The Stokes Phenomenon for Some Moment Partial Differential Equations
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Abstract
We study the Stokes phenomenon for the solutions of general homogeneous linear moment partial differential equations with constant coefficients in two complex variables under condition that the Cauchy data are holomorphic on the complex plane but finitely many singular or branching points with the appropriate growth condition at the infinity. The main tools are the theory of summability and multisummability, and the theory of hyperfunctions. Using them, we describe Stokes lines, antiStokes lines, jumps across Stokes lines, and a maximal family of solutions.
Keywords
Linear PDEs with constant coefficients MomentPDEs Borel summability Multisummability Maximal family of solutions Stokes phenomenon HyperfunctionsMathematics Subject Classification (2010)
35C10 35C20 35E15 40G101 Introduction
Such type of equations was previously investigated by the first author [15, 16, 17] and by A. Lastra, S. Malek, and J. Sanz [11], mainly in the context of multisummability in a given direction.
Now, we use the similar methods as in the abovementioned papers to the study of multisummable normalised formal solution \(\widehat {u}\) of Eq. 1. It means that \(\widehat {u}\) has to be multisummable in every direction but finitely many singular directions. For this reason, we assume that the Cauchy data have finitely many singular or branching points \(z_{0},\dots ,z_{n}\in \mathbb {C}\setminus \{0\}\) and are analytically continued to \(\mathbb {C}\setminus \bigcup _{j = 0}^{n}\{z_{j}t\colon t\geq 1\}\), and that satisfy the appropriate exponential growth condition at the infinity. Observe that by the linearity of Eq. 1, it is sufficient to consider the case when there is exactly one such point, say \(z_{0}\in \mathbb {C}\setminus \{0\}\). Therefore, we only consider the case when \(\varphi _{j}(z)\in \mathcal {O}(\widetilde {\mathbb {C}\setminus \{z_{0}\}})\).
Using such formal multisummable solution \(\widehat {u}\), for any nonsingular admissible multidirection d, we are able to construct its multisum u^{d}. This multisum is an actual solution of Eq. 1 as a holomorphic function in some sectorial neighbourhood of the origin.
The main purpose of this article is the description of these actual solutions and the study of the relations between them. To this end, we introduce the concept of maximal family of solutions. It is defined as the whole family of actual solutions, which can be obtained by the method of multisummability.
The relations between solutions are studied in the context of the Stokes phenomenon. It means that we find the Stokes lines, which separate different actual solutions constructed from the same multisummable formal power series solution. We also calculate the differences between actual solutions on such lines, which are called jumps across the Stokes lines. To study such jumps, we apply the Laplacetype hyperfunctions supported on the Stokes line.
In this way, we get the main result of the paper about the maximal family of solutions and the Stokes phenomenon for Eq. 1, which is given in Theorem 3.
In the special case when \(\partial _{m_{1},t}\) and \(\partial _{m_{2},z}\) are replaced by ∂_{t} and ∂_{z}, we get the description of the Stokes phenomenon for general linear PDEs with constant coefficients.
In this sense, the paper gives the application of theory of summability for PDEs to the description of maximal family of solutions and to the study of Stokes phenomenon for such equations.
Let us recall that the theory of summability of the formal solutions of PDEs has been recently intensively developed by such authors as M. Hibino [4]; K. Ichinobe and M. Miyake [7]; K. Ichinobe [5, 6]; A. Lastra, S. Malek and J. Sanz [12]; P. Remy [19], H. Tahara and H. Yamazawa [21]; H. Yamazawa and M. Yoshino [23]; M. Yoshino [24, 25]; and others.
The paper is organised as follows. Section 2 consists of basic notations. In Section 3, we recall Balser’s theory of moment summability. In particular, we introduce kernel functions and connected with them moment functions, Gevrey order, moment Borel and Laplace transforms, ksummability, and multisummability. In the next section, we recall the concept of moment differential operators and their generalisation to pseudodifferential operators. In Section 5, we recall the notion of Stokes phenomenon. We define Stokes lines and jumps across them for multisummable formal power series. We also introduce Laplacetype hyperfunction on Stokes lines, which allows us to describe these jumps. In Section 6, we introduce the idea of a maximal family of normalised actual solutions of nonKowalevskian equation. We describe such family of solutions of Eq. 1 in the case when formal solution \(\widehat {u}\) is multisummable (Theorem 1). In Section 7, we recall how to reduce the Cauchy problem (1) to a family of the Cauchy problems of simple pseudodifferential equations. Next, using the theory of moment summability, we find the integral representation of actual solutions of these simple pseudodifferential equations in the case when their formal solutions are summable (Proposition 5). It allows us to describe a maximal family of solutions of simple equations, Stokes lines, and jumps across them (Theorem 2). Finally, we return to the equation (1), and using the theory of multisummability, we get the main result of the paper, i.e. the description of a maximal family of solution, Stokes lines, and jumps across them for the equation (1), which is given in Theorem 3. In the last section, we present a few examples of special cases of moment partial differential equations with constant coefficients, where by using hyperfunctions we derive the form of jumps across obtained Stokes lines.
2 Notation
If a function f is holomorphic on a domain \(G\subset \mathbb {C}^{n}\), then it will be denoted by \(f\in \mathcal {O}(G)\). Analogously, the space of holomorphic functions of the variable \(z^{1/\gamma }=(z_{1}^{1/\gamma _{1}},\dots ,z_{n}^{1/\gamma _{n}})\) on a domain \(G\subset \mathbb {C}^{n}\) is denoted by \(\mathcal {O}_{1/\gamma }(G)\), where \(z=(z_{1},\dots ,z_{n})\in \mathbb {C}^{n}\), \(\gamma =(\gamma _{1},\dots ,\gamma _{n})\in \mathbb {N}^{n}\) and 1/γ = (1/γ_{1},…,1/γ_{n}). In other words, \(f\in \mathcal {O}_{1/\gamma }(G)\) if and only if the function w↦f(w^{γ}) is analytic for every \(w^{\gamma }=(w_{1}^{\gamma _{1}},\dots ,w_{n}^{\gamma _{n}})\in G\).
