An a posteriori KAM theorem for whiskered tori in Hamiltonian partial differential equations with applications to some ill-posed equations
The goal of this paper is to develop a KAM theory for tori with hyperbolic directions, which applies to Hamiltonian partial differential equations, even to some ill-posed ones.
The main result has an a-posteriori format, i.e., we show that if there is an approximate solution of an invariance equation which also satisfies some non-degeneracy conditions, then there is a true solution nearby. This allows, besides dealing with the quasi-integrable case, to validate numerical computations or formal perturbative expansions as well as to obtain quasi-periodic solutions in degenerate situations. The a-posteriori format also has other automatic consequences (smooth dependence on parameters, bootstrap of regularity, etc.). We emphasize that the non-degeneracy conditions required are just quantities evaluated on the approximate solution (no global assumptions on the system such as twist). Hence, they are readily verifiable in perturbation expansions.
The method of proof is based on an iterative method to solve a functional equation for the parameterization of the torus satisfying the invariance equations and for parametrization of directions invariant under the linearizatation. The iterative method does not use transformation theory or action-angle variables. It does not assume that the system is close to integrable. We do not even need that the equation under consideration admits solutions for every initial data. In this paper we present in detail the case of analytic tori when the equations are analytic in a very weak sense.
We first develop an abstract theorem. Then, we show how this abstract result applies to some concrete examples, including the scalar Boussinesq equation and the Boussinesq system so that we construct small amplitude tori for the equations, which are even in the spatial variable. Note that the equations we use as examples are ill-posed. The strategy for the abstract theorem is inspired by that in [FdlLS09b, FdlLS09a]. The main part of the paper is to study infinite dimensional analogues of dichotomies which applies even to ill-posed equations and which is stable under addition of unbounded perturbations. This requires that we assume smoothing properties. We also present very detailed bounds on the change of the splittings under perturbations.
- 1 Introduction
- 2 Overview of the method
3 The precise framework for the results
- 3.1 The evolution equation
- 3.2 Symplectic properties
- 3.3 Diophantine properties
- 3.4 Spaces of analytic mappings from the torus
- 3.5 Non-degeneracy assumptions
- 3.6 Statement of the results
- 4 The linearized invariance equation
- 5 Solutions of linearized equations on the stable and unstable directions
- 6 Perturbation theory of hyperbolic bundles in an infinite-dimensional framework
- 7 Solution of the cohomology equation on the center subspace
- 8 Uniqueness statement
- 9 Nash-Moser iteration
10 Construction of quasi-periodic solutions for the Boussinesq equation
- 10.1 Formal and geometric considerations
- 10.2 Choice of spaces
- 10.3 Linearization around
- 10.4 Verifying the smoothing properties of the partial evolutions of the linearization around
- 10.5 Construction of an approximate solution
- 10.6 Application of Theorem 3.5 to the approximate solutions. End of the proof of Theorem 3.7
- 11 Application to the Boussinesq system
The goal of this paper is to develop a KAM theory for tori with hyperbolic directions, which applies to Hamiltonian partial differential equations, even to some ill-posed ones. The main result, Theorem 3.5 is stated in an a-posteriori format, that is, we formulate invariance equations and show that approximate solutions that satisfy some explicit non-degeneracy conditions, lead to a true solution. This a-posteriori format leads automatically to several consequences (see Section 3.6.2) and can be used to justify numerical solutions and asymptotic expansions. We note that the results do not assume that the equations we consider define evolutions and indeed we present examples of quasi-periodic solutions in some well known ill-posed equations. See Sections 10, 11.
1.1. Some general considerations and relations with the literature
Some partial differential equations appear as models of evolution in time for Physical systems. It is natural to consider such evolutionary PDE’s as a dynamical system and try to use the methods of dynamical systems.
Adapting dynamical systems techniques to evolutionary PDE’s has to overcome several technical difficulties. For starters, since the PDE’s involve unbounded operators, the standard theory of existence, uniqueness developed for ordinary differential equations does not apply. As it is well known, by now, there are systematic ways of defining the evolution using e.g. semigroup theory [Sho97, Paz83, Gol85] and many dynamical systems techniques can be adapted in the generality of semigroups ( see the pioneering work of [Hen81] and more modern treatises [Hal88, Miy92, Tem97, CFNT89, Rob01, SY02, CV02, HMO02, CM12].) Besides the analytic difficulties, adapting ODE techniques to PDE’s has to face that several geometric arguments fail to hold. For instance, symplectic structures on infinite-dimensional spaces (see for instance [CM74, Bam99]) could lack several important properties. Hence, the techniques (e.g. KAM theory) that are based on geometric properties have to overcome several difficulties specially the methods based on transformation theory [Kuk93, Kuk94, Kuk00, Kuk06, KP03]. Some recent methods based on avoiding transformation theory are [CW93, CW94, Bou99, Ber07, Cra00]. When working near an equilibrium point, one also has to face the difficulty that the action angle variables are singular (even in finite dimensions) [KP03, GK14]. In the approach of this paper, we do not use action angle variables, which present difficulties even in finite dimensional fixed points and, much more in PDE’s.
One class of evolutionary equations that has not received much systematic attention is ill-posed equations. In ill-posed equations, one cannot define the evolution for all the initial data in a certain space (an equation may be ill-posed in a space and well posed in another) or the evolution is not continuous in this space. Nevertheless, it can be argued that even if one cannot find solutions for all the inital data, one can still find interesting solutions which provide accurate descriptions of physical phenomena. Many ill-posed equations in the literature are obtained as a heuristic approximation of a more fundamental equation. The solutions of the ill-posed equation may be approximate solutions of the true equation.
