Asymptotic Behavior of Solutions of the Cauchy Problem for a Hyperbolic Equation with Periodic Coefﬁcients II

: The paper is devoted to studying the behavior of solutions of the Cauchy problem for large values of time—more precisely, obtaining an asymptotic expansion characterizing the behavior of the solution of the Cauchy problem for a one-dimensional second-order hyperbolic equation with periodic coefﬁcients for large values of the time parameter t . To obtain this asymptotic expansion as t → ∞ , methods of the spectral theory of differential operators are used, as well as the properties of the spectrum of a non-positive Hill operator with periodic coefﬁcients


Introduction
The paper is a continuation of the paper by the authors [1], where to obtain the asymptotics of the solution of the Cauchy problem (1) and (2), the positive Hill operator (H 0 > 0) and the properties of the spectrum σ(H 0 ) of this operator were studied.
Here, we will consider the case when the Hill operator is non-positive (H 0 ≤ 0), which means that the left end of the spectrum σ(H 0 ) of the Hill operator H 0 on the complex plane of the variable λ coincides with zero or negative.
This paper also studies the behavior of the solution for |x| < b and t → ∞ of the following Cauchy: where the functions p(x) and q(x) are periodic with period 1, We also assume that the functions p(x) and q(x) are continuous or have a finite number of discontinuities of the first kind on the period, f ∈ C ∞ 0 (R), supp f ⊂ [0, 1]; b is an arbitrary fixed constant.
For the completeness of the description of these problems, we note that the study of the behavior as t → ∞ as a solution of the problems (1) and (2), and the corresponding multidimensional problems, provided that the potential differs from a constant by a finite function or is sufficient, rapidly tends to a constant at infinity; many papers and books are devoted to this area (see, for ex. [1][2][3][4][5][6][7][8][9]). Let us point out some papers in which the problems are studied, similarly to the problems (1) and (2), with different conditions on the potential and coefficients.
The behavior at large time t of the solution of the Cauchy problem for the hyperbolic equation was studied in [3,4]. In [4], under the assumption that the function q(x) tends in a certain way to the limit as x → ±∞, some estimates for the rate of decay of the solution as t → ∞, related to the stabilization rate q(x) at x → ±∞ were obtained; that is, the estimates xq (x) ≥ 0 and sup |x| s+1 |q (x)| < ∞ are satisfied for some s > 1 as x → ±∞.
Paper [5] studies that a perturbed Hill operator with an exponentially decreasing impurity potential has a resonance (or an odd number of resonances) at each sufficiently distant lacuna on the second ("non-physical") sheet.
In [6], for the one-dimensional perturbed Hill operator H, whose impurity potential has a finite first moment, the "Levinson series" is obtained. This series of relations generalize the well-known Levinson formula to the case when there is a periodic potential. The "Levinson series" is an effective tool for studying the discrete spectrum in lacunae (gap bands). In particular, it is shown that in the case of a reflectionless impurity potential with a finite second moment, there are no eigenvalues of the operator H in the distant lacunae of the spectrum.
In [7], for the Hill operator with a 1-periodic potential q(x) with the condition 1 0 q(x) dx = 0, estimates of the periodic potentials are established for gap lengths. We also note paper [8], in which the one-dimensional stationary Schrödinger equation with a quasi-periodic potential u(ω t) is studied. It is shown that if the frequency vector ω is large enough, the Schrödinger equation admits two linear independent Floquet solutions for a set of positive energy measure.
Note that in [9], the behavior of the solution of the Cauchy problem for a hyperbolic equation with a periodic potential q(x) is also studied, that is, a problem similar to the problems (1) and (2), which are considered in this paper with p(x) = 1.
In the case of periodic coefficients p(x) and q(x), the first results on the behavior of solutions of the Cauchy problem and the initial-boundary value problem of both homogeneous and inhomogeneous hyperbolic equations were obtained in [10,11].
In addition to the papers and books mentioned above, we also note more important literature, such as [12][13][14][15][16], which reflect the spectral properties of the Hill operator from different points of view. In particular, the papers [12,13] were also devoted to the definition of the Hill equation from its spectrum.
Let us present a scheme for studying the Cauchy problem (1) and (2), for the case of a non-positive Hill operator. To solve the Cauchy problem (1) and (2), in the case when the left end of the spectrum of the Hill operator coincides with zero or is negative, it is necessary to make a Fourier transform to reduce the Cauchy problem to a stationary problem. Then, we write the solution of this problem in terms of the resolvent of the Hill operator and apply the inverse Fourier transform. When the left end of the spectrum coincides with zero, at the point k = λ 0 = 0, the vertical cut in the lower half-plane of the variable k is not made. At the negative left end of the spectrum, the approach to solving the problem is the same as in [1].

