Abstract

We investigate Gevrey asymptotics for solutions to nonlinear parameter depending Cauchy problems with 2π-periodic coefficients, for initial data living in a space of quasiperiodic functions. By means of the Borel-Laplace summation procedure, we construct sectorial holomorphic solutions which are shown to share the same formal power series as asymptotic expansion in the perturbation parameter. We observe a small divisor phenomenon which emerges from the quasiperiodic nature of the solutions space and which is the origin of the Gevrey type divergence of this formal series. Our result rests on the classical Ramis-Sibuya theorem which asks to prove that the difference of any two neighboring constructed solutions satisfies some exponential decay. This is done by an asymptotic study of a Dirichlet-like series whose exponents are positive real numbers which accumulate to the origin.

1. Introduction

We consider a family of nonlinear Cauchy problems of the form for given initial data where is a complex parameter, , , , are some positive integers, is a finite subset of , and is a finite subset of that fulfills constraints (185). The coefficients of the linear part and of the nonlinear part are -periodic Fourier series in the variable with coefficients , in (which denotes the Banach space of bounded holomorphic functions in on some small polydisc centered at the origin in with supremum norm). We assume that all Fourier coefficients , have exponential decay in (see (177), (178)). Hence, and define bounded holomorphic functions on where is some strip of width .

The initial data are quasiperiodic in the variable and are constructed as follows: where and are real algebraic numbers (for some integer ) such that the family is -linearly independent and where the coefficients are bounded holomorphic functions on , where is a fixed open bounded sector centered at 0 and is a family of open bounded sectors centered at the origin and whose union forms a covering of , where denotes some bounded neighborhood of 0. These functions (4) are constructed in such a way that they define holomorphic functions on for some .

Recall that a function (where denotes some vector space) is said to be quasiperiodic with period , for some integer , if there exists a function such that, for all , the partial function is -periodic on , for all fixed when , which satisfies (see, e.g., [1] for a definition and properties of quasiperiodic functions). In particular, one can check that functions (4) are quasiperiodic with period in the variable.

Our main purpose is the construction of actual holomorphic functions to the problem (1), (2) on the domains for some small disc and the analysis of their asymptotic expansions as tends to zero on , for all . More precisely, we can state our main result as follows.

Main Statement. We take a set of directions , , such that , for , which are assumed to satisfy moreoverfor all , all , and all , for some fixed . We make the hypothesis that the coefficients of the initial data (4) can be expressed as Laplace transforms on along the half-line , where is a family of holomorphic functions which share the exponential growth constraints (152) with respect to , the uniform bound estimates (161), and the analytic continuation property (184).

Then, in Proposition 28, we construct a family of holomorphic and bounded functionswhich are quasiperiodic with period in the variable and which solve the problem (1), (2) on the products , where satisfies inequality (153) and for some small radius . Moreover, the differences satisfy the exponential decay (187) whose type depends on the constants , , and on the degree of any algebraic number field containing .

In Theorem 32, we show the existence of a formal serieswhose coefficients belong to the Banach space of bounded holomorphic functions on , which formally solves (1) and is moreover the Gevrey asymptotic expansion of order of on . In other words, there exist two constants such that for all and all .

Notice that the problem (1), (2) is singularly perturbed with irregular singularity (in the sense of Mandai, [2]) with respect to at provided that . It is of Kowalevski type if (meaning that the hypotheses of the classical Cauchy-Kowalevski theorem (see, e.g., [3], pp. 346–349) are fulfilled for (1)) and of mixed type when and are equal.

In a recent work [4], we have considered singularly perturbed nonlinear Cauchy problems of the form which carry both an irregular singularity with respect to at and a Fuchsian singularity (see [5] for a definition) with respect to at , for given initial data where is some linear differential operator with polynomial coefficients and is some polynomial. The initial data were assumed to be holomorphic on products . Under suitable constraints on the shape of (10) and on the initial data (11), we have shown the existence of a formal series with coefficients belonging to the Banach space of bounded holomorphic functions on (for some ) equipped with the supremum norm, solution of (10), which is the Gevrey asymptotic expansion of order of actual holomorphic solutions of (10), (11) on as -valued functions, for all .