More generally, if \(\mathbb {E}\) denotes a complex Banach space with a norm \(\\cdot \_{\mathbb {E}}\), then by \(\mathcal {O}(G,\mathbb {E})\) (resp. \(\mathcal {O}_{1/\gamma }(G,\mathbb {E})\)), we shall denote the set of all \(\mathbb {E}\)valued holomorphic functions (resp. holomorphic functions of the variables z^{1/γ}) on a domain \(G\subseteq \mathbb {C}^{n}\). For more information about functions with values in Banach spaces, we refer the reader to [2, Appendix B]. In the paper, as a Banach space \(\mathbb {E}\), we will take the space of complex numbers \(\mathbb {C}\) (we abbreviate \(\mathcal {O}(G,\mathbb {C})\) to \(\mathcal {O}(G)\) and \(\mathcal {O}_{1/\gamma }(G,\mathbb {C})\) to \(\mathcal {O}_{1/\gamma }(G)\)) or the space of functions \(E_{1/\gamma }(D):=\mathcal {O}_{1/\gamma }(D)\cap C(\overline {D})\) equipped with the norm \(\\varphi \_{E_{1/\gamma }(D)}:=\max _{z\in \overline {D}}\varphi (z)\).
The space of formal power series \({\sum }_{n = 0}^{\infty } a_{n} t^{n}\) with \(a_{n}\in \mathbb {E}\) is denoted by \(\mathbb {E}[[t]]\).
We use the “hat” notation (\(\widehat {u}\), \(\widehat {u}_{i}\), \(\widehat {f}\)) to denote the formal power series. If the formal power series \(\widehat {u}\) (resp. \(\widehat {u}_{i}\), \(\widehat {f}\)) is convergent, we denote its sum by u (resp. u_{i}, f ).
Definition 1
Suppose \(k\in \mathbb {R}\), S is an unbounded sector and \(u\in \mathcal {O}_{1/\gamma }(S,\mathbb {E})\). The function u is of exponential growth of order at most k, if for every proper subsector S^{∗}≺ S (i.e. \(\overline {S^{*}}\setminus \{0\} \subseteq S\)) there exist constants C_{1},C_{2} > 0 such that \(\u(x)\_{\mathbb {E}}\le C_{1}\*e^{C_{2}x^{k}}\) for every x ∈ S^{∗}. If this is so, one can write \(u\in \mathcal {O}_{1/\gamma }^{k}(S,\mathbb {E})\) and \(u\in \mathcal {O}_{1/\gamma }^{k}(\mathbb {C},\mathbb {E})\) for \(S=\mathbb {C}\).
More generally, if G is an unbounded domain in \(\mathbb {C}^{n}\) and \(u\in \mathcal {O}_{1/\gamma }(G,\mathbb {E})\), then \(u\in \mathcal {O}_{1/\gamma }^{k}(G,\mathbb {E})\) if for every set G^{∗} satisfying \(\overline {G^{*}}\subset \text {Int} G\) there exist constants C_{1},C_{2} > 0 such that \(\u(x)\_{\mathbb {E}}\le C_{1}\*e^{C_{2}x^{k}}\) for every x ∈ G^{∗}.
3 Kernel and Moment Functions, KSummability, and Multisummability
In this section, we recall the notion of moment methods introduced by Balser [2]. It allows us to describe moment Borel transforms, Gevrey order, Borel summability, and multisummability.
Definition 2 (see [2, Section 5.5])
 1.
\(e_{m}\in \mathcal {O}(S_{0}(\pi /k))\), e_{m}(z)/z is integrable at the origin, \(e_{m}(x)\in \mathbb {R}_{+}\) for \(x\in \mathbb {R}_{+}\) and e_{m} is exponentially flat of order k as z →∞ in S_{0}(π/k) (i.e., for every ε > 0 there exist A,B > 0 such that \(e_{m}(z)\leq A e^{(z/B)^{k}}\) for z ∈ S_{0}(π/k − ε)).
 2.
\(E_{m}\in \mathcal {O}^{k}(\mathbb {C})\) and E_{m}(1/z)/z is integrable at the origin in S_{π}(2π − π/k).
 3.The connection between e_{m} and E_{m} is given by the corresponding moment function m of order1/k as follows. The function m is defined by the Mellin transform of e_{m}and the kernel function E_{m} has the power series expansion$$ m(u):={\int}_{0}^{\infty}x^{u1} e_{m}(x)dx \quad \text{for} \quad \text{Re } u \geq 0 $$(2)$$ E_{m}(z)=\sum\limits_{n = 0}^{\infty}\frac{z^{n}}{m(n)} \quad \text{for} \quad z\in\mathbb{C}. $$(3)
 4.
Additionally, we assume that the corresponding moment function satisfies the normalisation property m(0) = 1.
Remark 1
Observe that by the inverse Mellin transform and by Eq. 3, the moment function m uniquely determines the kernel functions e_{m} and E_{m}.
In case k ≤ 1/2 the set S_{π}(2π − π/k) is not defined, so the second property in Definition 2 can not be satisfied. It means that we must define the kernel functions of order k ≤ 1/2 and the corresponding moment functions in another way. To this end, we use the ramification at z = 0.
Definition 3 (see [2, Section 5.6])
Remark 2
As in [16], we extend the notion of moment functions to real orders.
Definition 4
We say that m is a moment function of order 1/k < 0 if 1/m is a moment function of order − 1/k > 0.
We say that m is a moment function of order 0 if there exist moment functions m_{1} and m_{2} of the same order 1/k > 0 such that m = m_{1}/m_{2}.
By Definition 4 and by [2, Theorems 31 and 32] we have the following:
Proposition 1

m_{1}m_{2} is a moment function of order s_{1} + s_{2}.

m_{1}/m_{2} is a moment function of order s_{1} − s_{2}.