For example, many long wave approximations of water waves turn out to be ill-posed (e.g. the Boussinesq equations used as examples here, see Section 10) but several special solutions (e.g. traveling waves or the quasi-periodic solutions considered in this paper) of the long wave approximations can be constructed. These special solutions are such that, for them, the long wave approximation is rather accurate. Hence, the solutions obtained here for the long wave approximation provide approximate solutions of the original water wave equation and are physically relevant.
Note that the long-wave approximations are PDE’s while the water waves problem is a free boundary and many techniques are different, notably in numerical analysis. Being able to validate the numerical solutions is useful.
Of course, the straightforward adaptation of ODE methods for invariant manifolds to ill-posed equations present some challenges because some methods (e.g. graph transform, index theory methods, etc.), which are very useful in ODEs, require taking arbitrary initial conditions. Nevertheless, we will present rather satisfactory adaptations of some of the methods of hyperbolic dynamical systems.
In the present paper, we are concerned with the construction of quasi-periodic motions of PDEs. The method is very general. Some concrete examples of ill-posed equations to which the method applies will be presented in Sections 10 and 11.
The tori we consider are whiskered, that is the linearization has many hyperbolic directions, indeed, as many directions as it is possible to be compatible with the preservation of the symplectic structure. There is a rich KAM theory for whiskered tori [Gra74, Zeh76] or for lower dimensional tori will elliptic directions [Eli89, You99, LY05, Sev06]. A treatment of normally elliptic tori by methods similar to those here is in [LV11].
In PDE’s, where the phase space is infinite dimensional, the quasiperiodic solutions are very low dimensional. Nevertheless, most of the literature in PDE is concerned with normally elliptic tori, so that most of the small divisors come from the elliptic normal directions. The models considered here have no elliptic normal directions. On the other hand, the models we consider do not admit solutions for all initial conditions and present very severe unstable terms. Hence, methods based on transformation theory, normal forms etc. are very difficult in our case. We also deal with unbounded perturbations.
1.2. Overview of the method
We are going to follow roughly the method described in [FdlLS09b] and implemented in [FdlLS09a] for finite dimensional systems, in [LdlL09, FdlLS15] for infinite dimensional systems (but whose evolution is a smooth differential equation; the main difficuly overcome in [LdlL09] was the fact that the equations involve delays, a new difficulty in [FdlLS15] is the spatial structure). In this paper we overcome the difficulty that the evolution equations are PDE’s which are perturbed by unbounded operators. Hence, we have to overcome many problems (unbounded operators, regularity issues and spectral theory for instance ). Some results in KAM with unbounded perturbations by very different methods appear in [LY11].
The method we use is based on the solution of a functional equation whose unknown is a parameterization of the invariant torus and devicing a Newton method to solve these equations by quadratically convergent schemes. We assume that the linearized evolution admits an invariant splitting. In the hyperbolic directions we can use essentially soft functional analysis methods. There are subtleties such that we have to deal with unbounded perturbations and be very quantitative in the hyperbolic perturbation theory, and a center direction case, in which we have to deal with equations involving small divisors and use heavily the number theoretic properties of the equation and the symplectic geometry.
The method does not rely on methods that require the evolution for all initial data on a ball. Also, the symplectic geometry properties are used only sparingly. We certainly do no use action-angle variables. The solutions we construct are very unstable – indeed, some perturbations near them may lead to a solution of the evolution equation – but they are in some precise sense hyperbolic in the usual meaning of dynamical systems. We expect that one can define stable and unstable manifolds for them and we hope to come back to this problem. Fortunately, the analysis on the center is very similar to the analysis in the finite dimensional case. The bulk of the work is in the study of hyperbolic splittings with unbounded perturbations. We hope that the theory developed here can be used in other contexts.
Indeed, other theories of persistence of invariant splitting (having significant applications to PDE) have already been developed in [Hen81, PS99, CL95, CL96, HI11]. The main difference between Section 6 and [CL95, CL96] is that we take advantage of the smoothing properties and, hence, can deal with more singular perturbations. We also take advantage of the fact that the dynamics on the base is a rotation whereas [CL95, CL96] deal with more general dynamics. This allows us to obtain analyticity results which are false in the more general contexts considered in [CL95, CL96].
The method presented here applies even to some ill-posed equations. A fortiori, it applies also to well posed equations. Even then, it presents advantages, notably our main result has an a-posteriori format that can justify several expansions and deal with situations with weak hyperbolicity, bootstrap regularity, establish smooth dependence, etc. It also leads to efficient numerical algorithms. See Section 3.6.2. In a complementary direction, we point out that for finite dimensional problems the present methods leads to efficient algorithms (See [HdlLS12]). The case without center directions and no Hamiltonian structure has been considered in [CH15].
1.3. Organization of the paper
This paper is organized as follows: In Section 2 we present an overview of the method, describing the steps we will take, but ignoring some important precisions (e.g. domains of the operators), and proofs. In Section 3 we start developing the precise formulation of the results. We first present an abstract framework in the generality of equations defined in Banach spaces, including the abstract hypothesis. The general abstract results are stated in Section 3.6.1 and in Section 3.6.3 we discuss how to apply the results to some concrete examples. Some possible extensions are discussed in Section 3.6.2. The rest of the paper is devoted to the proof of the results following the strategy mentioned in the previous sections. One of the main technical results, which could have other applications is the persistence of hyperbolic evolutions with smoothing properties. See Section 6.