Preliminaries
is called a periodic (anti-periodic) solution of the Cauchy problems (1) and (2) if it satisfies the relation for any x ∈ R and t ≥ 0, with j = 0 and j = 1 in the case of periodic and anti-periodic solutions, respectively.

Spectrum and Green's Function of the Hill Operator
Continuing the function u(x, t) by zero in the region t < 0, and applying the Fourier transform with respect to the variable t in the Cauchy problem (1) and (2), for the function we obtain the equation For any function g(x) from L 2 (−∞, +∞), we define its norm in the same space ||g; L 2 || = ||g; L 2 (−∞, +∞)||.
If the function g(x) is defined on the entire axis (−∞, +∞), then byĝ(x), we denote the restriction of this function on the segment [0, 1].
Let us present some necessary facts from the spectral theory of differential equations. For any function g(x, k), we denote by g the derivative with respect to x and by g k the derivative with respect to k Let {y = θ(x, k), y = ϕ(x, k)} be the fundamental system of solutions of the homogeneous (for f (x) ≡ 0) Equation (3) such that It is known [17] that θ(x, k) and ϕ(x, k) are entire functions in k real on the real axis, and for |k| → ∞, we have the form . These expansions can be differentiated with respect to x and with respect to k.
, ϕ (k) and F(k) are even on the real axis of the complex plane of the variable k.
The Hill operator is the differential operator generated in the Hilbert space L 2 (R) by the operation where the functions p(x) and q(x) are periodic with period 1. The spectrum σ(H 0 ) of the Hill operator H 0 is absolutely continuous and is a finite or infinite sequence of isolated segments (zones) separated by lacunae going to infinity.
Note that the Hill operator has only a continuous spectrum, which lies on the real axis and is left semi-bounded [17]. Let us replace the spectral parameter λ by k 2 so that the spectrum σ(H 0 ) of the operator H 0 on the complex plane of the variable k consists of points for which H 0 − k 2 does not have a bounded inverse on an everywhere dense set in L 2 (R).
For a more detailed characterization of the spectrum σ(H 0 ) of the Hill operator H 0 , consider the following periodic (anti-periodic) Sturm-Liouville problems.
Continuing the functionv(x, λ n ) (orv(x, µ n )) to the entire real axis, in a periodic (or anti-periodic) way, we obtain a function, which we denote by v(x, λ n ) (or v(x, µ n )).
Since Equation (3) contains the parameter k 2 , by replacing the spectral parameter λ by k 2 , we can use the expression "complex plane of the variable k" instead of the expression "complex plane of the variable λ".
Denote by C the complex plane, as in [1], in which vertical cuts were made from the points ±λ 0 in the lower half-plane of the variable k.
In this article, the complex plane C will be defined separately, depending on the cases λ 0 = 0 and λ 0 < 0.

Auxiliary Statements
For the convenience of reading this article, we present the formulations of some Propositions and Lemmas from [1]. (1) and (2), the following representation is valid:

Proposition 1 ([1]). For the solution of the problems
As is known [17], if λ n and µ n are the ends of a lacuna, then λ n = λ 2 n is a simple eigenvalue of the periodic Sturm-Liouville problem (5), and µ n = µ 2 n is a simple eigenvalue of the anti-periodic Sturm-Liouville problem (6).
Therefore, we obtain the following system Since λ n = λ 2 n are simple eigenvalues of problem (5), then the determinant of the system (7) is equal to zero, and all coefficients of the system do not vanish simultaneously. Together with the equality F(k) ≡ θ(k) + ϕ (k) = 2 for k = λ n (this results from the fact that k 2 = λ 2 n ), which served as the definition of the numbers λ n , this leads to the fact that at the points, λ n satisfies one of the following relations: Note that for any x, ξ ∈ R, the functions θ(x, k), ϕ(x, k), h(x, ξ, k) and F 2 (k) − 4 are even on the real axis of the complex plane of variable k.