Example 1

\(e_{m}(z)=kz^{k}e^{z^{k}}\),

m(u) = Γ(1 + u/k),

\(E_{m}(z)={\sum }_{j = 0}^{\infty }\frac {z^{j}}{{\Gamma }(1+j/k)}=:\mathbf {E}_{1/k}(z)\), where E_{1/k} is the MittagLeffler function of index 1/k.
Example 2
The moment functions Γ_{s} will be extensively used in the paper, since every moment function m of order s has the same growth as Γ_{s}. Precisely speaking, we have the following:
Proposition 2 (see [2, Section 5.5])
Using Balser’s theory of general moment summability ([2, Section 6.5], in particular [2, Theorem 38]), we apply the moment functions to define moment Borel transforms, the Gevrey order and the Borel summability. We first introduce the following:
Definition 5
We define the Gevrey order of formal power series as follows:
Definition 6
Let \(s\in \mathbb {R}\). Then, \(\widehat {u}\in \mathbb {E}[[t]]\) is called a formal power series of Gevrey order s if there exists a disc \(D\subset \mathbb {C}\) with centre at the origin such that \(\widehat {\mathcal {B}}_{{\Gamma }_{s}}\widehat {u}\in \mathcal {O}(D,\mathbb {E})\). The space of formal power series of Gevrey order s is denoted by \(\mathbb {E}[[t]]_{s}\).
Remark 3
By Proposition 2, we may replace Γ_{s} in Definition 6 by any moment function m of the same order s.
Remark 4
If \(\widehat {u}\in \mathbb {E}[[t]]_{s}\) and s ≤ 0 then the formal series \(\widehat {u}\) is convergent, so its sum u is welldefined. Moreover, \(\widehat {u}\in \mathbb {E}[[t]]_{0} \Longleftrightarrow u\in \mathcal {O}(D,\mathbb {E})\) and \(\widehat {u}\in \mathbb {E}[[t]]_{s} \Longleftrightarrow u\in \mathcal {O}^{1/s}(\mathbb {C},\mathbb {E})\) for s < 0.
Definition 7
 If \(v\in \mathcal {O}^{k}(\widehat {S}_{d},\mathbb {E})\), then the integral operator T_{m,d} defined as follows:is called an mmoment Laplace transform in a direction d.$$(T_{m,d}v)(t):={\int}_{e^{id}\mathbb{R}_{+}}e_{m}(s/t)v(s)\frac{\text{ds}}{s} $$
 If \(v\in \mathcal {O}(S_{d}(\frac {\pi }{k}+\varepsilon ,R),\mathbb {E})\) for some ε,R > 0, then the integral operator \(T^{}_{m,d}\) is defined as follows:(where a path γ(d) is the boundary of a sector contained in \(S_{d}(\frac {\pi }{k}+\varepsilon ,R)\) with bisecting direction d, a finite radius, an opening slightly larger than π/k, and the orientation is negative) is called an inverse mmoment Laplace transform in a direction d.$$\left( T^{}_{m,d}v\right)(s):=\frac{1}{2\pi i}{\int}_{\gamma(d)}E_{m}(s/t)v(t)\frac{\text{dt}}{t} $$
Remark 5
Observe, that T_{m,d} (t^{n}) = m(n)t^{n} for every \(n\in \mathbb {N}_{0}\). Hence, \( T_{m,d}\widehat {\mathcal {B}}_{m}u=u\) for every \(u\in \mathcal {O}(D)\).
Now, we are ready to define the summability of formal power series.
Definition 8
Let k > 0 and \(d\in \mathbb {R}\). Then, \(\widehat {u}\in \mathbb {E}[[t]]\) is called ksummable in a direction d if there exist ε > 0 and a discsector \(\widehat {S}_{d}=\widehat {S}_{d}(\varepsilon )\) in a direction d such that \(v=\widehat {\mathcal {B}}_{{\Gamma }_{1/k}}\widehat {u}\in \mathcal {O}^{k}\left (\widehat {S}_{d},\mathbb {E}\right )\).
Definition 9
If \(\widehat {u}\in \mathbb {E}[[t]]\) is ksummable in all directions d but (after identification modulo 2π) finitely many directions d_{1},…,d_{n}, then \(\widehat {u}\) is called ksummable and d_{1},…,d_{n} are called singular directions of \(\widehat {u}\).
Next, we extend the notion of ksummable formal power series to that which are multisummable.
Definition 10
Remark 6
Admissibility of d with respect to k is equivalent to the inclusions I_{1} ⊆ I_{2} ⊆⋯ ⊆ I_{n}, where \(I_{j}:=(d_{j}\frac {\pi }{2k_{j}},d_{j}+\frac {\pi }{2k_{j}})\) for j = 1,…,n.
Definition 11

\(v_{n}(t):=\left (\widehat {\mathcal {B}}_{m_{n}}\cdots \widehat {\mathcal {B}}_{m_{1}}\widehat {u}\right )(t) =\sum \limits _{j = 0}^{\infty }\frac {u_{j}}{m_{1}(j){\cdots } m_{n}(j)}t^{j}\in \mathcal {O}^{\kappa _{n}}\left (\widehat {S}_{d_{n}}\right )\).

\(v_{j1}(t):=\left (T_{m_{j},d_{j}}v_{j}\right )(t)\in \mathcal {O}^{\kappa _{j1}}\left (S_{d_{j1}}\right )\) for j = n,n − 1,…,2.
Definition 12
If (d_{1},…,d_{n}) is an admissible multidirection and the functions v_{n},…,v_{j} all exist, but \(v_{j}\not \in \mathcal {O}^{\kappa _{j}}\left (S_{d_{j}}\right )\), then d_{j} is called a singular direction of \(\widehat {u}\) of level k_{j} (for j = 1,…,n).
Definition 13
If \(\widehat {u}\) has at most (after identification modulo 2π) finitely many singular directions of each level k_{j}, 1 ≤ j ≤ n, then \(\widehat {u}\) is called kmultisummable.