2. Overview of the method
In this section, we present a quick overview describing informally the steps of the method. We present the equations that need to be solved and the manipulations that need to be done ignoring issues such as domain of operators, estimates. These precisions will be taken up in Section 3. This section can serve as motivation for Section 3 since we use the formal manipulations to identify the issues that need to be resolved by a precise formulation.
One example to keep in mind and which has served as an important motivation for us is the Boussinesq equation
In Section 11, we will also consider the Boussinesq system. Other models in the literature which fit our scheme are the Complex Ginzburg-Landau equation and the derivative Complex Ginzburg-Landau equation for values of the parameters in suitable ranges.
There are several equations called the Boussinesq equation in the literature (in Section 11 we also present the Boussinesq system), notably the Boussinesq equation for fluids under thermal buoyancy. The paper [McK81] uses the name Boussinesq equation for and shows it is integrable in some sense made precise in that paper. Note that this equation is very different from (1) because of the sign of the fourth space derivative and (less importantly), the absence of the term with the second derivative. The sign of the fourth derivative term causes that the wave propagation properties of (1) and the equation in [McK81] are completely different.
We note that the fourth derivative in (1) is just the next term in the long wave expansion of the water wave problem (which is not a PDE, but rather a free boundary problem). Equations similar to (1) appear in many long wave approximations for waves. See [CGNS05, Cra08] for modern discussions.
The special solutions of (1) which are in the range of validity of the long wave approximation are good approximate solutions of the water wave problem, but they are analyzable by PDE methods rather than the free boundary methods required by the original problem. [CNS11, LM09]. Note that the solutions produced here lie in the regime (low amplitude, long wave) where the equation (1) was derived, so that they provide approximate solutions to the water wave problem.
2.1. The evolution equation
We consider an evolutionary PDE, which we write symbolically,
where will be a differential and possibly non-linear operator. This will, of course, require assumptions on domains etc. which we will take up in Section 3. For the moment, we will just say that is defined in a domain inside a Banach space . We will write
where is linear and is a nonlinear and possibly unbounded operator.
The differential equations will not be assumed to generate dynamical evolution for all initial conditions (we just assume that it generates forward and backward evolutions when restricted to appropriate subspaces). Of course, we will not assume that (2) defines an evolution either. Lack of solutions for all the initial conditions will not be a severe problem for us since we will only try to produce some specific solutions.
The meaning in which (2) is to hold may be taken to be the classical sense. As we will see we will take the space to consist of very differentiable functions so that the derivatives can be taken in the elementary classical sense. As intermediate steps, we will also find useful some solutions in the mild sense, satisfying some integral equations formally equivalent to (2). The mild solutions require less regularity in . Again, we emphasize that the solutions we try to produce are only special solutions.
We will assume that the nonlinear operator is “sub-dominant” with respect to the linear part. This will be formulated later in Section 3, but we anticipate that this means roughly that is of higher order than and that the evolution generated by when restricted to appropriate sub-spaces gains more derivatives than the order of . We will formulate all this precisely later.
We will follow [Hen81] and formulate these effects by saying that the operator is an analytic function from a domain – is a Banach space of smooth functions – to – a space corresponding to less smooth functions and that the evolution operators map back to with some quantitative bounds.
In the applications that we present in Sections 10 and 11, the equations we consider are polynomial111The equations we consider are taken from the literature of approximations of water waves. In these derivations, it is customary to expand the non-linearity and keep only the lower order terms but the method can deal with more general nonlinearities.
2.2. The linearized evolution equations
Note that, in this set up we can define a linearized evolution equation around a curve in , i.e.
The equations (4) are to be considered as evolution equations for while is given and fixed. The meaning of the term could be understood if is a differentiable operator from to .
Of course, when is solution of the evolution equation (2), equations (4) are the variational equations for the evolution. In our case, the evolution is not assumed to exist and, much less, the variational equations are assumed to provide a description of the effect of the initial conditions on the variation. We use these equations (4) even when is not a solution of the evolution equation (2) and we will show that they are indeed a tool to modify an approximate solution into a true solution.
Notice that (4) is non-autonomous, linear non-homogeneous, but that the existence of solutions is not guaranteed for all the initial conditions (even if the time dependent term is omitted).
In the finite dimensional case, equations of the form (4) even when is not a solution are studied when performing a Newton method to construct a solution; for example in multiple shooting. Here, we will use (4) in a similar way. We will see that (4) can be studied using that is dominant and has a splitting (and that is not too wild).
2.3. The invariance equation
Given a fixed that satisfies some good number theoretic properties (formulated precisely in Section 3.3), we will be seeking an embedding in such a way that
Note that if (5) holds, then, for any , will be solution of (2). Hence, when we succeed in producing a solution of (5), we will have a -parameter family of quasi-periodic solutions. The meaning of these parameters is the origin of the phase as is very standard in the theory of quasi-periodic functions.
2.4. Outline of the main result
is small enough. We will also assume that the linearized evolution satisfies some non-degeneracy assumptions. The conclusions is that there is a true solution close to the original approximate solutions. Theorems of these form in which we start from an approximate solution and conclude the existence of a true one are often called “ a posteriori” theorems.
In the concrete equations that we consider in the applications, the approximate solutions will be constucted using Lindstedt series.
The sense in which the error is small requires defining appropriate norms, which will be taken up in Section 3. The precise form of the non-degeneracy conditions will be motivated by the following discussion which specifies the steps we will perform for the Newton method for the linearized equation
The non-degeneracy conditions have two parts. We first assume that for each , the linearized equation satisfies some spectral properties. These spectral properties mean roughly that there are solutions of (7) that decrease exponentially in the future (stable solutions), others that decrease exponentially in the past (unstable solutions), and some center directions that can grow or decrease with a smaller exponential rate. The span of these three class of solutions is the whole space. We will also assume that the evolutions, when they can be defined, gain regularity.
