# Theory of Nonlinear Guided Electromagnetic Waves in a Plane Two-Layered Dielectric Waveguide

Authors: Valeria Yu. Kurseeva; Dmitry V. Valovik
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

Propagation of transverse electric electromagnetic waves in a homogeneous plane two-layered dielectric waveguide filled with a nonlinear medium is considered. The original wave propagation problem is reduced to a nonlinear eigenvalue problem for an equation with discontinuous coefficients. The eigenvalues are propagation constants (PCs) of the guided waves that the waveguide supports. The existence of PCs that do not have linear counterparts and therefore cannot be found with any perturbation method is proven. PCs without linear counterparts correspond to a novel propagation regime that arises due to the nonlinearity. Numerical results are also presented; the comparison between linear and nonlinear cases is made.


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## Body

## 1. Introduction

Theory of electromagnetic wave propagation in regular (planar, cylindrical, etc.) waveguides filled with linear dielectrics traditionally attracts attention [1–4]. This theory is interesting due to several reasons: first, such problems describe real physical processes that are of importance for applications; second, from the mathematical point of view, this theory is an affluent source of sophisticated and interesting mathematical problems.Theory of electromagnetic waves in nonlinear media has also attracted attention for decades [5–12]. There are a lot of topics in this field, for example, electromagnetic wave propagation in self-focusing and self-defocusing media, higher harmonic generation (especially second and third), and Raman scattering [6, 8, 10, 13].In the theory of nonlinear electromagnetic wave propagation, the most advanced results can be found for the case of monochromatic polarised (TE and TM) waves in planar layered dielectric waveguides. From the mathematical standpoint, similar problems for circle cylindrical waveguides are much more complicated. To the best of our knowledge, the first rigorous formulation of TE and TM wave propagation in plane and circle cylindrical waveguides with nonlinear filling had been proposed in [14], and since then these and similar problems have been studied very intensively [7, 10, 15–25]. Nevertheless, key results in the cases of TE and TM wave propagation in a layer with Kerr nonlinearity have been found only recently [23–25].It is worth noting that the development of the wave propagation theory in a single layer is a first step towards studying stratified (or multilayered) waveguide structures. Layered and periodic waveguides are of special interest for optical guiding industry. Since such structures play an important role in a number of applications in optics, then they compel attention of researchers [26–35].This paper focuses on the problem of monochromatic TE wave propagation in a plane two-layered dielectric waveguideΣ filled with Kerr media. The guided wave harmonically depends on one of the longitudinal coordinates and decays along the transverse coordinate. Perfectly conducted wall is located on one of the waveguide boundaries; on the opposite side, the waveguide is open and the half-space is filled with a homogeneous isotropic nonmagnetic medium having constant permittivity. We apply the approach developed in [36]. From the mathematical point of view, the problem under investigation is a nonlinear eigenvalue problem for an ordinary nonlinear autonomous differential equation of the second order with discontinuous coefficients and boundary and transmission conditions followed from electromagnetic theory. Eigenvalues of the problem arepropagation constants (PCs) of eigenwaves of the waveguide. The PCs are solutions to the so-calleddispersion equation (DE). We derive the DE in the general case. If one of the layers is nonlinear and the other one is linear, then the DE can be studied in detail [23, 36].