Remark 7
If k_{1} > ⋯ > k_{n} > 0, (d_{1},…,d_{n}) is an admissible multidirection and \(\widehat {u}_{j}\) is k_{j}summable in a direction d_{j} for j = 1,…,n, then, by [2, Lemma 20], \(\widehat {u}:=\widehat {u}_{1}+\cdots +\widehat {u}_{n}\) is kmultisummable in the multidirection d and \(\mathcal {S}_{\mathbf {k},\mathbf {d}}\widehat {u}(t)=\mathcal {S}_{k_{1},d_{1}}\widehat {u}_{1}(t)+\cdots +\mathcal {S}_{k_{n},d_{n}}\widehat {u}_{n}(t)\).
Moreover, if additionally \(\widehat {u}_{j}\) is k_{j}summable with n_{j} singular directions \(d_{j,1},\dots ,d_{j,n_{j}}\) (for j = 1,…,n) then \(\widehat {u}\) is kmultisummable and \(d_{j,1},\dots ,d_{j,n_{j}}\) are singular directions of \(\widehat {u}\) of level k_{j}.
4 Moment Operators
In this section, we recall the notion of moment differential operators constructed by Balser and Yoshino [3] and the concept of moment pseudodifferential operators introduced in the previous papers of the first author [15, 16].
Definition 14
Below, we present most important examples of moment differential operators. Other examples, including also integrodifferential operators, can be found in [16, Example 3].
Example 3
Immediately by the definition, we obtain the following connection between the moment Borel transform and the moment differentiation.
Proposition 3
 i)
\(\widehat {\mathcal {B}}_{m^{\prime }}\partial _{m,t}\widehat {u}=\partial _{\overline {m},t}\widehat {\mathcal {B}}_{m^{\prime }}\widehat {u}\) ,
 ii)
\(\widehat {\mathcal {B}}_{m^{\prime }}P(\partial _{m,t})\widehat {u}=P(\partial _{\overline {m},t})\widehat {\mathcal {B}}_{m^{\prime }}\widehat {u}\) forany polynomial P with constant coefficients.
Now, following [16], we generalise moment differential operators to a kind of pseudodifferential operators. Namely, we have the following:
Definition 15 ([16, Definition 13])
Definition 16
[15, Definition 9] Let λ(ζ) be an analytic function of the variable ξ = ζ^{1/γ} for ζ≥ r_{0} (for some \(\gamma \in \mathbb {N}\) and r_{0} > 0) of polynomial growth at infinity. Then, we define the pole order \(q\in \mathbb {Q}\) and the leading term \(\lambda _{0}\in \mathbb {C}\setminus \{0\}\) of λ(ζ) as the numbers satisfying the formula \(\lim _{\zeta \to \infty }\lambda (\zeta )/\zeta ^{q}=\lambda _{0}\). We write it also λ(ζ) ∼ λ_{0}ζ^{q}.
5 Stokes Phenomenon and Hyperfunctions
Now, we extend the concept of the Stokes phenomenon (see [18, Definition 7]) to multisummable formal power series \(\widehat {u}\in \mathbb {E}[[t]]\).
Definition 17
Assume that \(\widehat {u}\in \mathbb {E}[[t]]\) is kmultisummable with singular directions \(d_{j,1},\dots ,d_{j,n_{j}}\) of level k_{j}, 1 ≤ j ≤ n. Then, for every l = 1,…,n_{j} and j = 1,…,n, the set \(\mathcal {L}_{d_{j,l}}=\{t\in \tilde {\mathbb {C}}\colon \arg t=d_{j,l}\}\) is called a Stokes line of level k_{j} for \(\widehat {u}\).
Assume now that for fixed j ∈{1,…,n} the vector d = (d_{1},…,d_{n}) is an admissible multidirection with a singular direction d_{j} of level k_{j} and with nonsingular directions d_{l} of level k_{l} for l≠j, and let \(\mathbf {d_{j}^{\pm }}:=(d_{1},\dots ,d_{j}^{\pm },\dots ,d_{n})\) be the admissible multidirections, where \(d_{j}^{+}\) (resp. \(d_{j}^{}\)) denotes a direction close to d_{j} and greater (resp. less) than d_{j}, and let \(u^{\mathbf {d_{j}^{+}}}:=\mathcal {S}_{\mathbf {k},\mathbf {d_{j}^{+}}}\widehat {u}\) (resp. \(u^{\mathbf {d_{j}^{}}}:=\mathcal {S}_{\mathbf {k},\mathbf {d_{j}^{}}}\widehat {u}\)) then the difference \(J_{\mathcal {L}_{d_{j}},k_{j}}\widehat {u}:= u^{\mathbf {d_{j}^{+}}}u^{\mathbf {d_{j}^{}}}\) is called a jump for \(\widehat {u}\) across the Stokes line \(\mathcal {L}_{d_{j}}\) of level k_{j}.
Remark 8
Every Stokes line \(\mathcal {L}_{d_{j}}\) of level k_{j} for \(\widehat {u}\) determines also so called antiStokes lines \(\mathcal {L}_{d_{j}\pm \frac {\pi }{2k_{j}}}\) of level k_{j} for \(\widehat {u}\).
We will describe jumps across the Stokes lines in terms of hyperfunctions. The similar approach to the Stokes phenomenon one can find in [8, 13, 20]. For more information about the theory of hyperfunctions, we refer the reader to [9].
Moreover, it is natural to define the mmoment Laplace operator T_{m,d} acting on the hyperfunction G(s) as \(T_{m,d}G(t):=G(s)\left [\frac {e_{m}(s/t)}{s}\right ]\) for \(t\in S_{d}(\frac {\pi }{k}, r)\), where G(s)[φ(s)] is defined by Eq. 5. So, by Eq. 7, we may describe the jump in terms of the mmoment Laplace operator acting on the hyperfunction as \(J_{\mathcal {L}_{d}}\widehat {f}(t)=T_{m,d}G_{0}(t)\).
