see Figure