In the ODE case, this means that the linearized equation admits an exponential trichotomy in the sense of [SS76].
In the PDE case, there are some subtleties not present in the ODE case. For instance, the vector field is not differentiable and is only defined on a dense subset.
We will not assume that (7) defines an evolution for all time and all the initial conditions. We will however assume that (7) admits a solution forward in time for initial conditions in a space (the center stable space) and backwards in time for the another space (the center unstable space). We will furthermore assume that the center stable and center unstable spaces span the whole space, and they have a finite dimensional intersection (we will also assume that they have a finite angle, which we will formulate as saying that the projections are bounded). We emphasize that we will not assume that the evolution forward of (7) can be defined outside of the center stable space nor that the backward evolution can be defined outside of the center unstable space.
Furthermore, we will assume that the evolutions defined in these spaces are smoothing. Of course, these subtleties are only present when we consider evolutions generated by unbounded operators and are not present in the ODE case.
A crucial result for us is Lemma 6.1 which shows that this structure (the trichotomy with smoothing) is stable under the addition of unbounded terms of lower order. We also present very quantitative estimates on the change of the structure under perturbations. Note that the result is also presented in an a-posteriori format so that we can use just the existence of an approximate invariant splitting.
The smoothing properties along the stable directions overcome the loss of regularity of the perturbation. Hence, we can obtain a persistence of the spaces under unbounded perturbations of lower order. A further argument shows the persistence of the smoothing properties. The result in Lemma 6.1 can be considered as a generalization of the finite dimensional result on stability of exponential dichotomies to allowing unbounded perturbations. An important consequence is that, when is small enough (in an appropriate sense) we can transfer the hyperbolicity from to the approximate solution, which is the way that we construct the approximately hyperbolic solutions in the applications.
We will need to assume that in the center directions, there is some geometric structure that leads to some cancellations (sometimes called automatic reducibility). These cancellations happen because of the symplectic structure. We note that, in our case, we only need a very weak form of symplectic structure, namely that it can be made sense of in a finite dimensional space consisting of rather smooth functions. Note that the infinitesimal perturbations do not grow in the tangent directions. The preservation of the geometric structure also implies that some of the perpendicular directions evolve not faster than linearly. Hence, the tori we consider are never normally hyperbolic and that for -dimensional tori, the space of directions with subexponential growth is at least dimensional. We will assume that the tori are as hyperbolic as possible while preserving of the symplectic structure. That is, the set of directions with subexponential growth is precisely dimensional . These tori are called whiskered in the finite dimensional case.
We note that the geometric structure we need only requires to make sense as the restriction to an infinitesimal space and be preserved only in a set of directions. The geometric structure that appears naturaly in applications will be given by an unbounded form and many of the deeper features of symplectic structures in finite dimensions will not be available. Hence, it is important to note that the present method does not rely much in the symplectic structure. We do not rely on transformation theory we only use some geometric identities in finite dimensional spaces to construct a good system of coordinates in finite dimensions and to show that some (finite dimensional) averages vanish. In systems without the geometric structure, the system of coordinates and the averages would require adjusting external parameters.
We note that (7) is formally the variation equation giving the derivative of the flow of the evolution equation. This interpretation is very problematic since the equations we will be interested in do not define necesserally a flow.
An important part of the effort in Section 3 consists in defining these structures in the restricted framework considered in this paper when many of the geometric operations used in the finite dimensional case are not available.
We also need to make assumptions that are analogues of the twist conditions in finite dimensions. See Definition 3.4. The twist condition we will require is just that a finite dimensional matrix is invertible. The matrix is computed explicitly on the approximate solution and does not require any global considerations on the differential equation.
2.5. Overview of the proof
The method of proof will be to show that, under the hypotheses we are making, a quasi-Newton method for equation (5) started in the initial guess, converges to a true solution. We emphasize that the unknown in equation (5) is the embedding of into a Banach space . Hence, we will need to introduce families of Banach spaces of embeddings (the proof of the convergence will be patterned after the corresponding proofs [Mos66b, Zeh75]).
For simplicity, we will only consider analytic spaces of embeddings. Note that the regularity of the embedding as a function of their argument is different from the regularity of the functions . The term will be functions of the variable. The space encodes the regularity with respect to the variable . Indeed, we will consider also other Banach spaces consisting of functions of smaller regularity in .
The Newton method consists in solving the equation
and then, taking as an improved solution.
Clearly, (8) is a non-homogeneous version of (7). Hence, the spectral properties of (7) will play an important role in the solution of (8) by the variations of constants formula. Following [FdlLS09b, FdlLS09a], we will show that using the trichotomy, we can decompose (8) into three equations, each one of them corresponding to one of the invariant subspaces.
The equations along the stable and unstable directions can be readily solved using the variation of parameters formula also known as Duhamel formula (which holds in the generality of semigroups) since the exponential contraction and the smoothing allow us to represent the solution as a convergent integral.
The equations along the center direction, as usual, are much more delicate. We will be able to show the geometric properties to establish the automatic reducibility. That is, we will show that there is an explicit change of variables that reduces the equation along the center direction to the standard cohomology equations over rotations (up to an error which is quadratic – in the Nash-Moser sense – in the original error in the invariance equation). It is standard that we can solve these cohomology equations under Diophantine assumptions on the rotation and that we can obtain tame estimates in the standard meaning of KAM theory [Mos66b, Mos66a, Zeh75]. One geometrically delicate point is that the cohomology equations admit solutions provided that certain averages vanish. The vanishing of these averages over perturbations is related to the exactness properties of the flow. Even if this is, in principle, much more delicate in the infinite dimensional case, it will turn out to be very similar to the finite dimensional case, because we will work on the restriction to the center directions which are finite dimensional. The procedure is very similar to that in [FdlLS09b].