Lemma 1 ([1]
). For points ±λ n , n = 0, 1, 2, . . . , if λ n are the ends of lacunae (that is, simple zeros of the function F(k) − 2), then the equalities are satisfied, where the function v(x, λ n ) = v(x, −λ n ) is the eigenfunction of the periodic Sturm-Liouville problem, and the numbers C λ n depending on the cases (A 1 )-(A 3 ) have the form If on the complex plane of the variable λ, the left end of the spectrum σ(H 0 ) of the Hill operator H 0 coincides with zero, then on the complex plane of the variable k, the spectrum σ(H 0 ) of the Hill operator H 0 is merging segments:

Main Results
The point k = 0 is a two-fold zero of the function F(k) 2 − 4, and this means that the simple zero of the function G(k) = F(k) 2 − 4.
Denote by C and L the complex plane and contour, as in [1], for which vertical cuts were made from the points ±λ 0 in the lower half-plane of the variable k . In this case, no vertical cut is drawn from the point k = λ 0 = 0.
On the complex plane C , we consider the contour L, which can be represented as where L 1 and L 2 are defined in the same way as in [1], For the integrals J 1 and J 2 , which are defined in [1], estimates (17) and (18) and t > 0 at the point k = 0 can have a first-order pole, since k = 0 is a simple zero of the function F(k) 2 − 4. Let δ be some finite contour in C . Denote by J δ the integral (1) and (2), the following representation is

Proposition 3. For the solution of problems
where the function v 1 (x, t) for x ∈ [−b, b] and t > 0 satisfies the estimate: and the function U(x) has the form Proof. The proof of this statement is similar to the proof of Proposition 1 from [1]. Due to the fact that in this case, no vertical cut is made from the point k = λ 0 into the lower half-plane of the variable k, the term U(x) is separated in the expansion of the solution of the Cauchy problem (1) and (2). Furthermore, this term is also separated in the asymptotic representation of the periodic solution of the Cauchy problem.
where u 1 (x, t) is a periodic solution of the Cauchy problem, for which is a anti-periodic solution of the Cauchy problem, for which the function U(x) is defined as while the function v(x, t) for |x| < b and t > 0 satisfies the estimate The function v(x, 0) is obtained from the normalized eigenfunctionv(x, 0) of the periodic Sturm-Liouville problem (5) corresponding to the eigenvalue λ 0 = λ 2 0 = 0, if it is continued along the entire axis in a periodic way, b 0 is some constant defined by the formula (10) below.
Here, the summation is carried out only over those n for which λ n = λ 2 n (or µ n = µ 2 n ) are simple eigenvalues of the periodic (or anti-periodic) Sturm-Liouville problem.

The Case When the Left End of the Spectrum σ(H 0 ) of the Hill Operator H 0 Is Negative
Let us now consider the case when the left end of the spectrum σ(H 0 ) of the Hill operator H 0 is negative on the complex plane of the variable λ and coincides with the point (−λ 0 ).
If the left end of the spectrum σ(H 0 ) of the Hill operator H 0 coincides with the point (−λ 0 ), then a part of the spectrum of the Hill operator H 0 on the complex plane of the variable k is located on the imaginary axis. Then, the highest point of the spectrum on the imaginary axis will be iλ 0 , λ 0 > 0. At the point iλ 0 , the function G(k) = (θ(k) + ϕ (k)) 2 − 4 has a branch point, since the point iλ 0 is a simple zero of the function G(k) [17].
Let us cut the complex plane of the variable k along the vertical ray {k : Re k = 0, Im k ≤ α 0 } and denote the resulting domain by C .
Let us put where the branch of the root is determined by the condition F(k) 2 − 4 > 0 for k = 0. Note that the function F(k) 2 − 4 has branching only at the ends of the lacuna [17], so m 1 (k) and m 2 (k) are single-valued in C . Then, for any k, Im k > 0 We define Green's function of Equation (3) for k from the upper half-plane and, taking into account the identities (11) and the equality where The solution to problems (1) and (2) has the form where a is some positive constant. Note that Green's function Γ(x, ξ, k) for every x, ξ ∈ [−b, b] continues analytically to C .
To study the properties of the (13) integral, we introduce the following notation: L + = {k : Im k = a, a > λ 0 }, L λ 0 −ε = {k : Im k = λ 0 − ε} C , and q l is the segment Re k = lπ + π 3 , λ 0 − ε ≤ Im k ≤ a, l is any real number. Consider the integral From the relations (4), it follows that moreover, |l| can tend to infinity in any way, so in (14), one can replace the line L + by L λ 0 −ε . In addition, according to (4), we have Thus, Let us explore these integrals. where Let us examine the first term in (16). Consider the function For any fixed x ∈ [−b, b], we have w ∈ L 2 (−∞, +∞) and where C 1 does not depend on f and x.
For all x ∈ [−b, b], due to the Parseval equality for the Fourier transform, we have The second term of the equality (16) is studied in a similar way. Therefore, for any fixed where C 2 does not depend on f and x.
In article [18], an elegant method is presented that allows, in the case of a periodic potential q(x), to obtain a very simple Parseval formula for the problem in the entire space.
By the Cauchy-Schwartz inequality and the last inequality, from (15), we obtain |J (1) where C 3 depends only on b.
In the same way, we obtain |J (2) where C 4 depends only on b.
To investigate J It is easy to show that By the Cauchy-Schwartz inequality, we obtain the estimate where C 5 depends only on b.
From the estimates for J