Analogously, we calculate jumps across a Stokes line \(\mathcal {L}_{d_{j}}\) of level k_{j} for kmultisummable \(\widehat {u}\) satisfying \(\widehat {u}=\widehat {u}_{1}+\cdots +\widehat {u}_{n}\), where \(\widehat {u}_{i}\) is k_{i}summable (i = 1,…,n).
Remark 9
Since in general the moment differential operators are not invariant under translation, we are not able to use this method to describe the jumps for solutions of \(P(\partial _{m_{1},t},\partial _{m_{2},z})u = 0\) at any point z ∈ D.
6 A Maximal Family of Solutions
Since λ_{α}(ζ) are algebraic functions, we may assume that there exist \(\gamma \in \mathbb {N}\) and r_{0} < ∞ such that λ_{α}(ζ) are holomorphic functions of the variable ξ = ζ^{1/γ} (for ζ≥ r_{0} and α = 1,…,l) and, moreover, there exist \(\lambda _{\alpha }\in \mathbb {C}\setminus \{0\}\) and q_{α} = μ_{α}/ν_{α} (for some relatively prime numbers \(\mu _{\alpha }\in \mathbb {Z}\) and \(\nu _{\alpha }\in \mathbb {N}\)) such that \(\lambda _{\alpha }(\zeta )\sim \lambda _{\alpha }\zeta ^{q_{\alpha }}\) for α = 1,…,l. Observe that ν_{α}γ for α = 1,…,l.
Under the above assumption, by a normalised formal solution \(\widehat {u}\) of Eq. 8, we mean such solution of Eq. 8, which is also a solution of the pseudodifferential equation \(\widetilde {P}(\partial _{m_{1},t},\partial _{m_{2},z})\widehat {u}= 0\) (see [15, Definition 10]).
Since the principal part of the pseudodifferential operator \(\widetilde {P}(\partial _{m_{1},t},\partial _{m_{2},z})\) with respect to \(\partial _{m_{1},t}\) is given by \(\partial _{m_{1},t}^{N}\), the Cauchy problem (8) has a unique normalised formal power series solution \(\widehat {u}\in \mathcal {O}(D)[[t]]\). If we additionally assume that \(\widehat {u}\) is multisummable, then using the procedure of multisummability in nonsingular directions, we obtain a family of normalised actual solutions of Eq. 8 on some sectors with respect to t. This motivates us to introduce the following definitions.
Definition 18
In [18], we introduced a maximal family of solutions of Eq. 8 in the case when a formal power series solution is ksummable. It is a collection of all actual solutions of Eq. 8 constructed by the procedure of ksummability. Now, we generalise this definition to the multisummable case.
Definition 19
Assume that the normalised formal power series solution \(\widehat {u}\) of Eq. 8 is kmultisummable, \(\mathcal {J}\) is a finite set of indices, and V is a sector with an opening greater than π/k_{n} on the Riemann surface of \(t^{\frac {1}{q}}\) for some \(q\in \mathbb {Q}_{+}\).
 (a)
V_{i} ⊆ V is a sector of opening greater than π/k_{1} for every \(i\in \mathcal {J}\).
 (b)
\(\{V_{i}\}_{i\in \mathcal {J}}\) is a covering of V .
 (c)
\(u_{i}\in \mathcal {O}(V_{i}\times D)\) is a normalised actual solution of Eq. 8 for every \(i\in \mathcal {J}\).
 (d)
If V_{i} ∩ V_{j}≠∅ then u_{i}≢u_{j} on (V_{i} ∩ V_{j}) × D for every \(i,j\in \mathcal {J}\), i≠j.
 (e)
For every \(i\in \mathcal {J}\), there exists an admissible nonsingular multidirection d such that \(u_{i}= \mathcal {S}_{\mathbf {k},\mathbf {d}}\widehat {u}\) on \(\tilde {V}\times D\) for some nonempty sector \(\tilde {V}\subseteq V_{i}\).
 (f)
For every admissible nonsingular multidirection d, there exists \(i\in \mathcal {J}\) such that \(\mathcal {S}_{\mathbf {k},\mathbf {d}}\widehat {u}= u_{i}\) on \(\tilde {V}\times D\) for some sector \(\tilde {V}\subseteq V_{i}\).
Now, we are ready to describe a maximal family of solutions of Eq. 8 generalising our previous result [18, Theorem 3] to the multisummable case.
Theorem 1
Let \(\widehat {u}\) be a kmultisummable normalised formal power series solution of Eq. 8 with akmultisum in a nonsingular admissible multidirection d given by \(u^{\mathbf {d}}=\mathcal {S}_{\mathbf {k},\mathbf {d}}\widehat {u}\) and satisfying \(\widehat {u}=\widehat {u}_{1}+\cdots +\widehat {u}_{n}\), where \(\widehat {u}_{j}\) isk_{j}summable for j = 1,…,n. Assume that there exists \(q\in \mathbb {Q}_{+}\),which is the smallest positive rational number such thatu^{d}(t,z) = u^{d}(te^{2qπi},z) for every nonsingular multidirection d. Suppose that the set of singular directions of \(\widehat {u}\) of level k_{j} modulo 2qπ is given by \(\{d_{j,1},\dots ,d_{j,n_{j}}\}\), where \(0\leq d_{j,1} < \dots < d_{j,n_{j}}<2q\pi \) (j = 1,…,n).
 (i)for every \(\mathbf {l}\in \mathcal {J}\) there exists an admissible multidirection d = (d_{1},…,d_{n}) satisfyingfor which the function \(u_{\mathbf {l}}:=\mathcal {S}_{\mathbf {k},\mathbf {d}}\widehat {u}\) is welldefined.$$ d_{j}\in (d_{j,l_{j}},d_{j,l_{j}+ 1}),\quad j = 1,\dots,n, $$(10)
 (ii)For every sufficiently small ε > 0, there exists r > 0 such that \(u_{\mathbf {l}}\in \mathcal {O}(V_{\mathbf {l}}(\varepsilon ,r)\times D)\) for every \(\mathbf {l}\in \mathcal {J}\), whereand W_{r} = {t ∈ W : 0 < t < r} with W being the Riemann surface of \(t\mapsto t^{\frac {1}{q}}\).$$V_{\mathbf{l}}(\varepsilon,r):=\left\{t\in W_{r}\colon \left( \arg t\frac{\varepsilon}{2},\arg t +\frac{\varepsilon}{2}\right)\subseteq I_{1,l_{1}}\cap \dots\cap I_{n,l_{n}}\right\} $$
 (iii)
\(\{u_{\mathbf {l}}\}_{\mathbf {l}\in \mathcal {J}}\) is a maximal family of solutions of Eq. 8 on W_{r} × D.