We will not solve the linearized equations in center direction exactly. We will solve them up to an error which is quadratic in the original error. The resulting modified Newton method, will still lead to quadratically small error in the sense of Nash-Moser theory and can be used as the basis of a quadratically convergent method.
Once we have the Newton-like step under control we need to show that the step can be iterated infnitely often and it converges to the solution of the problem.
A necessary step in the strategy is to show stability of the non-degeneracy assumptions. The stability of the twist conditions is not difficult since it amounts to the invertibility of a finite dimensional matrix, depending on the solution. The stability of spectral theory is reminiscent of the standard stability theory for trichotomies [SS76, HPS77] but it requires significant more work since we need to use the smoothing properties of the evolution semigroups to control the fact that the perturbations are unbounded. Then, we need to recover the smoothing properties to be able to solve the cohomology equations. For this functional analysis set up, we have found very inspiring the “two spaces approach” of [Hen81] and some of the geometric constructions of [Hen81, PS99, CL95, CL96]. Since the present method is part of an iterative procedure, we will need very detailed estimates of the change.
We note that rather than presenting the main result as a persistence result, we prove an a-posteriori result showing that an approximate invariant structure implies the existence of a truly invariant one and we bound the distance between the original approximation and the truly invariant one. This, of course, implies immediately the persistence results.
3. The precise framework for the results
In this section we formalize the framework for our abstract results. As indicated above, we will present carefully the technical assumptions on domains, etc. of the operators under consideration, and the symplectic forms. We will formulate spectral non-degeneracy conditions and the twist non-degeneracy assumption.
In Section 3.6 we will state our main abstract result, Theorem 3.5. The proof will be obtained in the subsequent Sections. Then, in Sections 10 and 11 we will show how the abstract theorem applies to several examples. The abstract framework has been chosen so that the examples fit into it, so that the reader is encouraged to refer to these sections for motivation. Of course, the abstract framework has been formulated with the goal that it applies to other problems in a more or less direct manner. We leave these to the reader.
We note that the formalism we use is inspired by the two-space formalism of [Hen81]. We consider two Hilbert spaces and . The differential operators, which are unbounded from a space to itself will be very regular operators considered as operators from to . Some evolutions will have smoothing properties and map to with good bounds.
3.1. The evolution equation
H1 There are two complex Hilbert spaces
with continuous embedding. The space (resp. ) is endowed with the norm (resp. )
We denote by the space of bounded linear operators from to .
We will assume furthermore that is dense in . We will assume in applications that and are such that they map real functions into real functions; it will be part of the conclusions that the solutions of the invariance equations we obtain are then real.
H2 The non-linear part of (3) is an analytic function from to .
We recall that the definition of an analytic function is that it is locally defined by a norm convergent sum of multilinear operators. Since we will be considering an implicit function theorem, it suffices to consider just one small neighborhood and a single expansion in multi-linear operators. The examples in Sections 10 and 11 have nonlinearities which are just polynomials (finite sums of multilinear operators).
In our case, it seems that some weaker assumptions would work. It would suffice that is analytic for any analytic embedding . In many situations this is equivalent to the stronger definition [HP74, Chapter III]. In the main examples that we will consider and in other applications, the vector field is a polynomial.
It also seems possible that one could deal with finite differentiable problems. For the experts, we note that there are two types of KAM smoothing techniques: either smoothing only the solutions in the iterative processs (single smoothing)[Sch60, CdlL10b] or smoothing also the problems (double smoothing) [Mos66b, Zeh75]. In general, double smoothing techniques produce better differentiability in the results. On the other hand, in this case, the approximation of the problems seems fraught with difficulties (how to define smoothings in infinite dimensional spaces, also for unbounded operators). Nevertheless, single smoothing methods do not seem to have any problem. Of course, if the non-linearities have some special structure (e.g. they are obtained by composing with a non-linear function) it seems that a double smoothing could also be applied.
Note that the structure of assumed in (3) allows us to estimate always the errors in , even if the unknown are in .
This is somewhat surprising since the loss of derivatives from to is that of the subdominant term . We expect that the results of applying to elements in does not lay in .
Nevertheless, using the structure in (3) and the smoothing properties we will be able to show by induction that if the error is in at one step of the iteration, we can estimate the error in subsequent steps of the iteration. Note that the new error is the error in the Taylor approximation of , which is the error in the Taylor approximation of .
Of course, we also need to ensure that the initial approximation satisfies this hypothesis. In the practical applications, we will just take a trigonometric polynomial.
3.2. Symplectic properties
We will need that there is some exact symplectic structure. In our method, this does not play a very important role. We just use the preservation of the symplectic structure to derive certain identities in the (finite dimensional) center directions. These are called automatic reducibility and use the exactness to show that some (finite dimensional) averages vanish (vanishing lemma) so that we can prove the result without adjusting parameters.
We will assume that there is a (exact) symplectic form in the space and that the evolution equation (2) can be written in Hamiltonian form in a suitable weak sense, which we will formulate now.
