Research Article | Open Access
On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity
We study a singularly perturbed PDE with quadratic nonlinearity depending on a complex perturbation parameter . The problem involves an irregular singularity in time, as in a recent work of the author and A. Lastra, but possesses also, as a new feature, a turning point at the origin in . We construct a family of sectorial meromorphic solutions obtained as a small perturbation in of a slow curve of the equation in some time scale. We show that the nonsingular parts of these solutions share common formal power series (that generally diverge) in as Gevrey asymptotic expansion of some order depending on data arising both from the turning point and from the irregular singular point of the main problem.
In this work, we consider a family of nonlinear singularly perturbed equations of the formwhere , , , and are polynomials with complex coefficients and is an analytic function in the vicinity of the origin with respect to and in and holomorphic with respect to on a horizontal strip in of the form for some .
Here we consider the case when vanishes identically near 0. The point is known to be called a turning point in that situation; see [1, 2] for a more detailed description of this terminology in the linear and nonlinear settings. Let us recall the definition of the valuation of an analytic function near as the smallest integer with the factorization for an analytic function near with . The most interesting case examined in this work is when the valuation of with respect to is larger than the valuation or since the problem cannot be reduced to the case by dividing (1) by a suitable power of and ; see Remark 13.
In our previous study , we already have considered a similar problem which corresponds to the situation when for our equation (1). Namely, we focused on the following problem:for given vanishing initial data , where , , , and are polynomials with complex coefficients and is a forcing term constructed as above. Under appropriate assumptions on the shape of (2), we established the existence of a family of actual bounded holomorphic solutions , , for some integer , defined on domains , for some fixed bounded sector with vertex at 0 and , a set of bounded sectors whose union covers a full neighborhood of 0 in . These solutions are obtained by means of Laplace and inverse Fourier transforms. On each sector , they share with respect to a common asymptotic expansion which defines a formal series with bounded holomorphic coefficients on . Moreover, this asymptotic expansion is shown to be of Gevrey order (at most) that appears in the highest order term of the operator which is of irregular type in the sense of  outlined as , for some integer and a polynomial with complex coefficients. Conjointly, since the aperture of the sectors can be chosen slightly larger than , the functions can be viewed as -sums of the formal series as defined in .
In this work, our goal is to achieve a similar statement, namely, the existence of sectorial holomorphic solutions and asymptotic expansions as tends to 0. However, the main contrast with problem (2) is that, due to the presence of the turning point, our solutions are no longer bounded in the vicinity of the origin, being meromorphic in both time variable and parameter . Namely, we build a set of actual meromorphic solutions to problem (1) of the form where , are some rational numbers, is an integer, and is a nonidentically vanishing root of a second-order algebraic equation with polynomial coefficients related to the polynomials , , see (70), and where is a bounded holomorphic function on products similar to the ones mentioned above, which can be expressed as a Laplace transform of some order and Fourier inverse transform along some half line , for some positive rational number , where represents a function with at most exponential growth of order on a sector containing with respect to , with exponential decay with respect to on and with analytic dependence on near 0 (see Theorem 19). Furthermore, we show that these functions own with respect to a common asymptotic expansion which represents a formal series with bounded holomorphic coefficients on . We specify also the nature of this asymptotic expansion which turns out to be of Gevrey order (at most) . Besides, since the aperture of the sectors may be selected slightly larger than , the functions can be identified as -sums of the formal series (Theorem 21). By construction, the integer shows up in the highest order term of the operator which is of irregular type of the form , with , for some integers , and a polynomial with complex coefficients. The rational number is built with the help of the integers , , , and and the rational numbers , ; see (86). According to the fact that , are mainly related to constraints assumed on the polynomials and (see (66), (67)), we observe that the Gevrey order of the asymptotic expansion involves information coming both from the highest irregular term and from the two polynomials and that shape the turning point at , whereas, in our previous contribution , the Gevrey order was exclusively stemming from the irregular singularity at .
