Research Article | Open Access
On Complex Singularity Analysis for Some Linear Partial Differential Equations in
We investigate the existence of local holomorphic solutions Y of linear partial differential equations in three complex variables whose coefficients are holomorphic on some polydisc in outside some singular set . The coefficients are written as linear combinations of powers of a solution X of some first-order nonlinear partial differential equation following an idea, we have initiated in a previous work (Malek and Stenger 2011). The solutions Y are shown to develop singularities along with estimates of exponential type depending on the growth's rate of X near the singular set. We construct these solutions with the help of series of functions with infinitely many variables which involve derivatives of all orders of X in one variable. Convergence and bounds estimates of these series are studied using a majorant series method which leads to an auxiliary functional equation that contains differential operators in infinitely many variables. Using a fixed point argument, we show that these functional equations actually have solutions in some Banach spaces of formal power series.
In this paper, we study a family of linear partial differential equations of the form for given initial data , , where is a subset of and is an integer which satisfies the constraints (175). The coefficients are holomorphic functions on some domain where is some singular set of (where denotes the disc centered at 0 in with radius ) and the initial data are assumed to be holomorphic functions on the polydisc .
In order to avoid cumbersome statements and tedious computations, the authors have chosen to restrict their study to (1) that involves at most first-order derivatives with respect to and but the method proposed in this work can also be extended to higher order derivatives too.
There exists a huge literature on the study of complex singularities and analytic continuation of solutions to linear partial differential equations starting from the fundamental contributions of Leray in . Many important results are known for singular initial data and concern equations with bounded holomorphic coefficients. In that context, the singularities of the solution are generally contained in characteristic hypersurfaces issued from the singular locus of the initial conditions. For meromorphic initial data, we may refer to [2–5] and for more general ramified multivalued initial data, we may cite [6–9]. In our framework, the initial data are assumed to be nonsingular and the coefficients of the equation now carry the singularities. To the best knowledge of the authors, few results have been worked out in that case. For instance, the research of the so-called Fuchsian singularities in the context of partial differential equations is widely developed; we provide [10–13] as examples of references in this direction. It turns out that the situation we consider is actually close to a singular perturbation problem since the nature of the equation changes nearby the singular locus of it coefficients.
This work is a continuation of our previous study . In , the authors focused on linear partial differential equations in . They have constructed local holomorphic solutions with a careful study of their asymptotic behaviour near the singular locus of the initial data. These initial data were chosen to be polynomial in , and a function satisfying some nonlinear differential equation of first order on some punctured disc and owning an isolated singularity at which is either a pole or an algebraic branch point according to a result of Painlevé. Inspired by the classical tanh method introduced in , they have considered formal series solutions of the form where are holomorphic functions on where is a small disc centered at 0. They have given suitable conditions for these series to be well defined and holomorphic for in a sector with vertex and moreover as tends to the solutions are shown to carry at most exponential bounds estimates of the form for some constants .
In this work, the coefficients are constructed as polynomials in some function with holomorphic coefficients in , where is now assumed to solve some nonlinear partial differential equation of first order and is asked to be holomorphic on a domain and to be singular along the set . The class of functions in which one can choose the coefficients is quite large since it contains meromorphic and multivalued holomorphic functions in (see the example of Section 2.1).
In our setting, one cannot achieve the goal only dealing with formal expansions involving the function like (2) since the derivatives of with respect to or cannot be expressed only in terms of . In order to get suitable recursion formulas, it turns out that we need to deal with series expansions that take into account all the derivatives of with respect to . For this reason, the construction of the solutions will follow the one introduced in a recent work of Tahara and will involve Banach spaces of holomorphic functions with infinitely many variables.
In , Tahara introduced a new equivalence problem connecting two given nonlinear partial differential equations of first order in the complex domain. He showed that the equivalence maps have to satisfy the so-called coupling equations which are nonlinear partial differential equations of first order but with infinitely many variables. It is worthwhile saying that within the framework of mathematical physics, spaces of functions of infinitely many variables play a fundamental role in the study of nonlinear integrable partial differential equations known as solitons equations as described in the theory of Sato. See  for an introduction.
The layout of the paper is a follows. In a first step described in Section 2.2, we construct formal series of the form solutions of some auxiliary nonhomogeneous integrodifferential equation (17) with polynomial coefficients in . The coefficients , , are holomorphic functions on some polydisc in that satisfy some differential recursion (Proposition 2).
In Section 2.3, we establish a sequence of inequalities for the modulus of the differentials of arbitrary order of the functions denoted by for all nonnegative integers with (Proposition 3). In the next section, we construct a sequence of coefficients which is larger than the latter sequence for any nonnegative integers with and whose generating formal series satisfies some integrodifferential functional equation (51) that involves differential operators with infinitely many variables (Propositions 5 and 6). The idea of considering recursions over the complete family of derivatives and the use of majorant series which lead to auxiliary Cauchy problems were already applied in former papers by the authors of this work; see [14, 18–21].