Motivated by the examples in Sections 10 and 11 and others in the literature, we will assume that the symplectic form is just a constant operator over the whole space (notice that we can identify all the tangent spaces). We will not consider the fact that the symplectic form depends on the position. Note that heuristically, the fact that the symplectic form is constant ensures and, because we are considering a Banach space, Poincaré lemma would give . We will need only weak forms of these facts. General symplectic forms in infinite dimensions may present surprising phenomena not present in finite dimensions [CM74, Bam99, KP03]. Fortunately, we only need very few properties in finite dimensional subspaces in a very weak sense.
H3 There is an anti-symmetric bounded operator taking real values on real vectors.
The operator is assumed to be non-degenerate in the sense that , implies .
will be refered to as the symplectic form.
As we mentioned above, we are assuming that the symplectic form is constant.
In some of the applications, could be a differential operator or the inverse of a differential operator. When is a differential operator, the fact that is bounded only means that we are considering a space consisting of functions with high enough regularity. The form could be unbounded in or in spaces consisting of functions with lower regularity than the functions in .
Notice that given a embedding of to we can define the pull-back of by the customary formula
The form is a form on . If is as a mapping form to (in our applications it will be analytic), the form will be .
H3.1 We will assume that is exact in the sense that, for all embeddings we have
with a one-form on the torus.
In the applications we will have that for some 1-form in . Note that if is not constant, we will need that depends on the position.
H4 There is an analytic function such that for any path , we have
Note that H4 is a weak form of the standard Hamilton equations . We take the Hamiltonian equations and integrate them along a path to obtain (11).
A consequence of H3 and H4 is we have that for any closed loop with image in
3.2.1. Some remarks on the notation for the symplectic form
The symplectic form can be written as
where is a Hilbert space and denotes the inner product in and is a (possibly unbounded) operator in – but bounded from to .
Once we have defined the operator , we can talk about the operator if it is defined in some domain.
The evolution equations can be written formally
where is the gradient understood in the sense of the metric in . In the concrete applications here, we will take , , for large enough . Of course, in well posed systems we can take .
We recall that the definition of a gradient (which is a vector field) requires a metric to identify differentials with vector fields. This is true even in finite dimensions. In infinite dimensions, there are several more subtleties such as the way that the derivative is to be understood. Hence, we will not use much the gradient notation and the operator except in Section 7, which is finite dimensional.
In the Physical literature (and in the traditional calculus of variations) it is very common to take to be always , even if the functions in the space or are significantly more differentiable. In some ways the space is considered as fixed and the spaces are mathematical choices. So that the association of the symplectic form to a symplectic operator is always done with a different inner product . The book [Neu10] contains a systematic treatment of the use of gradients associated to Sobolev inner products.
3.3. Diophantine properties
We will consider frequencies that satisfy the standard Diophantine properties.
Given and , we define as the set of frequency vectors satisfying the Diophantine condition:
where . We denote
It is well known that when , the set has full Lebesgue measure.
3.4. Spaces of analytic mappings from the torus
We will denote the complex strip of width , i.e.
We introduce the following -norm for with values in a Banach space
Let be a Banach space and consider the set of continuous functions on , analytic in with values in . We endow this space with the norm
is well known to be a Banach space. Some particular cases which will be important for us are when the space is a space of linear mappings (e.g. projections).
We will also need some norms for linear operators. Fix and consider a continuous linear operator from into , two Banach spaces. Then we define as
where denotes the Banach space of linear continuous maps from into endowed with the supremum norm.
Let and where is some Banach space. We denote its average on the -dimensional torus, i.e.
Of course, in the previous definition, since might be an infinite-dimensional space, the above integral, in principle, has to be understood as a Dunford integral. Nevertheless, since we will consider rather smooth functions, it will agree with simple approaches such as Riemann integrals.
3.5. Non-degeneracy assumptions
This section is devoted to the non-degeneracy assumptions associated to approximate solutions of (6). We first deal with the spectral non degeneracy conditions. The crucial quantity is the linearization equation around a map given by
where is an operator mapping into .
Roughly, we want to assume that there is a splitting of the space into directions on which the evolution can be defined either forwards or backwards and that the evolutions thus defined are smoothing. We anticipate that in Section 6, we will present other conditions that imply Definition 3.3. We will just need to assume approximate versions of the invariance.
Spectral non degeneracy
We will say that an embedding is spectrally non degenerate if for every in , we can find a splitting
with associated bounded projection and where are in such a way that:
SD1 The mappings are in (in particular, analytic).
SD2 The space is finite dimensional with dimension . Furthermore the restriction of the operator to denoted induces a symplectic form on which is preserved by the evolution on (see below).
SD3 We can find families of operators
SD3.1 The families are cocycles over the rotation of angle (cocycles are the natural generalization of semigroups for non-autonomous systems)
SD3.2 The operators are smoothing in the time direction where they can be defined and they satisfy assumptions in the quantitative rates. There exist , and independent of such that the evolution operators are characterized by the following rate conditions:
(19) (20) (21)
with and .
SD3.3The operators are fundamental solutions of the variational equations in the sense that
We are not aware of any general argument that would show that:
follow from the other assumptions. Needless to say, we would be happy to hear about one.
One can, however, clearly have that since so that the semigroups are exponentially decreasing for large .
We remark that when the equation preserves a symplectic structure, we can have without loss of generality
We anticipate that the results in Section 6 on persistence of trichotomies (a fortiori dichotomies) with smoothing are developed without assuming that the equation is Hamiltonian and, hence apply also to dissipative equations. Similarly, the solutions of linearized equations in the hyperbolic directions developed in Section 5 are obtained without using the Hamiltonian structure. The Hamiltonian structure is used only to deal with the linearized equations in the center direction in Section 7.
Let us comment on the previous spectral non-degeneracy conditions.