## 2. Statement of the Problem

We consider the propagation of a monochromatic TE waveE,He-iωt, where ω is the circular frequency, in a lossless two-layered plane dielectric waveguide Σ=Σ1∪Σ2, where (1)Σ1=x,y,z:0⩽x<h1,y,z∈R2,Σ2=x,y,z:h1⩽x⩽h1+h2,y,z∈R2.The TE wave is described as follows:(2)E=0,Eyxeiγz,0⊤,H=Hxxeiγz,0,Hzxeiγz⊤,where E and H are the complex amplitudes [14] and γ is an unknown real propagation constant (spectral parameter).In the half-spacex<0, the permittivity is constant and is equal to ε~s=ε0εs, where εs⩾1 and ε0>0 is the permittivity of free space. There are no sources in the entire space. Everywhere, μ=μ0, where μ0 is the permeability of free space.The waveguideΣ is characterised by the permittivity ε~=ε0ε, where (3)ε=ε1+α1E2,x,y,z∈Σ1,ε2+α2E2,x,y,z∈Σ2and α1,α2>0 (see Figure 1). In what follows, we assume that εs<ε1<ε2 are real constants. There is a perfectly conducted wall σ at the boundary x=h1+h2.Figure 1
Geometry of the problem. Hereε~i=ε0εi, ε~s=ε0εs, and α~i=ε0αi.Complex amplitudes (2) satisfy Maxwell’s equations,(4)rotH=-iωε~E,rotE=iωμ0H,and decay as Ox-1 when x→-∞; tangential components of the fields are continuous on the boundaries x=0 and x=h1; tangential component of the electric field vanishes on the boundary x=h1+h2. It is assumed that the value Eyx=0≠0 is prescribed.If it does not lead to misunderstanding, the explicit dependence onx and γ is omitted.Substituting (2) into (4), one gets(5)iγHxx-Hz′x=-iωε0εEyx,-iγEyx=iωμ0Hxx,Ey′x=iωμ0Hzx.Let k02=ω2μ0ε0. Expressing Hx and Hz from the second and third equations in (5) and substituting the results into the first equation, one obtains(6)Ey′′=γ2-k02εEy.DenotingEy by y1 and y2 in the layers Σ1 and Σ2, respectively, one obtains the equations(7)y1′′=-κ12+β1y12y1,y1≡y1x;γ,x∈0,h1,(8)y2′′=-κ22+β2y22y2,y2≡y2x;γ,x∈h1,h1+h2,where κi2≔k02εi-γ2 and βi≔k02αii=1,2; κi2 are not necessarily positive.Equation (6) is linear in the half-space x<0. Taking into account conditions at infinity, one obtains its solution in the form(9)Eyx=Aeκsx,where κs2≔γ2-k02εs>0. This solution results in the condition γ2>k02εs.The continuity condition for the tangential field components results in the continuity ofEy and Ey′ at x=0 and x=h1. Using solution (9) and the continuity of Ey and Ey′, one obtains y10=A and y1′0=κsA. Since Ey corresponds to the tangential component of the electric field, then it vanishes at x=h1+h2. In view of this, one gets the following conditions:(10)y1′0-κsy10=0,y2h1+h2=0,(11)y1h1-0-y2h1+0=0,y1′h1-0-y2′h1+0=0,where y10=A≠0 is supposed to be known (without loss of generality, A>0).Thus, the original wave propagation problem is reduced to theproblemPβ1,β2, which is to determine PCs γ=γ^, such that there exists a nontrivial function (12)yx;γ^=y1x;γ^,x∈0,h1;y2x;γ^,x∈h1,h1+h2, which satisfies (7)-(8), conditions (10)-(11), and (13)y∈C10,h1+h2∩C20,h1∩C2h1,h1+h2.The problemPβ1,β2 can be treated as an eigenvalue problem for a nonlinear differential equation of the second order with discontinuous coefficients on a segment with mixed boundary conditions and transmission conditions at the point x=h1.