Compared to this former result [4], the singularity nature of (1) does not come from the divergence of the formal series. This divergence rather emerges from the quasiperiodic structure of the solution space which produces a small divisor problem (as we will see below) and its Gevrey type depends not only on the type of space of our initial data but also on the shape of (1). It is worth noticing that a similar phenomenon has been observed in the paper [6] for the steady Swift-Hohenberg equation where the authors have constructed formal series solutions where the coefficients belong to some weighted Sobolev space (for well-chosen real number ) of quasiperiodic Fourier expansions in of the form where with is the so-called quasilattice in for some integer . They have shown that this formal series (13) is actually at most of Gevrey order (for a suitable integer depending on ) as series in the Hilbert space . Their main purpose was actually to use this result in order to construct approximate smooth quasiperiodic solutions of (12) up to an exponential small order by means of truncated Laplace transforms.

In a more general setting, the Cauchy problem (1), (2) we consider in this work comes within the framework of asymptotic analysis of solutions to differential equations or to partial differential equations with periodic or quasiperiodic coefficients which is a domain of intense research in these last years.

In the category of differential equations most of the results concern nonlinear equations of the form where the forcing term contains periodic or quasiperiodic coefficients. These statements deal with the construction of formal solutions which are called Lindstedt series in the literature. For convergence properties of these series, we quote the seminal work [7] and the overview [8], and for Borel resummation procedures applied more recently, we mention [9]. For applications in KAM theory for nearly integrable finitely dimensional Hamiltonian systems, we may refer to [10, 11].

In the context of partial differential equations, for existence results of quasiperiodic solutions to general families of nonlinear PDE containing a small real parameter, we indicate [12] and for the construction of periodic solutions to abstract second order nonlinear equations, we notice [13]. Concerning KAM theory results in the context of PDE such as small nonlinear perturbations of wave equations or Schrödinger equations we mention the fundamental works [14–16].

Now, we explain our main result and the principal arguments needed in its proof. The first step consists (as in [4]) of transforming (1) by means of the linear map into an auxiliary regularly perturbed nonlinear equation (149). The drawback of this transformation is the appearance of poles in the coefficients of this new equation with respect to at 0.

The approach we follow is the same as in our previous works [4, 17] and is based on a Borel resummation procedure applied to formal expansions of the form where are formal series in , which formally solves the auxiliary equation (149) for well-chosen initial data (165). It is worth pointing out that this resummation method known as -summability already enjoys a large success in the study of Gevrey asymptotics for analytic solutions to linear and nonlinear differential equations with irregular singularity; see, for instance, [18–24]. We show that the formal Borel transform of with respect to given by where , formally solves a nonlinear convolution integrodifferential Cauchy problem with rational coefficients in and is holomorphic with respect to near the origin and with respect to in some strip and meromorphic in with a pole at 0; see (171), (172).

For appropriate initial data satisfying conditions (152), (184), and (161), we show (in Proposition 20) that the formal series actually defines a holomorphic function on the product , for some , and where is some unbounded open sector with small aperture and with bisecting direction (as described above in the main statement). The functions have exponential growth rate with respect to meaning that there exist two constants such that for all , . Moreover, we show that, for all , the formal series actually define holomorphic functions on domains , where is a Riemann type sequence of the form , for some constant , which tends to 0 as tends to infinity, and share the same exponential growth rate, namely, that there exist constants , , with for all , . We point out that the occurrence of a radius of convergence shrinking to zero for the coefficients near the origin of the Borel transform is due to the presence of a small divisor phenomenon in the convolution Cauchy problem (171), (172) mentioned above. In our previous study [4], a similar outcome was caused by a leading term in the main equation (10) containing a Fuchsian operator . In this analysis, the denominators arise from the function space where the solutions are found, especially from their Fourier exponents which may tend to zero but not faster than a Riemann type sequence as follows from Lemma 13.

In order to get the estimates described above, we use a majorizing technique described in Propositions 17, 18, and 19 which reduces the investigation for bounds (19) to the study of a Cauchy-Kowalevski type problem (114), (115) in several complex variables for which local analytic solutions are found in Section 2.1; see Proposition 5. On the way we make use of estimates in weighted Banach spaces introduced in Section 2.2; see Propositions 9, 10, and 12 and Corollary 11, which are very much like those already seen in the work [4].