, it follows that
Likewise, for the integral we obtain the estimate Thus, we learn that the integrals J 1 and J 2 decrease exponentially as t → ∞. Let us choose the number ε > 0 in two ways: (i) ε = λ 0 2 , if on the imaginary axis, except for the point iλ 0 , there are no other branch points of the function G(k) = (θ(k) + ϕ (k)) 2 − 4; (ii) ε = d 2 , where d > 0 is the distance from the point iλ 0 to the nearest branch point of the function G(k) = (θ(k) + ϕ (k)) 2 − 4 located on the imaginary axis.
Denote by l λ 0 the contour going from the point i(λ 0 − ε) along the left edge of this cut to the point iλ 0 , and then from the point iλ 0 along to the right edge of the cut up to the point i(λ 0 − ε).
Green's function is defined by formula (12), where the single-valued branch of the root G(k) = (θ(k) + ϕ (k)) 2 − 4 is determined by the condition G(k)

Proposition 4.
For the solution of the problems (1) and (2), the following representation is valid: where the function v 2 (x, t) for x ∈ [−b, b] and t > 0 satisfies the estimate Proof. This statement is proved similarly to the proof of Proposition 1 from [1], where J M is considered instead of J L from [1], and k = σ + i(λ 0 − ε) with k ∈ L λ 0 −ε .
Proof. The proof of this statement is similar to the proof of Proposition 2 from [1], where J L λ 0 −ε is considered instead of J L 3 from [1], and k = σ + i(λ 0 − ε) with k ∈ L λ 0 −ε .

Theorem 2.
If the left end of the spectrum σ(H 0 ) of the Hill operator H 0 is negative on the complex plane of the variable λ and coincides with the point (−λ 0 ), then the solution of the Cauchy problems (1) and (2) for x ∈ [−b, b] and t → ∞ has the form where while the function v(x, t) for |x| < b and t → ∞ satisfies the estimate The function v(x, 0) is obtained from the normalized eigenfunctionv(x, 0) of the periodic Sturm-Liouville problem (5) corresponding to the eigenvalue λ 0 = λ 2 0 , if it is continued along the entire axis in a periodic way.
Proof. To complete the proof of the theorem, by virtue of Propositions 2 and 3, it remains to study the following integral: Let B(λ 0 ) = {k : |k − iλ 0 | ≤ 3 2 ε}. It is easy to show that for k ∈ B(λ 0 ) ∩ C , the following representation holds: G(k) = k − iλ 0 · g 0 (k), |g 0 (k)| ≥ c > 0, and the branch of the root √ k − iλ 0 is selected from the condition of its positivity for positive values of k − iλ 0 .
For k ∈ l λ 0 we set k = iτ, where λ 0 − ε ≤ τ ≤ λ 0 . It is easy to see that if k belongs to the left side of the contour l λ 0 , then the following equality holds and if k belongs to the right side of the contour l λ 0 , then the root has the opposite sign. The values of the function g 0 (k) coincide at the corresponding points of the left and right parts of the contour l λ 0 .
construct the asymptotics of this equation, we can use the method given in this article and the methods of resurgent analysis, similar to how it was conducted in papers [24,25].
Author Contributions: H.A.M. is responsible for the formulation of the problem and the method of its solution. M.V.K. is responsible for the generalization of the method for studying such problems using, among other things, the method of resurgent analysis. V.A.V. is responsible for the application of these problems in mechanics, aerospace and technical physics. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Data Availability Statement: Not applicable.