In Section 3, we solve the functional equation (51) by applying a fixed point argument in some Banach space of formal series with infinitely many variables (Proposition 19). The definition of these Banach spaces (Definition 7) is inspired from formal series spaces introduced in our previous work . The core of the proof is based on continuity properties of linear integrodifferential operators in infinitely many variables explained in Section 3.1 and constitutes the most technical part of the paper.
Finally, in Section 4, we prove the main result of our work. Namely, we construct analytic functions , solutions of (1) for the prescribed initial data, defined on sets for any compact set with precise bounds of exponential type in terms of the maximum value of over (Theorem 21). The proof puts together all the constructions performed in the previous sections. More precisely, for some specific choice of the nonhomogeneous term in (17), a formal solution (3) of (17) gives rise to a formal solution of (1) with the given initial data that can be written as the sum of the integral and a polynomial in having the initial data as coefficients. Owing to the fact that the generating series of the sequence , solution of (51), belongs to the Banach spaces mentioned above, we get estimates for the holomorphic functions with precise bounds of exponential type in terms of the radii of the polydiscs where they are defined; see (196). As a result, the formal solution is actually convergent for near the origin and for belonging to any compact set of . Moreover, exponential bounds are achieved; see (197). The same properties then hold for .
2. Formal Series Solutions of Linear Integrodifferential Equations
2.1. Some Nonlinear Partial Differential Equation
We consider the following nonlinear partial differential equation: where is some integer and the coefficients , are holomorphic functions on some polydisc such that is not identically equal to zero on .
Notice that (5) can be solved by using the classical method of characteristics which is described in some classical textbooks like [22, page 118] or [23, page 100]. However, the solutions of (5) cannot in general be expressed in closed form. Nevertheless, we can mention some general results concerning qualitative properties of holomorphic solutions to (5) and even to more general first-order partial differential equations of the form for where is some holomorphic function and an integer. For the construction of holomorphic functions to (6) with singularities located on some specific hypersurfaces (like ), see [24, 25]. For the existence of local multivalued holomorphic solutions ramified around some singular sets, we may refer to [26, 27]. Concerning the study of the analytic continuation of singular solutions bounded on some hypersurface, we cite  and with prescribed upper estimates, we quote [29, 30].
In this work, we make the assumption that (5) has a holomorphic solution on where is some set of ( will be called a singular set in the sequel).
In the next example, we show that a large class of functions can be obtained as solutions of equations of the form (5).
Example 1. Let be an integer and let be a holomorphic function which is not identically equal to zero. We consider which defines a holomorphic function on . Then, the function is a holomorphic solution of the equation on where is the singular set defined by and is some half-line with depending on the choice of the determination of the logarithm.
2.2. Composition Series
Let be as in the previous subsection. In the following, we choose a compact subset with nonempty interior of for some and we consider a real number such that Let be a compact set with nonempty interior . From the Cauchy formula, there exists a real number such that for all integers . For all integers , we denote . We consider a sequence of functions which are holomorphic and bounded on the polydisc , for all .
We define the formal series in the variable as For all , we consider a holomorphic and bounded function on the product . We define the formal series Let be a finite subset of and let be an integer which satisfies the property that for all . For all , , and all integers , we define a function which is holomorphic on and satisfies estimates of the following form. There exist two constants , and an integer such that for all , with all . In particular, each function is a polynomial of degree at most for all . Finally, for all , , we consider the series which define holomorphic functions on , for any .
Proposition 2. Assume that the sequence of functions satisfies the following recursion: for all , all , all , for . Then, the formal series satisfies the following integrodifferential equation: for all , where denotes the -iterate of the usual integration operator .
Proof . We have that and we also see that with for all . We also get that with for all . Now, from (5) and the classical Schwarz’s result on equality of mixed partial derivatives, we get that and from the Leibniz formula, we can write for all . Finally, gathering all the equalities above and using the recursion (16), one gets the integrodifferential equation (17).
2.3. Recursion for the Derivatives of the Functions ,
We consider a sequence of functions , , which are holomorphic and bounded on some polydisc for some real numbers and and which satisfy the equalities (16). We introduce the sequence for all , all , , for all . We define also the following sequences: for and . We put for all , and , . We define the sequences for all , all , all , , for all . We also recall the definition of the Kronecker symbol which is equal to 0 if and equal to 1 if .
Proposition 3. The sequence satisfies the following inequality: for all , all for .
Proof. In order to get the inequality (30), we apply the differential operator on the left and right hand side of the recursion (16) and we use the expansions that are computed below.
From the Leibniz formula, we deduce that Moreover, we can write with with By construction, we have for all . Again, by the Leibniz formula, we get that Inside the formula (37), we can rewrite the relations (34) and with In the same way, one gets the following equalities: with the factorizations We recall that for all and we deduce that Inside the formula (44), we can rewrite the relations (41) and with the factorization (39).
2.4. Majorant Series and a Functional Equation with Infinitely Many Variables
Definition 4. One denotes by the vector space of formal series in the variables of the form where for all .
We keep the notations of the previous section and we introduce the following formal series: for