Nanopteron solutions of diatomic FermiPastaUlamTsingou lattices with small massratio
Abstract.
Consider an infinite chain of masses, each connected to its nearest neighbors by a (nonlinear) spring. This is a FermiPastaUlamTsingou lattice. We prove the existence of traveling waves in the setting where the masses alternate in size. In particular we address the limit where the mass ratio tends to zero. The problem is inherently singular and we find that the traveling waves are not true solitary waves but rather “nanopterons”, which is to say, waves which asymptotic at spatial infinity to very small amplitude periodic waves. Moreover, we can only find solutions when the mass ratio lies in a certain open set. The difficulties in the problem all revolve around understanding Jost solutions of a nonlocal Schrödinger operator in its semiclassical limit.
Key words and phrases:
FPU, FPUT, nonlinear hamiltonian lattices, periodic traveling waves, solitary traveling waves, solitons, singular perturbations, homogenization, heterogenous granular media, dimers, polymers, nanopeteronsArrange infinitely many particles on a horizontal line, each attached to its nearest neighbors by a spring with a nonlinear restoring force. Constrain the motion of the particles to be within the line. This system is called a FermiPastaUlam (FPU) or (more recently [8]) a FermiPastaUlamTsingou (FPUT) lattice and it is one of the paradigmatic models for nonlinear and dispersive waves. In this article, we consider the existence of traveling waves in a diatomic (or “dimer”) FPUT lattice when the ratio of the masses is nearly zero. By this we mean that the masses of the particles alternate between and along the chain and
The springs are all identical materially. The force they exert, when stretched by an amount from their equilibrium length, is where and . See [6] for an overview of this problem’s history and [19] for a discussion of technological applications of such a system.
Newton’s second law gives the equations of motion. After nondimensionalization, these read
(0.1) 
Here and When is odd and when is even, . In the above, is the nondimensional displacement from equilibrium of the th particle. See Figure 1 for a schematic.
In the monatomic case (where ), this system famously possesses localized traveling wave solutions. Formal arguments suggesting their existence date back to [17]. The first rigorous proofs can be found in [22] (for a very special alternate nonlinearity) and [14] (for more general convex nonlinearities). The articles [10] [11] [12] [13] demonstrate that these traveling waves are asymptotically stable.
Putting amounts to removing the smaller masses but leaving the springs attached. Which is to say that we have a monatomic lattice with a modified spring force. The results in [14] and [10] apply in this setting as well and so there is a localized traveling wave solution for (0.1) when .
The central question of this article is this: does the traveling wave solution persist when ? In (0.1) the small parameter multiplies a second derivative and as such the system is singularly perturbed. An attempt to answer the question by way of regular perturbation theory (i.e. an implicit function theorem argument) is doomed to failure. Nonetheless, we have an answer: the localized traveling wave at perturbs into a nanopteron^{1}^{1}1A nanopteron is the superposition of a localized function and an extremely small amplitude spatially periodic piece. for in an open set of postive numbers whose closure contains zero.
Several recent articles, specifically [23] and [18], have carried out detailed formal asymptotics and performed careful numerics for this problem. They strongly indicate traveling waves solutions for (0.1) are nanopterons, at least for most values of the mass ratio ; this article represents a rigorous mathematical validation of those predictions. We will in particular comment on the results of [23] below in Remark 3.2 in Section 3.
Nanopteron solutions are one of the many outcomes one may find for singularly perturbed systems of differential equations [5]. The “usual” way to prove their existence for a set of ordinary differential equations is through either geometric singular perturbation theory or matched asymptotics [15]. However, our problem is infinite dimensional which complicates using those sorts of tools. The method by which we prove our main result is a modification of one developed by Beale in [3] to study the existence of traveling waves in the capillarygravity problem.^{2}^{2}2That is, onedimensional free surface water waves which are acted by the restoring forces of gravity and surface tension. His method, which is functional analytic in nature, was subsequently deployed to show the existence of nanopteron solutions in several other singularly perturbed problems (e.g. [1]), including one closely related to the one here in [9].^{3}^{3}3In that article, the existence of traveling waves of nanopteron type for (0.1) is established but in a rather different limit: is fixed but the the wavespeed is taken just above the lattice’s speed of sound.
The common feature of problems with nanopteron solutions is that the singular perturbation manifests itself as a high frequency solution of the linearization. This in turn implies that a certain solvability condition must be met by solutions of the nonlinear problem. The key difference between what transpires here versus in [3] [1] [9] is that in our problem the high frequency linear solution is not a pure sinusoid but rather a Jost solution for a nonlocal Schrödinger operator. This is no minor thing: the ability to meet the solvability condition is more subtle and, as a consequence, the method only gives solutions for in the aforementioned open set (which we call ) as opposed to for all sufficiently close to zero. As we shall demonstrate, is an infinite union of finite open intervals which aggregate at .
This article is organized as follows:

Section 2 sets up the function analytic framework we work in and contains a number of simple estimates we will use repeatedly.