## 3. Linear Problem

Here we consider the caseα1=α2=0, which corresponds to the linear problem denoted by P0,0. In this case, (7) and (8) are linear:(14)y1′′x=-κ12y1x,x∈0,h1,y2′′x=-κ22y2x,x∈h1,h1+h2.Using (10), solutions to (14) are written in the following form:(i) Forγ2>k02ε2,(15)y1=Aκsκ~1sinh⁡κ~1x+cosh⁡κ~1x,y2=Bsinh⁡κ~2x-h1-h2,where κ~12≔-κ12 and κ~22≔-κ22.(ii) Fork02ε1<γ2<k02ε2,(16)y1=Aκsκ~1sinh⁡κ~1x+cosh⁡κ~1x,y2=Bsin⁡κ2x-h1-h2,where κ~12≔-κ12.(iii) Fork02εs<γ2<k02ε1,(17)y1=Aκsκ1sin⁡κ1x+cos⁡κ1x,y2=Bsin⁡κ2x-h1-h2.Using solutions (15)–(17) and (11), the DE for the problem P0,0 is written in the following form:(i) Forγ2>k02ε2,(18)κssinh⁡κ~1h1+κ~1cosh⁡κ~1h1κscosh⁡κ~1h1+κ~1sinh⁡κ~1h1=-κ~1sinh⁡κ~2h2κ~2cosh⁡κ~2h2,where κ~12=-κ12 and κ~22=-κ22.(ii) Fork02ε1<γ2<k02ε2,(19)κssinh⁡κ~1h1+κ~1cosh⁡κ~1h1κscosh⁡κ~1h1+κ~1sinh⁡κ~1h1=-κ~1sin⁡κ2h2κ2cos⁡κ2h2,where κ~12=-κ12.(iii) Fork02εs<γ2<k02ε1,(20)κssin⁡κ1h1+κ1cos⁡κ1h1κscos⁡κ1h1-κ1sin⁡κ1h1=-κ1sin⁡κ2h2κ2cos⁡κ2h2.It is clear that (18) has no solutions. Indeed, the left-hand side is positive and right-hand side is negative. Thus, solutions to the linear problem satisfy the condition(21)k02εs<γ2<k02ε2.Let us consider (19). Introduce the function(22)Fγ2=κ2cos⁡κ2h2κssinh⁡κ~1h1+κ~1cosh⁡κ~1h1+κ~1sin⁡κ2h2kscosh⁡κ~1h1+κ~1sinh⁡κ~1h1.Letsin⁡κ2h2=0; therefore γ2=k02ε2-πn/h22=γ∗2, where n⩾1 is an integer, such that k02ε2-πn/h22>k02ε1 [see formula (19)]. Thus, one obtains (23)Fγ∗2=πnh2-1nκssinh⁡κ~1h1+κ~1cosh⁡κ~1h1.Choosing the lowest possible n=1, one gets Fγ∗2<0. The inequality ε2-π2/k02h22>ε1 must be fulfilled.Now letcos⁡κ2h2=0; therefore γ2=k02ε2-π+2πm/2h22=γ∗2, where m⩾0 is an integer, such that k02ε2-π+2πm/2h22>k02ε1 [see formula (19)]. Thus, one obtains (24)Fγ∗2=κ~1-1mκscosh⁡κ~1h1+κ~1sinh⁡κ~1h1.Choosing the lowest possible m=0, one gets Fγ∗2>0. The inequality ε2-π2/4k02h22>ε1 must be fulfilled.SinceF is continuous for γ∈γ∗,γ∗ and Fγ∗2Fγ∗2<0, then there is γ~∈γ∗,γ∗ such that Fγ~=0. The inequality(25)h2>πk0ε2-ε1is a sufficient condition of existence of solutions to (19).Let us pass to (20). Introduce the function(26)Fγ2=κ2cos⁡κ2h2κssin⁡κ1h1+κ1cos⁡κ1h1+κ1sin⁡κ2h2κscos⁡κ1h1-κ1sin⁡κ1h1.Letsin⁡κ2h2=0; therefore γ2=k02ε2-πn/h22=γ∗2, where n⩾1 is an integer, such that k02εs<k02ε2-πn/h22<k02ε1 [see formula (20)]. Thus, one obtains (27)Fγ∗2=πnh2-1nκssin⁡κ1h1+κ1cos⁡κ1h1=πnh2-1nκ12+κs2sin⁡κ1h1+ϕ,where sin⁡ϕ=κ1/κ12+κs2 and cos⁡ϕ=κs/κ12+κs2. Choosing the lowest possible n=1, one gets Fγ∗2<0 subject to 0<κ1h1+ϕ<π. Since the inequality εs<ε2-π/k0h22<ε1 must be fulfilled, then π/k0ε2-εs<h2<π/k0ε2-ε1.Now letcos⁡κ2h2=0; therefore γ2=k02ε2-π+2πm/2h22=γ∗2, where m⩾0 is an integer, such that k02εs<k02ε2-π+2πm/2h22<k02ε1 [see formula (20)]. Thus, one obtains (28)Fγ∗2=κ1-1mκscos⁡κ1h1-κ1sin⁡κ1h1=κ1-1mκ12+κs2cos⁡κ1h1+ϕ.Choosing the lowest possible m=0, one gets Fγ∗2>0 subject to κ1h1+ϕ<π/2. Since the inequality εs<ε2-π/2k0h22<ε1 must be fulfilled, then π/2k0ε2-εs<h2<π/2k0ε2-ε1.It follows from the above that the inequalities(29)πk0ε2-εs<h2<π2k0ε2-ε1,h1<1κ1π2-ϕgive a sufficient condition of existence of solutions to (20).Thus, we formulate the following.Statement 1.
The problemP0,0 does not have more than a finite number of PCs γ~i, i=1,l¯. For any i=1,l¯, it is true that k02εs<γ~i2<k02ε2.Proof.
The existence of solutions to (19) and (20) subject to conditions (25) and (29), respectively, results from the analysis given above. These conditions are only sufficient. It is also clear how conditions (25) and (29) change when the number of solutions increases.
The functionsF depend analytically on γ. Since γ belongs to a finite interval [see formula (21)], then each of (19) and (20) does not have more than a finite number of isolated solutions inside the interval.