Remark 10
Observe that \(\mathcal {L}_{d_{j,l}}\) and \(\mathcal {L}_{d_{j,l}\pm \frac {\pi }{2k_{j}}}\), with d_{j,l} satisfying the assumptions of Theorem 1, are respectively Stokes and antiStokes lines of level j for l = 1,…,n_{j} and j = 1,…,n. They play an important role in our description of the maximal family of solutions of Eq. 8.
Proof of Theorem 1
Finally, we prove (iii). Since the inequality \(I_{1,l_{1}}\cap \dots \cap I_{n,l_{n}}>\frac {\pi }{k_{1}}\) holds for every \(\mathbf {l}\in \mathcal {J}\), we are able to take such small ε > 0 that the opening of V_{l}(ε,r) (V_{l} for short) is greater than \(\frac {\pi }{k_{1}}\).
By the moment version of [1, Theorem 6.2], we conclude that the space of kmultisummable series in a multidirection d is a moment differential algebra over \(\mathbb {C}\). It means that it is a linear space, which is also closed under multiplication and moment differentiations, and which for any kmultisummable series \(\widehat {f}\) and \(\widehat {g}\) satisfies: \(\mathcal {S}_{\mathbf {k},\mathbf {d}}(\widehat {f}+\widehat {g})=\mathcal {S}_{\mathbf {k},\mathbf {d}}\widehat {f}+\mathcal {S}_{\mathbf {k},\mathbf {d}}\widehat {g}\), \(\mathcal {S}_{\mathbf {k},\mathbf {d}}(\widehat {f}\cdot \widehat {g})=\mathcal {S}_{\mathbf {k},\mathbf {d}}\widehat {f}\cdot \mathcal {S}_{\mathbf {k},\mathbf {d}}\widehat {g}\), \(\mathcal {S}_{\mathbf {k},\mathbf {d}}(\partial _{m_{1},t}\widehat {f})= \partial _{m_{1},t}(\mathcal {S}_{\mathbf {k},\mathbf {d}}\widehat {f})\), and \(\mathcal {S}_{\mathbf {k},\mathbf {d}}(\partial _{m_{2},z}\widehat {f})= \partial _{m_{2},z}(\mathcal {S}_{\mathbf {k},\mathbf {d}}\widehat {f}\)).
By the construction of the family \(\{u_{\mathbf {l}}\}_{\mathbf {l}\in \mathcal {J}}\), the last two conditions in Definition 19 are also satisfied, which completes the proof. □
7 General Linear Moment Partial Differential Equations with Constant Coefficients
We will study the Stokes phenomenon and the maximal family of solutions for the normalised formal solution \(\widehat {u}\) of Eq. 8. Let us recall that we may reduce the Cauchy problem (8) of a general linear moment partial differential equation with constant coefficients to a family of the Cauchy problems of simple moment pseudodifferential equations. Namely, we have
Proposition 4
Moreover, if q_{α} is a pole order of λ_{α}(ζ) and \(\overline {q}_{\alpha }=\max \{0,q_{\alpha }\}\), then \(\widehat {u}_{\alpha \beta }\in \mathcal {O}_{1/\gamma }(D)[[t]]_{\overline {q}_{\alpha }s_{2}  s_{1}}\).
We start from the following representation of summable solutions of Eq. 11.
Proposition 5
Proof
Since φ(z) satisfies (12), by [16, Lemma 4], we conclude that \(v(t,z)\in \mathcal {O}_{1,1/\gamma }^{K}(\widehat {S}_{d}(\varepsilon )\times D)\). So, for every \(\tilde {d}\in (d\frac {\varepsilon }{2}, d+\frac {\varepsilon }{2})\), the function \(u^{\tilde {d}}(t,z):=T_{m,\tilde {d}}v(t,z)\) is welldefined and by the definitions of kernel functions (Definitions 2 and 3) for every \(\tilde {\varepsilon }\in (0,\varepsilon ),\) there exists r > 0 such that \(u^{\tilde {d}}\in \mathcal {O}_{1,1/\gamma }(S_{\tilde {d}}(\pi /K\tilde {\varepsilon },r)\times D)\). □
Now, we are ready to describe the Stokes phenomenon and the maximal family of solutions of the simple moment pseudodifferential equation (11) with the Cauchy data having the separate singular point at \(z_{0}\in \mathbb {C}\setminus \{0\}\).
Theorem 2
Let \(\widehat {u}\) be a formal solution of Eq. 11 with \(\varphi \in \mathcal {O}_{1/\gamma }^{qK}(\widetilde {\mathbb {C}\setminus \{z_{0}\}})\) for some \(z_{0}\in \mathbb {C}\setminus \{0\}\). Set K := (qs_{2} − s_{1})^{− 1}, \(\delta _{l}:=q\arg z_{0}+\frac {2l\pi }{\nu }\arg \lambda _{0}\) and \(u_{l}:=u^{\tilde {d}}\) for \(\tilde {d}\in (\delta _{l},\delta _{l + 1})\mod 2q\pi \) (forl = 0,…,μ − 1 with δ_{μ} := δ_{0} + 2qπ), where \(u^{\tilde {d}}\) is given by Eq. 13. Finally, let W_{r} = {t ∈ W : 0 < t < r} for r > 0, where W is the Riemann surface of thefunction \(t\mapsto t^{\frac {1}{q}}\).