The kind of equations with quadratic nonlinearity we investigate in this work is strongly related to singularly perturbed ODEs which are nonsingular at the origin of the form for some analytic functions , small complex parameter , and a complex additional parameter , described in the seminal joint paper by Canalis-Durand et al., see , where they study asymptotic properties of actual overstable solutions near a slow curve (meaning that ) in the case when the Jacobian is not invertible at . The main notable difference is that we assume the origin to be at the same time a turning point and an irregular singularity. More precisely, with the rescaling map the transformed equation (64) possesses a rational slow curve and remains a turning point and an irregular singularity for this new equation.
The construction of the distinguished solution performed in Section 4 and the parametric Borel/Laplace summable character of these solutions shown in Section 7 are also intimately linked to recent developments of exact WKB analysis of formal and analytic solutions to second-order linear ODEs of Schrödinger type. Namely, letbe a singularly perturbed ODE where is a small complex parameter and is some polynomial with complex coefficients. WKB solutions of (5) are known as special solutions that are described as an exponential where the expression satisfies a so-called Riccati equationThis last equation possesses formal power series solutions , where satisfies the quadratic equation . Once is fixed, we get two formal solutions , where for any . Notice that solves the first-order Riccati equation with turning points at the roots of . Our main PDE (1) resembles this last one provided that is a polynomial and with the significant distinction that our equation only involves differential operators with irregular singularity at . An essential feature of the theory is that the formal series are -summable in suitable directions with respect to (that are related to the function ) for any fixed . Different proofs of this fact can be found in [7–10]. Our second main statement, Theorem 21, can be considered as a similar contribution for some higher order PDEs of this latter result. Furthermore, in our study we are also able to describe the behaviour of our specific solutions near .
For more recent and advanced works related to WKB analysis and local/global studies of solutions to linear ODEs near turning points, we refer to contributions related to the 1D Schrödinger equation with simple poles , with merging pairs of simple poles and turning points , and with merging triplet of poles and turning points [13, 14] and for analytic continuation properties of the Borel transform (resurgence) of WKB expansions in the problem of confluence of two simple turning points we quote . Concerning the structure of singular formal solutions to singularly perturbed linear systems of ODEs with turning points we point out  solving an old question of Wasow. We mention also preeminent studies on WKB analysis for higher order differential equations which reveal new Stokes phenomena giving rise to so-called virtual turning points [17, 18].
In the framework of linear PDEs, normal forms for completely integrable systems near a degenerate point where two turning points coalesce have been obtained in , which is a first step toward the so-called Dubrovin conjecture which concerns the question of universal behaviour of generic solutions near gradient catastrophe of singularly Hamiltonian perturbations of first-order hyperbolic equations; see . We mention also that sectorial analytic transformations to normal forms have been obtained for systems of singularly perturbed ODEs near a turning point with multiplicity using the recent approach of composite asymptotic expansions developed in ; see .
The paper is organized as follows. In Section 2, we recall the definition introduced in the work  of some weighted Banach spaces of continuous functions with exponential growth on unbounded sectors in and with exponential decay on . We analyze the continuity of specific multiplication and linear/nonlinear convolution operators acting on these spaces.
In Section 3, we remind the reader of basic statements concerning -Borel-Laplace transforms, a version of the classical Borel-Laplace maps already used in previous works [3, 22, 23] and Fourier transforms acting on exponentially flat functions.
In Section 4, we display our main problems and explain the leading strategy in order to solve them. It consists in four operations. In a first step, we restrict our inquiry for the sets of solutions to time rescaled function spaces; see (63). Then, we consider candidates for solutions to the resulting auxiliary problem (64) that are small perturbations of a so-called slow curve which solves a second-order algebraic equation and which may be singular at the origin in . In a third step, we search again for time rescaled functions solutions for the associated problem (84) solved by the small perturbation of the slow curve; see (85). In the last step, we write down the convolution problem (95) solved by a suitable -Borel transform of a formal solution to the attached problem (87).
In Section 6, we provide a set of actual meromorphic solutions to our initial equation (61) by executing backwards the operations described in Section 4. In particular, we show that our singular functions actually solve problem (164) which is a factorized part of (61) with a more restrictive forcing term. Furthermore, the difference of any two neighboring solutions tends to 0 as tends to 0 faster than a function with exponential decay of order .
In Section 7, we show the existence of a common asymptotic expansion of Gevrey order for the nonsingular parts of these solutions of (61) and (164) based on the flatness estimates obtained in Section 6 using a theorem by Ramis and Sibuya.