Section 4 is the first technical part of the proof and contains a “refinement” of the monatomic approximation. This is the first building block of the nanopteron solutions.

Section 6 we put together the refined leading order limit and periodic solutions and derive the first form of the governing equations for the nanopterons.

Section 7 deals with the singular part of the linearization. It is in this section where the ultimate form of the nanopteron equations appear.
1. The main result.
1.1. The equations of motion.
Before we state our main theorem, we reformulate (0.1). Let
(1.1) 
To be clear, is defined only for odd and for even. Computing in terms of and gives
where is the “shift by ” map, specifically .
If then the second equation in (1.2) is satisfied when . We want to change variables in a way that exploits this, so we set
Notice is defined for even integers . Plugging this into (1.2) we get
(1.4) 
In computing the above we have made judicious use of the identity .
On the right hand side of (1.4), always appears with at least one applied. Thus we define
Like , is defined for even integers . With this, if we apply to the first equation we get
Next we use the identity on the right hand side and we arrive at
(1.5) 
This system, which is posed for in the even lattice , is equivalent to the equations of motion (0.1).
The formulas for the variables and may seem to be somewhat nonintuitive, but they in fact have simple physical meanings. A chase through their definitions shows that and are found from the “stretch” variables by
Since is an even number in the above, we see that is simply half the distance from a heavy mass to the next heavy mass. Which is to say it is the distance to the midway point between heavy masses. And measures how far the light mass in between those heavy ones is from this midpoint. See Figure 2.
This point of view is why we will sometimes refer to as the “heavy variable”; it is determined fully by the locations of the heavy particles alone. We will also call first equation in (1.5) the “heavy equation.” On the other hand, specifies the location of the lighter particles and so we call it the “light variable” and the second equation in (1.5) the “light equation.”
1.2. Our result.
The system (1.5) has a simple structure at : the second equation reduces to
(1.6) 
which can be solved by taking . Physically this means the light (in this case, massless) particles are located exactly halfway between their bigger brethren. The heavy equation is then simply
(1.7) 
which (see for instance [10]) coincides with the equations of motion for a monatomic lattice with restoring force given by .
Since the problem is equivalent to a monatomic FPUT lattice, we can summon the results of [14] and [10] to get an exact traveling wave solution to it.
Theorem 1.1 (Friesecke & Pego).
There exists , where
(1.8) 
for which implies the existence of a positive, even, smooth, bounded and unimodal function, , such that satisfies (1.7). Moreover is exponentially localized in the following sense: there exists such that, for any and , we have
(1.9) 
Now we can state our main thereom. It says that this solution perturbs into a nanopteron for ; see Figure 3.
Theorem 1.2.
For there exists an open set whose closure contains zero and for which implies the following. There exist smooth functions , , and such that putting
and
solves (1.5). Moreover, and are exponentially localized and have amplitudes of . Finally, and are high frequency (specifically ) periodic functions of whose amplitudes are small beyond all orders of .
We do not prove this theorem in the coordinates and , but rather we make an additional near identity change of variables.
1.3. “Almost diagonalization”
Denoting^{4}^{4}4Generally, we represent maps from by bold letters, for instance or . Likewise, the first and second components of such functions, as shown here, will be represented in the regular font with a subscript “” and “”, respectively. if we put
(1.11) 
then (1.5) is equivalent to
(1.12) 
Observe that is upper triangular when . We can make a simple change of variables that “almost diagonalizes” the linear part of (1.12); this will be advantageous down the line. Let
(1.13) 
Since is a small perturbation of the identity, we will continue to ascribe to and the physical meaning of and ; that is is “heavy” and (which is in fact exactly ) is “light.”
With this, (1.12) becomes
Since is invertible for all and is invertible for , the above is equivalent to
(1.14) 
where^{5}^{5}5This product defining is rather painful to multiply out, so we omit showing the details; it can be computed formally by replacing with and with and asking a computer algebra system to carry out the product. Then you just replace all the cosines with and sines with . The reason that this works is that and . Which is to say the product is easier on the Fourier side.
(1.15) 
and
As we advertised above,
is a diagonal operator and thus is nearly diagonal.
At last we make the traveling wave ansatz:
where and is the wave speed. With this we get the following equation for :^{6}^{6}6As before, we think of as being the heavy variable and as being the light one.
(1.16) 
The primes denote differentiation with respect to , the independent variable of . Of course the operators and (out of which are constructed and ) act on functions of just as they do on functions of . Note that is bilinear and symmetric in its arguments.
At we have . Thus the line of reasoning that lead to (1.10) tells us that putting^{7}^{7}7We use and to denote the usual unit vectors in .
(1.17) 
1.4. Some symmetries of
We now point out two symmetries possessed by the mapping . The first is that if is an even function of and is odd, then the components of are, respectively, even and odd. This is a consequence of the following simple facts:

Both and map even functions to odd ones and viceversa.

The map takes odd functions to odd functions and even to even.

If is even and is odd then and are even while is odd.
With these in hand, showing that maintains “even odd” symmetry amounts to just scanning through its definition. Henceforth we assume that the function and its descendents will have this symmetry.
Moreover, annihilates constants, just as does. And, also just like , we have
In the first integral we assume that is going to zero quickly enough as and in the second that is periodic with period . Both integrals read as a sort of “meanzero” condition for and so we say that is a “meanzero function.” Also observe that each term in the first row of has at least one factor of exposed. Which is to say that the first component of has no constant term, or equivalently, that it is meanzero.
We summarize these symmetries in the informal statement
(1.18) 
It is worth pointing out that , and , each on their own, have this same property. Now that we have our traveling wave equation spelled out, and we understand it well at , we take the next section to lay out our function spaces, key definitions as well as prove some rudimentary estimates.
2. Functions spaces, notation and basic estimates
2.1. Periodic functions
We let be the usual “” Sobolev space of periodic functions. We denote and . Put
(2.1) 
We will also make use of
(2.2) 
That is to say, meanzero even periodic functions. By we mean the space of times differentiable periodic functions and is the space of smooth periodic functions.
2.2. Functions on
We let be the usual “” Sobolev space of functions defined on . For put
(2.3) 
These are Banach spaces with the naturally defined norm. If we say a function is “exponentially localized” we mean that it is in one of these spaces with .
Put , and and denote . We let
(2.4) 
For instance the monatomic wave profile , by virtue of (1.9), is in for all . We will also make use of
(2.5) 
This is another space of meanzero even functions.
2.3. Big and big notation.
Many of our quantities will depend on the mass ratio , the wavespeed and a decay rate . Of these, is the chief but will play an important role as well; we generally view as being fixed and as such we do not usually track dependence on it. In order to simplify the statements of many of our estimates we employ the following conventions. Any exceptions/restrictions will be clearly noted.
Definition 2.1.
The point is this: if an estimate depends in a bad way on the decay rate then we adorn it with the subscript . Moreover, our definition indicates that increases as .
We will also occasionally write either or , as as opposed to or . We use this to indicate that we can control from above and below by . Specifically
Definition 2.2.
Suppose that and . We write
(2.8) 
if implies there is and such that
for all .
Lastly, we will encounter terms which are small “beyond all orders of .” By this we mean the following.
Definition 2.3.
Suppose . We write
if for all .
2.4. General estimates for
We take for granted the containment estimate:
Likewise, there is the generalization of the Sobolev embedding estimate:
We have the following simple estimate:
This in turn implies
(2.9) 
and, using the Sobolev estimate
(2.10) 
Note that (2.9) holds for but (2.10) only when . The important feature of these estimates is that the weight on the left hand side can be shared between the two functions and more or less however we wish. It will even be necessary at several stages for us to have . Note also that (2.10) implies that, if and , that is an algebra.
We also have the containment when and . These follow from the following estimate involving Hölders inequality
(2.11) 
where
2.5. Simple operator estimates
The following estimates follow from substitution and the definition of the shift operator :
(2.12) 
That is to say is a bounded operators on all of the function spaces we use here; we will treat them as such without comment. These estimates imply
(2.13) 
where is either or . In turn these, after a quick glance at the definitions of and , give
(2.14) 
where is either or . Note that the estimates in (2.14) hold for all .
2.6. The bilinear map .
We have not written out in full detail; there are very many terms and ultimately not much would be learned. But we can give a useful and relatively simple collection of estimates for it. For functions , define the “windowed absolute value” as
Here is the just the Euclidean norm on . From the esimates for in (2.12), we can conclude that
(2.15) 
where or .
Examining (1.11) and (1.13) gives
(2.16) 
Since is made out of the shifts , this formula tells us that for all
(2.17) 
and
(2.18) 
Similarly, for all , we have
(2.19) 
Here is why. The estimates (2.14) and the definition of in (1.11) demonstrate that extracting from leaves only terms with at least one exposed power of . The operator is bilinear and this is why we have the product of and on the right. And we have the windowed absolute value because and are constructed out of and , which themselves are made from . Tracking through the definitions shows that the most shifts that could land on one function is ten: two in , two in and six in . This is why the window has a radius of ten.
Putting (2.15), (2.17), (2.18) and (2.19) together with (2.9) and various Sobolev estimates yields the following collection of estimates:
(2.20) 
(2.21) 
(2.22) 
(2.23) 
(2.24) 
and
(2.25) 
In the above, and are free to be anything. In (2.22)(2.25) we need , but suffices for (2.20) and (2.21). All of the estimates (2.20)(2.25) hold for all .
3. The strategy
Here we outline our approach to the proof of Theorem 1.2. The first thing we do is we put into (1.16). For the time being, we are thinking of as being small and localized function, for instance in . (That is to say , like , has the even odd symmetry.) Using (1.17) and some algebra shows that
(3.1) 
The left hand side consists of all the leading order terms, plus the singularly perturbed term . The right hand side is everything else. Its exact form is not germaine at this point but it is made up of:

terms which are linear in but have at least one prefactor of (for instance ),

residuals (for instance ) and

terms which are quadratic in .
That is to say, the right hand side is “small.”
If we can invert the linear operator of on the left hand side we would have our system written as a where is an (ostensibly) small operator and then we could use a contraction mapping argument to solve this fixed point equation. A look at the page count of this article makes it clear that we could not get such a strategy to work.
To see why, we write out (3.1) componentwise. The first (or “heavy”) component reads
(3.2) 
The operator is closely related to an operator that appears in the analysis of the stability of monatomic FPUT solitary waves [10]. The following result, which states that is (more or less) invertible in the class of even functions, can be inferred from results there; we carry out the details in Appendix A.
Proposition 3.1.
The following holds for ^{8}^{8}8The constants and in this proposition may be smaller than their counterparts with the same names in Theorem 1.1, but since that theorem remains true with the smaller values here, we act here (and throughout) as if the constants were the same to begin with.. The map is a homeomorphism of and for any and . Specifically we have
(3.3) 
Note that the result states that the range of is the meanzero even functions as opposed to all even functions. The symmetry properties described in (1.18) imply that is in fact an even meanzero function. Thus we can write (3.2) as and the right hand side of this can be shown to meet the hypotheses of the contraction mapping theorem in the localized spaces. That is, the heavy component of (3.1) poses no problem.
Writing out the second (“light”) component of (3.1) gives
(3.4) 
This operator is a gardenvariety second order Schrödinger operator^{9}^{9}9Note that since , we are considering this operator in the “semiclassical limit.” with potential functiom . Since we are working with , standard undergraduate differential equations theory tells us that there are two linearly independent solutions of . Call them and . Since is an even function, we can arrange it so that is even and is odd. And since is positive and converges to zero at infinity we can infer that these functions converge, as , to solutions of . Which is to say they are asymptotic as to a linear combination of and , with . Thus, since is small, these have very high frequencies.
The functions and are the Jost solutions^{10}^{10}10Or, at least, they are linear combinations of them, see [21]. for the operator . Note that since these functions do not converge to zero at infinity they are not in the localized spaces . And we also know from ODE theory that all solutions of will be linear combinations of and . All of this implies that the only function which solves is . In this way we can conclude , viewed as a map from to , is injective, since it has a trivial kernel.
We remark now that an alternate way to prove the injectivity of on the localized spaces is by way of the following coercive estimate: if then
(3.5) 