Then, for every \(\tilde {\varepsilon }>0\), there exists r > 0 such that \(u_{l}\in \mathcal {O}_{1,1/\gamma }(S_{\delta _{l}+\frac {\pi }{\nu }}((K^{1}+\frac {2}{\nu })\pi  \tilde {\varepsilon }, r)\times D)\) (l = 0,…,μ − 1) and {u_{0},…,u_{μ− 1}} is a maximal family of solutions of Eq. 11 onW_{r} × D.
Proof
First observe that, if d≠δ_{l} mod 2qπ for l = 0,…,μ − 1 then φ satisfies the assumption (12) for sufficiently small ε > 0. Hence, by Proposition 5, \(\widehat {u}\) is Ksummable in a direction \(\tilde {d}\in \mathbb {R}\), \(\tilde {d}\neq \delta _{l}\mod 2q\pi \) for l = 0,…,μ − 1 and its Ksum \(u^{\tilde {d}}(t,z)\) satisfies (13).
Observe that \(u^{\tilde {d}}(t,z)=u^{\tilde {d}}(te^{2q\pi i},z)\) and q is the smallest positive rational number for which this equality holds. Moreover, the set of singular directions of \(\widehat {u}(t,z)\) modulo 2qπ is given by {δ_{l} mod 2qπ: l = 0,…,μ − 1}. Hence by [18, Theorem 3], for every \(\tilde {\varepsilon }>0\), there exists r > 0 such that {u_{0},…,u_{μ− 1}} with \(u_{l}\in \mathcal {O}_{1,1/\gamma }(S_{\delta _{l}+\frac {\pi }{\nu }}((K^{1}+\frac {2}{\nu })\pi  \tilde {\varepsilon }, r)\times D)\) (l = 0,…,μ − 1) is a maximal family of solutions of Eq. 11. Moreover, Stokes lines for \(\widehat {u}\) are the sets \(\mathcal {L}_{\delta _{l}}\) and antiStokes lines for \(\widehat {u}\) are the sets \(\mathcal {L}_{\delta _{l}\pm \frac {\pi }{2K}}\).
Theorem 3
 (a)The formal solution \(\widehat {u}\) is Kmultisummable, \(\widehat {u}=\widehat {u}_{0}+\widehat {u}_{1}+\dots +\widehat {u}_{n}\), where \(\widehat {u}_{0}\) is a convergent power series solution ofand \(\widehat {u}_{i}\) is a K_{i}summable power series solution of$$ \left( \prod\limits_{i=n + 1}^{\tilde{n}}\prod\limits_{\alpha= 1}^{l_{i}}(\partial_{m_{1},t}\lambda_{i\alpha}(\partial_{m_{2},z}))^{N_{i\alpha}}\right)u_{0}= 0 $$(16)Moreover,$$ \left( \prod\limits_{\alpha= 1}^{l_{i}}(\partial_{m_{1},t}\lambda_{i\alpha}(\partial_{m_{2},z}))^{N_{i\alpha}}\right)u_{i}= 0\quad\text{for}\quad i = 1,\dots,n. $$(17)for any admissible nonsingular multidirection d = (d_{1},…,d_{n}).$$ \mathcal{S}_{\mathbf{K},\mathbf{d}}\widehat{u}=u_{0}+\mathcal{S}_{K_{1},d_{1}}\widehat{u}_{1}+\dots+\mathcal{S}_{K_{n},d_{n}}\widehat{u}_{n} $$(18)
 (b)
For every \(\mathbf {l}\in \mathcal {J}\), the function \(u_{\mathbf {l}}(t,z):=\mathcal {S}_{\mathbf {K},\mathbf {d}}\widehat {u}\) is a welldefined actual solution of Eq. 8, where d is an admissible nonsingular multidirection satisfying \(d_{i}\in (\delta _{i,l_{i}},\delta _{i,l_{i}+ 1})\) for i = 1,…,n,.
 (c)For every ε > 0, there exists r > 0 such that \(u_{\mathbf {l}}\in \mathcal {O}_{1,1/\gamma }(V_{\mathbf {l}}(\varepsilon ,r)\times D)\), where$$V_{\mathbf{l}}(\varepsilon,r):=\left\{t\in W_{r}\colon \left( \arg t  \frac{\varepsilon}{2},\arg t + \frac{\varepsilon}{2}\right)\subseteq I_{1,l_{1}}\cap\dots\cap I_{n,l_{n}}\right\}. $$
 (d)
\(\{u_{\mathbf {l}}\}_{\mathbf {l}\in \mathcal {J}}\) is a maximal family of solutions of Eq. 8.
 (e)
For every i ∈{1,…,n}, the sets \(\mathcal {L}_{\delta _{i,j}}\) (resp. \(\mathcal {L}_{\delta _{i,j}\pm \frac {\pi }{2K_{i}}}\)), j = 1,…,n_{i} are Stokes lines (resp. antiStokes lines) of level K_{i}.