The first observation is that, if we assume that the spaces are Sobolev spaces of high enough index (so that the functions in them are for high enough) then we have that (22) holds in a classical sense if it holds in the sense of mild solutions (the sense of integral equations). In the applications we have in mind, it is always possible to take the spaces , that have arbitrarily high derivatives.
Then, (22) is just a form of
Making sense of the integrals in (22) is immediate after some reflection. Our conditions just require the existence of an evolution for positive and negative times on certain subspaces. The important conditions on these evolutions are the characterization of the splitting by rates (19)-(20), expressing the fact that the operators are bounded and smoothing from into (recall that ). If the system were autonomous, such properties would hold under some spectral assumptions on the operator (bisectoriality or generation of strongly continuous semi-groups, see [Paz83]).
Since the spaces and are finite dimensionals and of the same dimension, the evolution can be considered as an operator from to .
In the finite dimensional case (or in the cases where there is a well defined evolution), property SD.1 follows from the contraction rates assumption SD.3 by a fixed point argument in spaces of analytic functions. See [HdlL06]. In our case, we have not been able to adapt the finite dimensional argument, that is why we have included it as an independent assumption (even if may end up be redundant). We note that SD.1, SD.3 are clearly true when and in this paper we will show it is stable under perturbations, hence SD.3 will hold for all small enough . This suffices for our purposes, so we will not pursue the question of whether SD.1 can be obtained from SD.3 in general.
The fact that is non-dengenerate (which is a part of SD.2) follows from the rate conditions SD.3 as we show in Lemma 7.3.
One situation when all the above abstract properties are satisfied is when the evolution is given just by the linear part , i.e. . The assumptions of our set up are verified if the spectrum of is just eigenvalues of finite multiplicity and the spectrum is the union of a sector around the positive axis, another sector around the negative axis and a finite set of eigenvalues of finite multiplicity around the imaginary axis. Then, the stable space is the spectral projection over the sector in the negative real axis, the unstable space will be the spectral projection over the sector along the positive axis and the center directions will be the spectral space associated to the eigenvalues in the finite set. There are many examples of linear operators satisfying these properties.
It will be important that the main result of Section 6 is these structures persist when we add a lower order perturbation which is small enough. Indeed, we will show that if we find splittings that satisfy them approximately enough, there is true splitting nearby. This would allow to validate numerical computations, formal expansions, etc.
3.5.1. The twist condition
As it is standard in KAM theory, one has to impose another non-degeneracy assumption, namely the twist condition. This is the object of the next definition. Notice that it amounts to a finite dimensional matrix being invertible. It is identical to the conditions that were used in the finite dimensional cases [dlLGJV05, FdlLS09a].
Denote the matrix such that
Let stand for restriction of symplectic operator to . We will show in Lemma 7.3 that the form is non-degenerate so that the operator is invertible.
We now define the twist matrix (the motivation will become aparent in Section 7, but it is identical to the definition in the finite dimensional case in [dlLGJV05, FdlLS09a]). The average of the matrix
We note that the matrix in (27) is a very explicit expression that can be computed out of the approximate solution of the invariant equation and the invariant bundles just taking derivatives, projections and performing algebraic operations. So that it is easy to verify in applications when we are given an approximate solution.
As it will become apparent in the proof, the twist condition has a very clear geometric meaning, namely that the frequency of the quasiperiodic motions changes when we change the initial conditions in a direction (conjugate to the tangent to the torus).
Note that, given an invariant torus, we can consider it as an approximate solution for similar frequencies and that the twist condition also holds.
Using the a-posteriori theorem shows that under the conditions, we have many tori with similar frequencies near to the torus.
3.5.2. Description of the iterative step
Once the two non-degeneracy conditions are met for the initial guess of the modified Newton method, the iterative step goes as follows:
We project the cohomological equations with respect to the invariant splitting.
We then solve the equations for the stable and unstable subspaces.
We then solve the equation on the center subspace. This involves small divisor equations. We note that solving the equation in the center requires to use the exactness so that we can show that the equations are solvable.
To be able to iterate we will need to show that the corrections also satisfy the non-degeneracy conditions (with only some slightly worse quantitative assumptions). This amounts to showing the stability of the spectral non-degeneracy conditions, and developing explicit estimates of the changes in the properties given the changes on the embedding.
3.6. Statement of the results
3.6.1. General abstract results
The following Theorem 3.5 is the main result of this paper. It provides the existence of an embedding for equation (5) under some non-degeneracy conditions for the initial guess. We stress here that Theorem 3.5 is in an a posteriori format (an approximate solution satisfying nondegeneracy conditions implies the existence of a true solution close to it). As already pointed out in the papers [FdlLS09b, FdlLS09a, FdlLS15], this format allows to validate many methods that construct approximate solutions, including asymptotic expansions or numerical solutions. We also note that it has several automatic consequences presented in Section 3.6.2.
Suppose assumptions are met; let for some and . Assume that
We assume that the range of acting on a complex extension of the torus is well inside of the domain of analyticity of introduced in H2. More precisely:
That is, if , and , then .
Define the initial error
Then there exists a constant depending on , , , , , , , (where and are as in Definition 3.4 replacing by ) and the norms of the projections such that, if satisfies the estimates
where is fixed, then there exists an embedding such that
Furthermore, we have the estimate
3.6.2. Some consequences of the a-posteriori format
The a-posteriori format leads inmediately to several consequences. When we have systems that depend on parameters, observing that the solution for a value of the parameter is an approximate solution for similar values of the parameters, one obtains Lipschitz dependence on parameters, including the frequency.