2. Banach Spaces with Exponential Growth and Exponential Decay
We denote by the open disc centered at with radius in and by its closure. Let be an open unbounded sector in direction and be an open sector with finite radius , both centered at in . By convention, these sectors do not contain the origin in .
We first give definitions of Banach spaces which already appear in our previous work .
Definition 1. Let and be real numbers. We denote by the vector space of functions such that is finite. The space endowed with the norm becomes a Banach space.
As a direct consequence of Proposition 5 from , we notice the following.
Proposition 2. The Banach space is a Banach algebra for the convolution product Namely, there exists a constant (depending on ) such that for all .
Definition 3. Let and , be real numbers. Let and be integers. Let . We denote by the vector space of continuous functions on , which are holomorphic with respect to on and such thatis finite. One can check that the normed space , is a Banach space.
In the next lemma, we check that some parameter depending functions with polynomial growth with respect to the variable and exponential decay with respect to the variable , which will appear later on in our study (Section 5), belong to the Banach spaces described above.
Lemma 4. Let , be integers. Let be a polynomial that belongs to such that for all . We take a function located in and we consider a continuous function on , holomorphic with respect to on , such that for all , all .
Then, the function belongs to . Moreover, there exists a constant (depending on and ) such thatfor all .
Proof. By definition of the norm and bearing in mind the constraint on the polynomial , we can writewhich yields the lemma since an exponential grows faster than any polynomial.
The next proposition provides norm estimates for some linear convolution operators acting on the Banach spaces introduced above. These bounds are more accurate than the one supplied in Proposition 2 from . These new estimates will be essential in Section 5 in order to solve problem (95). The improvements are due to the use of thorough upper bounds estimates of a generalized Mittag-Leffler function described in the proofs of Propositions 1 and 5 from .
Proposition 5. Let , , be real numbers with . Let and be polynomials with complex coefficients such that and with for all . We consider a continuous function on , holomorphic with respect to on , such that for all , all . We make the following assumptions:(1) If , then there exists a constant (depending on and ) such thatfor all .
(2) If and , then there exists a constant (depending on , , , , and , ) such thatfor all .
Proof. By definition of the norm, we can writewhere Again by the definition of the norm of and by the constraints on the polynomials , , we deduce thatwhereWe perform the change of variable inside the integral which is a part of that yieldsAs a result, we obtain the boundswhereWe now proceed as in Proposition 1 of . We split the function into two pieces and study them separately. Namely, we decompose , where We first provide estimates for .
(a) Assume that . We see that for all , for . Hence, from the first constraint of (16), we get for all . Subsequently, we obtainwhich is finite due to the second assumption of (16).
(b) Assume that . We notice that for all , for . Therefore, again from the first constraint of (16) we get for all . Consequently, we obtainwhich is finite due to the second assumption of (16).
In a second step, we study .
We see that for all . Hence,where for all . Taking account of the estimates in  which are deduced from the asymptotic behaviour for large of the generalized Mittag-Leffler function , for , we get a constant (that depends on , , , ) such thatfor all , provided the first and last constraints of (16) hold.
(1) We consider the first case when .
Bearing in mind (31) and (33), we deduce thatwhich is finite. On the other hand, when , we make the change of variable inside and, taking (31) into account, we getwhich is finite provided that the constraints (16) are fulfilled.
(2) We examine the second case when and .
We use this time the fact that for all and the bounds (33) in order to getOn the other hand, the bounds on the domain have already been treated above owing to (35).
Finally, gathering (21), (24), (28), (30), (34), (35), and (36) yields the statement of Proposition 5.
The forthcoming proposition presents norm estimates for some bilinear convolution operators acting on the aforementioned Banach spaces.
Proposition 6. There exists a constant (depending on and ) such thatfor all .
Proof. We follow the same guidelines as in the proof of Proposition 3 from . By definition of the norm, we can writewhereBy definition of the norms of and and according to the triangular inequality for all , we deduce thatwhereWe provide upper bounds that can be split in two parts,whereis finite under the condition that according to Lemma 4 of  andWe carry out the change of variable inside the integral piece of which yields the bounds