 (f)For every i ∈{1,…,n} and j ∈{1,…,n_{i}}, the jump across the Stokes line \(\mathcal {L}_{\delta _{i,j}}\) of level K_{i} is given bywhere \(u_{i}^{\delta _{i,j}^{\pm }}= \mathcal {S}_{K_{i},\delta _{i,j}^{\pm }}\widehat {u}_{i}\), \(\overline {m}_{i}\) is a moment function of order 1/K_{i}, F_{i,j}(s,0) is a hyperfunction on \(\mathcal {L}_{\delta _{i,j}}\) defined by \(F_{i,j}(s,0):=[v_{i}(s,0)]_{\delta _{i,j}}\), \(v_{i}(s,z):=\widehat {\mathcal {B}}_{\overline {m}_{i}}\widehat {u}_{i}(s,z)\) and \(\mathbf {l},\mathbf {l}^{\prime }\in \mathcal {J}\) satisfy \(l_{i}^{\prime }=j1\) in the case when l_{i} = j and j > 1, \(l_{i}^{\prime }=n_{i}\) in the case when l_{i} = 1, and \(l_{\alpha }=l_{\alpha }^{\prime }\) for α≠i.$$J_{\mathcal{L}_{\delta_{i,j}},K_{i}}\widehat{u}(t,0)=u_{\mathbf{l}}(t,0)u_{\mathbf{l}^{\prime}}(t,0)=u_{i}^{\delta_{i,j}^{+}}(t,0)u_{i}^{\delta_{i,j}^{}}(t,0) =F_{i,j}(s,0)\left[\frac{e_{\overline{m}_{i}}(s/t)}{s}\right], $$
 (f’)For every i ∈{1,…,n}, j ∈{1,…,n_{i}} and z ∈ D, the jump across the Stokes line \(\mathcal {L}_{\delta _{i,j}}\) of level K_{i} is given by the following:where \(u_{i}^{\delta _{i,j}^{\pm }}= \mathcal {S}_{K_{i},\delta _{i,j}^{\pm }}\widehat {u}_{i}\), \(\overline {m}_{i}\) is a moment function of order 1/K_{i}, F_{i,j}(s,z) is a hyperfunction on \(\mathcal {L}_{\delta _{i,j}(z)}\) defined by \(F_{i,j}(s,z):=[v_{i}(s,z)]_{\delta _{i,j}(z)}\), δ_{i,j}(z) := δ_{i,j} + q_{i}(arg(z_{0} − z) − arg z_{0}), \(v_{i}(s,z):=\widehat {\mathcal {B}}_{\overline {m}_{i}}\widehat {u}_{i}(s,z)\) and \(\mathbf {l},\mathbf {l}^{\prime }\in \mathcal {J}\) satisfy the same conditions as in (f).$$J_{\mathcal{L}_{\delta_{i,j}},K_{i}}\widehat{u}(t,z)=u_{\mathbf{l}}(t,z)u_{\mathbf{l}^{\prime}}(t,z)=u_{i}^{\delta_{i,j}^{+}}(t,z)u_{i}^{\delta_{i,j}^{}}(t,z) =F_{i,j}(s,z)\left[\frac{e_{\overline{m}_{i}}(s/t)}{s}\right], $$
Proof
Hence, by Proposition 5 and Theorem 2, we see that \(\widehat {u}_{i\alpha \beta }\) is K_{i}summable with the singular directions given by \(q_{i}\arg z_{0} + \frac {2j\pi }{\nu _{i}}\arg \lambda _{i\alpha }\mod 2\pi q_{i}\) for j = 0,…μ_{i} − 1. Consequently, \(\widehat {u}\) is Kmultisummable in any nonsingular admissible multidirection d = (d_{1},…,d_{n}). Since a formal power series \(\widehat {u}_{0}\) is convergent, its sum u_{0} is welldefined and by Remark 7, we conclude that Kmultisum \(\mathcal {S}_{\mathbf {K},\mathbf {d}}\widehat {u}\) of \(\widehat {u}\) is given by Eq. 18, so (a) holds.
Since \(\widehat {u}\) is Kmultisummable, using Theorem 1 we conclude that (b), (c), and (d) hold.
Since the set of singular directions of order K_{i} is given by Λ_{i}, we get the description of Stokes lines \(\mathcal {L}_{\delta _{i,j}}\) and antiStokes lines \(\mathcal {L}_{\delta _{i,j}\pm \frac {\pi }{2K_{i}}}\) of level K_{i} for δ_{i,j} ∈Λ_{i} and i = 1,…,n, so (e) is also satisfied.
Finally, to obtain (f) by Theorem 2, we calculate the jumps for \(\widehat {u}\) across the Stokes lines \(\mathcal {L}_{d_{i}}\) of level K_{i}. Using Remark 9, we get (f’). □
Let us illustrate our theory on the following simple example.
Example 4
Now, we are ready to describe the Stokes phenomenon and the maximal family of solutions of Eq. 19. By Theorem 2, \(\widehat {u}_{1}\) is 1summable with singular directions d_{1,l} := 2arg z_{0} + 2lπ and \(\widehat {u}_{2}\) is 1/2summable with singular directions d_{2,l} := 3arg z_{0} + 2lπ for \(l\in \mathbb {Z}\). Hence \(\widehat {u}\) is (1,1/2)summable, the set of singular directions (modulo 12π) of level 1 is given by {d_{1,0},…,d_{1,5}), and the set of singular directions (modulo 12π) of level 1/2 is given by {d_{2,0},…,d_{2,5}). It means that \(\mathcal {L}_{d_{1,l}}\) and \(\mathcal {L}_{d_{1,l}\pm \frac {\pi }{2}}\) are respectively Stokes and antiStokes lines of level 1, and analogously \(\mathcal {L}_{d_{2,l}}\) and \(\mathcal {L}_{d_{2,l}\pm \pi }\) are respectively Stokes and antiStokes lines of level 1/2 (l = 0,…,5).
Hence, \(\mathcal {J}=\{(j,k)\colon \ 0\leq j,k\leq 5,\ jk\leq 1\}\) and \(\{u_{(j,k)}\}_{(j,k)\in \mathcal {J}}\) is a maximal family of solutions of Eq. 19 on the Riemann surface of \(t\mapsto t^{\frac {1}{6}}\), where u_{(j,k)} := u_{1,j} + u_{2,k}, \(u_{1,j}:={u_{1}^{d}}=\mathcal {S}_{1,d}\widehat {u}_{1}\) for d ∈ (d_{1,j},d_{1,j+ 1}) = (2arg z_{0} + 2πj,2arg z_{0} + 2π(j + 1)) and \(u_{2,k}:={u_{2}^{d}}=\mathcal {S}_{2,k}\widehat {u}_{2}\) for d ∈ (d_{2,j},d_{2,j+ 1}) = (3arg z_{0} + 2πk,3arg z_{0} + 2π(k + 1)).
8 Moment Partial Differential Equations—Special Cases
In this section, we will consider certain special cases of moment partial differential equations. We derive Stokes lines and jumps across these Stokes lines in terms of hyperfunctions.
Notes
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