If one can obtain Lindstedt expansions in the parameters, one can obtain Taylor expansions. If the parameter ranges over , this is the hypothesis of the converse Taylor theorem [AR67, Nel69] so that one obtains smooth dependence on parameters. In the case that the parameters range on a closed set, we obtain one of the conditions of the Whitney extension theorem. Some general treatments are [Van02, CCdlL15].
In many perturbative solutions, one gets that the twist condition is small but that the error is much smaller. Note that in the main result, we presented explictly that the smallness conditions on the error are proportional to the square of the twist condition. Hence, we obtain the small twist condition. Note also that the twist condition required is not a global condition on the map, but rather a condition that is computed on the approximate solution. Indeed, we will take advantage of this feature in the sections on applications.
The abstract theorem can be applied to several spaces. Some spaces of low regularity (e.g. ) and others with high regularity (e.g. analytic). The existence results are more powerful in the high regularity spaces and the local uniqueness is more powerful in the low regularity spaces.
Given a sufficiently regular solution, one can obtain an analytic approximate solution by truncating the Fourier series, which leads to an analytic solution, which has to be the original one. Hence, one can bootstrap the regularity. See [CdlL10a] for an abstract version.
3.6.3. Results for concrete equations
Consider the following one-dimensional Boussinesq equation subject to periodic boundary conditions, i.e.
Looking for solutions of the linearization of the form we obtain the eigenvalue relation
We see that for large , . Hence, the Fourier modes may grow at an exponential rate and the rate is quadratic in the index of the mode. So that even analytic functions evolving under the linearized equation leave instaneously even spaces of distributions. The non-linear term does not restore the well posedness. (See Remark 10.1.) The previous equation (32) is Hamiltonian on . Indeed, we introduce first the skew-symmetric operator
Therefore, equation (32) writes
where has to be understood w.r.t. the inner product in . Note, however that when is small enough, there are several values of , which for which is real. We denote by the vector whose components are all the real frequencies that appear
We can think of as the frequency vector of the motions for very small amplitude.
Note that the equation (32) conserves the quantity (called the momentum). Hence (the center of mass) evolves linearly in time.
We can always change to a system of coordinates in which . Hence, in this system . By adding the constant we can assume without loss of generality that .
Hence we will assume (without loss of generality) that
We emphasize that the two parts of (36) are not two independent equations. The first one is just a derivative with respect to time of the second. Even if the relation is formal, it makes sense when we are dealing with polynomial approximate solutions.
We also note that the equation (32) leaves invariant the space of functions which are symmetric around (it does not leave invariant the space of functions antisymmetric around ). Hence, we can consider the equation as defined on the space of general functions or in the space of symmetric functions.
The main difference between the symmetric and the general case is that center space is of different dimension.
We introduce the following Sobolev-type spaces for and being the space of analytic functions in such that the quantity
is finite, and where are the Fourier coefficients of . Let
We state the following conjecture.
Consider a parameter in (32) such that the center space has dimension and fix a Diophantine exponent , a regularity exponent and a positive analyticity radius .
Then there exist three explicit functions such that
in such a way that: for sufficiently small, denote by the ball of radius around and let .
Then, there exists , an analytic function from solving (5) with frequency .
The mapping that given produces is Lipschitz when are given the topology of analytic embeddings from to when .
In Section 10.5 we present a complete proof of the following result.
Conjecture 3.6 is true under the extra assumption that which amounts to take .
Informally, following the standard Lindstedt procedure, for small we find families of approximate solutions up to an error which is smaller than an arbitrarily large power of .
We can also verify that the non-degeneracy assumptions hold with a condition number which is a fixed power of . If is very small one can allow frequencies with a large Diophantine constant, and obtain that the functions are analytic in a very large domain. As we will see in the proof, we can take the functions to be just powers.
The first step of constructing very approximate solutions is accomplished for all values of as in Conjecture 3.6.
To verify the non-degeneracy conditions, it suffices to compute the determinant of an explicit matrix and checking it is not zero. This is the only step we are missing to verify Conjecture 3.6. This calculation is, not very hard, but it is tedious. Of course, there may be insights that make it possible to verify it. In the present paper, we will concentrate on to check this condition.
We expect that Theorem 3.7 can be greatly expanded (a wider range of parameters, removing the symmetry conditions) by just performing longer calculations using the Lindstedt method. We hope to come back to this problem in future work.
Similar results will be proved for other equations such as the Boussinesq system of water waves (see Section 11). The system under consideration is
where and System (39) has a Hamiltonian structure given by:
In this case, one has to take
The elementary linear analysis around the equilibrium has been performed in [dlL09]. The dispersion relation is given by
We take the principal determination of the square root . We denote by the vector whose components are all the real frequencies that appear
Similarly to the Boussinesq equation, we state
Fix a Diophantine exponent and a regularity exponent large. Then there exist three explicit functions such that
in such a way that: for sufficiently small, denote by the ball of radius around and let .
Conjecture (3.8) is true provided that , i.e. .
4. The linearized invariance equation
The crucial ingredient of the Newton method is to solve the linearized operator around an embedding . This is motivated because one can hope to improve the solution of (2). Notice the appearence of the linearized evolution does not have a dynamical motivation. The linearized equation does not appear as measuring the change of the evolution with respect to the initial conditions, it appears as the linearization of (2).
Let us denote
Clearly, the invariance equation (5) can be written concisely as .
We prove the following result.
Consider the linearized equation
Then there exists a constant that depends on , , , , , and the hyperbolicity constants such that assuming that satisfies
There exists an approximate solution of (43), in the following sense: there exits a function such that solves exactly
with the following estimates: for all