Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions

1 Introduction

The boundary value problems for ordinary and partial differential operators on metric graphs describe a wide variety of physical processes: vibration and diffusion in networks, wave propagation in waveguide networks, expansion of signals in neurons etc. Currently, there is increasing interest in models on graphs, in particular, as a reaction to a great deal of progress in fabricating graph-like structures of the semiconductor materials, (see the survey [1] for details). The idea to investigate the quantum dynamics of particles confined to metric graphs originated with the study of free electron models of organic molecules [2–4]. Among the systems those were successfully modeled by graphs we also mention e.g., single-mode acoustic and electro-magnetic waveguide networks [5], the Anderson transition [6], fraction excitations in fractal structures [7], and mesoscopic quantum systems [8]. This resulted into the significant intensification of development of ordinary differential equations as well as PDEs on the metric graphs for the last three decades and numerous publications respectively; we refer the reader to e.g. [9–13].

It is worth pointing out that the boundary value problems for hyperbolic operators of the second order on metric graphs as well as for hyperbolic systems of conservation laws on graphs describe a diversity of physical processes. Such problems arise for example in the modelling of transversal vibrations of networks, gas transportation networks, traffic flow on road networks, supply chain management, water flow in open canals etc. (see [14–16]). The d’Alembert operator ◻=∂t2−Δ and the associated Klein-Gordon operator □ + m² on metric graphs play an important role in relativistic quantum theories [17, 18].

Recently, there has also been a growing interest in the singularly perturbed problems on metric graphs. This is partly motivated by importance of such models for many applied problems in classical and quantum mechanics, theory of non-homogeneous media, scattering theory etc. The differential equations on graphs with small or large parameters in their coefficients represent the natural models of various complicated devices with the irregular “ramified” geometry and heterogeneous properties (see e.g. [19] and [20]). Also, the asymptotic analysis of Schrödinger operators with singular perturbed potentials is one of the most natural ways to define the Hamiltonians corresponding to point interactions supported by a discrete set [21, 22] as well as the Hamiltonians on quantum graphs with point interactions at vertices [23–25].

This paper can be viewed as a natural continuation of our work [26], where the vibrations of star-shaped network of strings with vanishingly small stiffness were treated. In [26], the boundary value problem for hyperbolic equation containing a small parameter multiplying the second space derivative was studied within the framework of singular perturbation theory, and asymptotics of solutions were constructed. The main objective of the present paper is to describe the vibrations of networks with the essentially different physical properties, e.g., the stiffness coefficients of the strings have a different order of smallness as a small parameter goes to zero. We study the asymptotic behaviour of solutions to the initial boundary value problem for hyperbolic differential operator on a star-shaped metric graph with the Dirichlet type boundary conditions. In contrast to the previous paper, where the case of total degeneration of the elliptic part was treated, we consider here the partial degeneracy of a hyperbolic operator with different degeneracy factors on subgraphs. Since the coefficients of the operator depend on a small parameter in the singular way, we show that this leads to a relatively complicated behavior of solutions. We apply the boundary layer methods [27–29] in order to construct and justify the asymptotics of solutions. Note these results are readily extended to the case of more general finite graphs. It is worth to mention that different models of vibrating systems with the singularly perturbed stiffness were studied in [30–32].

The paper is organized as follows. In the next section, we recall the basic notions of metric graphs and PDEs on graphs, and introduce the main object of the paper–the hyperbolic problem on a star-like graph depending on a small parameter. In Sections 3 and 4 we construct the formal asymptotic expansion of a solution for the singularly perturbed problem. Finally, in the last section we justify the asymptotics constructed above.

2 Statement of problem

A non-oriented finite graph is a pair 𝒢 = (𝒱,𝓔), where 𝒱 is a finite set of vertices and 𝓔 is a finite set of edges. We consider a star-shaped planar graph 𝒢 with n edges, i.e., 𝒱 = {a, a₁, …, a_n} and 𝓔 = {(a, a₁), …, (a, a_n} The edge e_j :=(a, a_j)can be considered as a piece of line connecting two points a and a_j in ℝ². We will endow the graph with the metric structure. Any edge e_j ∈ 𝓔 will be associated with an interval [0, ℓ_j] as follows. Set ℓ_j = ∥ a_j − a∥ and suppose that the map π_j:[0, ℓ_j] → e_j is given by

The map is a parametrization of e_j by the arc length parameter τ such that π_j(0) = a and π_j(ℓ_j) = a_j. Hence, there is a canonical distance function d(x, y), (x, y ∈ 𝒢)making the graph a metric space or a metric graph.