In the next step, for given suitable initial data (150) satisfying (158), we construct actual solutions of (149), where each function can be written as a Laplace transform of the function with respect to along a half-line . For each , the function is bounded and holomorphic on a sector with aperture larger than , with bisecting direction and with radius for some constant . In Proposition 23, we show that the function itself turns out to define a holomorphic function on for some and where satisfies (153).

We observe that, for all , the functions defined as actually solve our initial Cauchy problem (1), (2) on the products and bear representation (7) as a quasiperiodic function whose Fourier coefficients decay exponentially in . It is worthy to mention that spaces of quasiperiodic Fourier series with exponential decay were also recently used in [25] in order to find global in time and quasiperiodic in space solutions to the KdV equation.

In Proposition 28, we show moreover that the difference of two neighboring solutions and has exponentially small bounds of order , uniformly in , as tends to 0 on . We observe that for each the difference for the Fourier coefficients has exponential decay of order but its type is proportional to and therefore tends to 0 as tends to infinity. This small denominator phenomenon is the reason for the decreasement of the order to . As in our previous study [4], the bulk of the proof rests on a thorough estimation of a Dirichlet like series of the form for and with small. This kind of series appears in the context of almost periodic functions introduced by . Bohr; see, for instance, the textbook [26]. These estimates (187) are crucial in order to apply a cohomological criterion known in the literature as the Ramis-Sibuya theorem (Theorem RS) which leads to the main result of this paper, namely, the existence of a formal series with coefficients in the Banach space of holomorphic and bounded functions on , which formally solves (1) and which is, moreover, the Gevrey asymptotic expansion of order of the functions on , for all .

The layout of the paper reads as follows.

Section 2.1 is dedicated to the study of a version of the Cauchy-Kowalevski theorem for nonlinear PDEs in analytic spaces of functions with precise control on the domain of existence of their solutions in terms of norm estimates of the initial data. In Section 2.2 we establish some continuity properties of several integrodifferential and multiplication operators acting on weighted Banach spaces of holomorphic functions. These results are applied in Section 2.3 when looking for global solutions with growth constraint at infinity for a parameter depending nonlinear convolution differential Cauchy problem with singular coefficients.

We recall briefly the classical theory concerning the Borel-Laplace transform and we show some commutation formulas with multiplication and integrodifferential operators in Section 3.1; then we center our attention on finding solutions of an auxiliary nonlinear Cauchy problem obtained by the linear change of variable from our main Cauchy problem in Section 3.2. The link between this Cauchy problem and the one solved in Section 2.3 is performed by means of Borel-Laplace transforms on the corresponding solutions.

In Section 4.1, we construct actual holomorphic solutions , , of our initial problem and we show exponential decay of the difference of any two of these solutions with respect to on the intersection of their domain of definition, uniformly in the other variables. Finally, in Section 4.2, we conclude with the main result of the work, that is, the existence of a formal power series with coefficients in an appropriate Banach space, which asymptotically represents the functions with a precise control on the Gevrey order on the sectors , for all .

2. A Global Cauchy Problem in Holomorphic and Quasiperiodic Function Spaces

2.1. A Cauchy-Kowalevski Theorem in Several Variables

In this section, we recall the well-known Cauchy-Kowalevski theorem in some spaces of analytic functions for which the size of the domain of existence of the solution can be controlled in terms of some supremum norm of the initial data.

The next Banach spaces are natural extensions to the several variables cases of the spaces used in [27].

Definition 1. Let be some integer. Let be positive real numbers. We denote by the space of formal series that belong to such that is finite. One can show that equipped with the norm are Banach spaces.

In the next two lemmas we show continuity properties for some linear integrodifferential operators acting on the aforementioned Banach spaces.

Lemma 2. Let with Then, for any given , the operator is a bounded linear map from into itself. Moreover, for all .

Proof. Let of form (23). By definition, we can write From (25), we have for all . Estimate (26) follows.