## 4. Nonlinear Problem: Dispersion Equations and Theorem of Equivalence

In the following, we need auxiliary results given below by Statements2 and 3.Statement 2.
The Cauchy problem for (7) with initial data(30)y10=A,y1′0=Aκs,where A>0 and κs=γ2-k02εs>0 are constants, has a unique continuous solution y1≡y1x;γ defined globally on 0,x∗, where x∗>0 is an arbitrary real point. This solution depends continuously on γ for all γ2>k02εs.Proof.
First integral of (7) takes the form(31)y1′2+κ12y12+12β1y14=C1,where, using conditions (30), one finds(32)C1=k02ε1-k02εs+12β1A2A2.It is clear that C1 does not depend on γ, and C1>0 if β1⩾0.
Introduce new variables:(33)τ1x=y12x,η1x=y1′xy1x.
Equation (7) can be rewritten as a system:(34)τ1′=2τ1η1,η1′=-η12+κ12+β1τ1.
First integral (31) takes the form(35)12β1τ12+η12+κ12τ1=C1.
Solving (35) with respect to τ1, taking into account the fact that τ1⩾0, and substituting the result into the right-hand side of the second equation in (34), one obtains(36)η1′=-w1η1;γ,where w1η1;γ=η12+κ122+2β1C1 and the radicand is positive for all real η1 and γ.
Using conditions (30), one finds(37)η10=κs>0.
Sinceη1′<0, η1 monotonically decreases for x>0. In the general case, y1x can have zeros at some points on the interval 0,x∗. Suppose that y1x has n1 zeros x1,…,xn1∈0,x∗. Then η1 has n1 break points x1,…,xn1∈0,x∗. If n1=0, then y1x does not become zero for any x∈0,x∗ and, therefore, η1 is continuous for x∈0,x∗. It is clear that y1′xi≠0 for all i=1,n1¯. Formula (36) implies that(38)η1xi-0=-∞,η1xi+0=+∞,i=1,n1¯.
Thereby, solutions to (36) are sought on each of the intervals 0,x1, x1,x2,…,xn1,x∗:(39)∫ηxηx1-0dsw1s;γ=x+c0,0⩽x<x1;-∫ηxi+0ηxdsw1s;γ=x+ci,xi<x<xi+1,i=0,n1-1¯;-∫ηxn1+0ηxdsw1s;γ=x+cn1,xn1<x<x∗.
Substitutingx=0, x=xi+1-0, and x=x∗ into the first, second, and third equations, respectively, in (39), one determines (40)c0=∫η0ηx1-0dsw1s;γ;ci=-∫ηxi+0ηxi+1-0dsw1s;γ-xi+1,i=1,n1-1¯;cn1=-∫ηxn1+0ηx∗dsw1s;γ-x∗.
Using the foundci, one can rewrite (39) in the following form: (41)∫ηxηx1-0dsw1s;γ=x+∫η0ηx1-0dsw1s;γ,0⩽x<x1;-∫ηxi+0ηxdsw1s;γ=x-∫ηxi+0ηxi+1-0dsw1s;γ-xi+1,xi<x<xi+1,i=1,n1-1¯;-∫ηxn1+0ηxdsw1s;γ=x-∫ηxn1+0ηx∗dsw1s;γ-x∗,xn1<x<x∗.
By substitutingx=x1-0, x=xi+0, and x=xn+0 into the first, second, and third equations, respectively, of previous equation, one obtains(42)0=x1+∫η0ηx1-0dsw1s;γ,0=xi-∫ηxi+0ηxi+1-0dsw1s;γ-xi+1,i=1,n1-1¯;0=xn1-∫ηxn1+0ηx∗dsw1s;γ-x∗.
Taking into account (37) and (38), one finds, from (42),(43)0<x1=∫-∞κsdsw1s;γ,0<xi+1-xi=∫-∞+∞dsw1s;γ,i=1,n1-1¯;0<h1-xn1=∫ηx∗+∞dsw1s;γ.