We call f : 𝒢 → ℝ the function on 𝒢. Here and subsequently, f_e denotes the restriction of f to the edge e and f_e(a)stands for the limit values lime∋x→afe(x). The function f is continuous on the star-shaped metric graph 𝒢 if it is continuous on each edge e ∈ 𝓔 and at the central vertex a. The continuity f at the vertex a means that all limit values f_e(a)along edges e ∈ 𝓔 equal the value of f at the vertex. We also introduce the differentiation of functions along edges. Set

where x = π_e(τ). Let C(𝒢)be the space of continuous functions on 𝒢. Let C^s(e) denote the space of functions f such that f^(j):e → ℝ is continuous on edge e for all j = 0, 1, …, s. We also define the spaces

We note that the derivatives of f at the central vertex a can not be defined; we must be content with the set of limit values fe′(a) along the edges. In other words, the first derivative f′ of a C¹(𝒢)-function f is generally discontinuous at x = a. Remark that the value f′(a)does not matter in our considerations against the set { fe′(a) }_{e ∈ 𝓔}.

Let us divide the set 𝓔 of edges into non-empty disjoint subsets 𝓔₀, 𝓔₁, …, 𝓔_k. Thereafter the graph 𝒢 breaks into k + 1 star subgraphs 𝒢₀, 𝒢₁, …, 𝒢_k, as is shown in Fig. 1. We will denote by ∂ 𝒢 the set of vertices {a₁, …, a_n}. Then ∂G=⋃i=0k∂Gi, where ∂ 𝒢_i is a collection of the vertices of 𝒢_i minus a. Set also G∗=⋃i=1kGi. Consequently, 𝒱_* = (∂𝒢∖ ∂ 𝒢₀)∪{a} and 𝓔_* = 𝓔 ∖ 𝓔₀.

Fig. 1

The star graph 𝒢 and the subgraphs 𝒢_i.

We consider the function b^ε:𝒢→ ℝ that is constant on each subgraph 𝒢_i. Set b^ε(x) = ε^2m_i for x ∈ 𝒢_i, where m₀ = 0 and m₁, …, m_k are positive integers such that m₁ < m₂ < ⋯ < m_k, and ε is a small positive parameter. The function b^ε can be considered as the stiffness coefficient of a bundle of strings, which possesses significantly different values on the subsystems 𝒢₀, …, 𝒢_k as ε → 0. Remark that the case of rational powers m₁, …, m_k can be reduced to the case under consideration by a suitable change of the small parameter ε ↦ ε^α. Let I=Rt+ be the positive time half-line and 𝒬 = 𝒢 × 𝓘.

We study the asymptotic behaviour of solution u^ε:𝒬 → ℝ of the boundary value problem

where f:𝒬 → ℝ, μ:∂𝒢 × 𝓘 → ℝ and q, φ, ψ:𝒢 → ℝ are given functions. By ∂xj we mean the operator of differentiation along an edge as in (1). Since b^ε is constant on each subgraph 𝓔_i, (2) is actually the set of differential equations

where e ∈ 𝓔_i and i ∈{0, …, k}. Condition (6) is usually called the Kirchhoff vertex condition.

In order to obtain both a smooth enough solution u^ε of (2)–(6) for each positive ε and the complete asymptotic expansion of u^ε, it is necessary to put some restrictions on the input data. We suppose that

Moreover, the following compatibility conditions

hold.

3 Formal asymptotic expansion of solution: leading terms

3.2 Boundary layers in a vicinity of the central vertex

First we will modify the approximation (23) in the area of degeneration of the hyperbolic equation in order to satisfy the continuity condition (5).

Given a subgraph 𝒢_i, we introduce the new variables ξ = ε^−m_i(x − a). In the auxiliary space ℝ² with the variables ξ we consider the star graph Giε , which is the image of 𝒢_i under the expanding mapping x ↦ ξ. For all positive ε this image lies on the noncompact star graph Gi∞=({O},Ei∞), as shown in Fig. 2.

Fig. 2

The graphs Giε and Gi∞ .

Therefore the star-like graph Gi∞ has the same number of edges as 𝒢_i, but its edges are noncompact, i.e., the edges are rays with the origin at point O of the auxiliary plane ℝ². Let Ei∞ be the set of edge-rays ej∞ of Gi∞. All such edges ej∞ possess the natural parametrisation

πj∞:[0,+∞)→ej∞,πj∞(τ)=aj−a∥aj−a∥τ.

Since ∂_ξ = ε^m_i ∂_x, in terms of the new variables the homogeneous PDE (2) on 𝒢_i can be written as

∂t2vε−∂ξ2vε+q(a+εmiξ)vε=0inGiε×I (24)

for all i = 1, …, k.

The coefficient b^ε in (2) is infinitely small on 𝒢_* as ε → 0, therefore we can use b^ε as a “small parameter” in this subgraph. Let us introduce the fast variables yε=(x−a)/bε(x) on the whole graph 𝒢_*, which coincide with ξ = ε^−m_i(x−a) on each subgraph 𝒢_i, and modify our approximation

uε(x,t)≈U0(x,t)if (x,t)∈Q0,u0(x,t)+v0(yε,t)if (x,t)∈Q∗. (25)

We define the function v₀ as follows. Fix a number i∊ {1, 2,…, k} and an edge e∊ 𝓔_i, and let e^∞ be the corresponding edge-ray of Ei∞ in ℝ² with the coordinates ξ = ε^−m_i(x−a) (see Fig. 2). In view of (24) and the formal Taylor expansion

qe(a+εmiξ)=qe(a)+εmiqe′(a)ξ+12ε2miqe″(a)ξ2+⋯, (26)

we assume that the restriction of v₀ to the edge e^∞ solves the initial boundary value problem

∂t2v−∂ξ2v+qe(a)v=0in e∞×I,v=0,∂tv=0on e∞×{0},v=U0(a,⋅)−u0e(a,⋅)on {O}×I. (27)

The value U₀(a, t) is uniquely defined, since U₀ is continuous at the vertex a. Hence, the approximation (25) satisfies (5), since u_0e(a, t) + v_0e(O, t) = U₀(a, t) for all e∊ 𝔼_* and t ∊ 𝓘. Note that the initial data of (27) satisfy the compatibility conditions:

limt→0v(O,t)=U0(a,0)−u0e(a,0)=φ(a)−φ(a)=0,limt→0∂tv(O,t)=∂tU0(a,0)−∂tu0e(a,0)=ψ(a)−ψ(a)=0,

which follows from (16), (20) and continuity of φ and ψ.

Lemma 3.2

Let v₀ be a solution of (27). Given T > 0, the composition

Vε(x,t)=v0x−abε(x),t

represents a boundary layer function near the central vertex as ε→ 0, i. e., V_ε is different from zero in the bε t-neighbourhood of the vertex a only.

Proof

Problem (27) is a standard initial boundary value problem for hyperbolic equation with constant coefficients in a quarter-plane [39, II.2] of the form

∂t2v−∂s2v−ϑv=0in {(s,t)∈R2:s>0,t>0},v(s,0)=α(s),∂tv(s,0)=β(s)for s>0,v(0,t)=ν(t)for t>0. (28)

It admits a unique solution, provided the C¹ compatibility conditions

ν(0)=α(0),ν′(0)=β(0) (29)

hold. Under the characteristic s − t = 0 passing through the origin, where the boundary value condition ν have no effect on the solution, we have

v(s,t)=12(α(s+t)+α(s−t))+12∫s−ts+t(k(t,s,y)β(y)+∂tk(t,s,y)α(y))dy, (30)

where k(t,s,y)=J0(ϑ((s−y)2−t2)) and J₀ is the Bessel function (see [39, II.5]). The last formula is valid for all real ϑ, in particular, for ϑ = 0 it turns into d’Alembert’s formula.

Hence, for s − t > 0 the solution v(s, t) depends on the initial data α and β in the interval [s − t, s+t] ⊂ ℝ₊ only. From this we deduce that a solution v₀ of (27) is equal to zero for s > t, because it satisfies the homogeneous initial conditions. In other words, v₀ can be different from zero in the t-neighbourhood of the origin 𝒪 only. Then V_ε is nonzero on the set {(x,t):t>0,x−a<bε(x)t} only, i.e., in the bε t-neighbourhood of the vertex a only. This region is small as ε → 0 uniformly on t∊ (0, T).  □

3.3 Improvement of the asymptotics near the boundary ∂ 𝒢_*

We now modify asymptotics (25) near the boundary vertices a_j∊ ∂ 𝒢_*, in the same way as we did it near the central vertex a. Let e be the edge of subgraph 𝒢_i connecting the vertices a and a_j. We introduce the new variables

z=ε−mi(x−aj) for x∈e.

Suppose that the function w_0,e solves the initial boundary value problem in the quarter-plane

∂t2w−∂z2w+qe(aj)w=0 in {(z,t):z<0,t>0},w(z,0)=0,∂tw(z,0)=0 for z<0,w(0,t)=μe(t)−u0e(aj,t) for t>0. (31)

In view of (8) and (20), the C¹ compatibility conditions for the problem hold. As in the earlier proof of Lemma 3.2, one likewise deduces that w_0,e is nonzero above the characteristic z + t = 0 only.

We define w0ε :𝒬_*→ ℝ as follows. For any e_j = (a, a_j) such that e_j ∊ 𝓔_i, i = 1,…, k, the restriction of w0ε to the set e_j × 𝓘 coincides with the function

w0,ej(ε−mi(x−aj),t). (32)

Hence, w0ε is a boundary layer function, which describes the singular behaviour of u^ε in a neighbourhood of boundary ∂ 𝒢_*. Note the boundary conditions at z = 0 have been chosen in the manner that u₀ + w0ε satisfies (4).

We will from now on assume that the time variable t belongs to a finite interval (0, T), and introduce the notation 𝓘_T = (0, T), 𝒬_T = 𝒢× 𝓘_T, Q0T=G0×IT, and Q∗T=G∗×IT. Then if ε is small enough and t∊(0, T), the functions v₀(y_ε, ⋅) and w0ε are actually boundary layers localized near the vertex of 𝒢_*(see the second plot in Fig. 3).

Fig. 3

The boundary layers near the vertices a and a_j on the edge e_j of subgraph 𝒢_*⋅ The function v₀ is a solution of (27) and w0ε is defined by (31), (32).

Hence, we set

uε(x,t)≈U0(x,t)if (x,t)∈Q0T,u0(x,t)+v0(yε,t)+w0ε(x,t)if (x,t)∈Q∗T. (33)

We will refer to (33) as the leading behavior or leading terms of the asymptotics. At this point, this approximation produces the largest error of order ε^m₁ in the Kirchhoff condition (6). We end the section by construction of the first correctors on 𝒢₀, although we already solved the task declared in the title of this section. The problems for these correctors are slightly different from the problem for U₀.

4 Formal asymptotic expansions: general terms

Taking into account our work in the previous section, we will look for a complete asymptotic expansion of u^ε in the form

We also assume that all v_s and wsε are functions of the boundary layer type, i.e., for small ε the functions v_s(y_ε, t) are different from zero in a small neighbourhood of the central vertex and wsε can possess non-zero values in a vicinity of the boundary ∂ 𝒢_* only. The validity of these assumptions will be considered further when we will construct the asymptotics. Recall that {bε(x)}s2=εsmi for x∈Gi.

This may happen that some terms in (39) as well as in (35) have the same order of smallness. This is because there are equal numbers among the integers sm_i. Let us introduce the sets

for each natural p. Of course, some of them are empty. All sets Λ(p) are finite, therefore the terms of the same order in (39) can be aggregated in the final form of approximation:

Substituting (39) and (40) into (2)–(6) and collecting powers of ε give a sequence of problems for the terms of series. First, the functions Us(i) are solutions to the problems

for s ≥ 2 and i ∊ {1, …, k}. In our approximation the terms Us(i) reduce errors in the Kirchhoff condition (6), but without modification of continuity condition (5) as well as the initial and boundary value conditions (3), (4). Recall that each subgraph 𝒢_i = {𝒱_i, 𝓔_i} is associated with the non-compact graph Gi∞={Vi∞,Ei∞}, obtained by the dilatation ξ = ε^−m_i(x−a) as ε → 0. The point O is the origin of plane ℝ² with the variables ξ.

The functions u_s are terms of the regular asymptotics on subgraphs 𝒢₁, …, 𝒢_k. Since the hyperbolic equation (2) becomes degenerate for ε = 0, we can find u_s on each edge e ∊ 𝓔_* by solving the Cauchy problems for ordinary differential equations with respect to time

for s ≥ 1, where u₀ is a solution of (20) and u₋₁ = 0. The task of u_s is to exhaust the residual in the right-hand side of (2) on more flexible graphs 𝒢_i, i ≥ 1.

5 Justification of the asymptotic expansions

In this section we will prove that formal series (39), (40) actually solve the singularly perturbed problem (2)–(6) in the sense of asymptotic approximation.