Formulas (43) give explicit expressions for distances between zeros of y1. Moreover, since the left-hand sides in (43) are finite, the right-hand sides are also finite. Therefore, the improper integrals on the right-hand sides converge.
Summing up all the terms in (43), one gets(44)x1+x2-x1+x3-x2+⋯+xn1-1-xn1-2+xn1-xn1-1+x∗-xn1=∫-∞κsw1ds+n1-1∫-∞+∞w1ds+∫ηx∗+∞w1ds.
From the last formula, one finds(45)∫ηx∗κsw1ds+n1∫-∞+∞w1ds=x∗.
Formula (45) shows that the solution of the Cauchy problem to (7) with initial conditions (30) exists and is defined globally at any segment 0,x∗. The uniqueness of this solution and its continuity with respect to γ follows from smoothness of the right-hand side of (7) with respect to y1 and γ [37].Let us consider the functionpγ≔y1′h1;γ/y1h1;γ. Since the right-hand side of (7) depends on γ analytically, the solution y1≡y1x;γ of the considered Cauchy problem depends on γ analytically as well [38] and therefore y1 and y1′ depend analytically on γ. Since y1 and y1′ do not vanish simultaneously, p≡pγ is an analytical function that can have only poles of the first order.Passing to the limitx∗→h1-0 in (45), one gets(46)Φ1γ;n1,p≡∫pκsw1ds+n1∫-∞+∞w1ds=h1,where n1=0,1,2,…. We notice that if p>κs, then necessarily n1⩾1.Statement 3.
The Cauchy problem for (8) with initial data(47)y2h1+h2=0,y2′h1+h2=B,where B is a real constant, has a unique continuous solution y2≡y2x;γ defined globally on x∗,h1+h2, where x∗<h1+h2 is an arbitrary real point. This solution depends continuously on γ for all γ2>k02εs.Proof.
Now let us consider (8). First integral of (8) has the form(48)y2′2+κ22y22+12β2y24=C2,where, using condition (47), one calculates(49)C2=B2>0,where B depends on γ. The positivity of C2 is essential.
Introduce new variables:(50)τ2x≔y22x,η2x≔y2′xy2x.
Equation (8) can be rewritten as a system:(51)τ2′=2τ2η2,η2′=-η22+κ22+β2τ2.First integral (48) takes the form(52)12β2τ22+η22+κ22τ2=C1.
Solving (52) with respect to τ2, taking into account the fact that τ2⩾0, and substituting the result into the right-hand side of the second equation in (51), one obtains(53)η2′=-w2η2;γ,where wη2;γ=η22+κ222+2β2C2 and the radicand is positive for all real η2 and γ.
Using condition (47) and the fact that η2′<0, one finds(54)η2h1+h2-0=-∞.
Sinceη2′<0, η2 monotonically decreases for x∈x∗,h1+h2. However, η2 is continuous if and only if y2x does not vanish for all x∈x∗,h1+h2. In the general case, y2x can have zeros at some points on the interval h1,h1+h2. Suppose that y2x has n2 zeros x1,…,xn2∈x∗,h1+h2. Then η2 has n2 break points x1,…,xn2∈x∗,h1+h2. It is clear that y2′xi≠0 for all i=1,n2¯. Formula (53) implies that(55)η2xi-0=-∞,η2xi+0=+∞,i=1,n2¯.
Thereby, solutions to (53) are sought on each of the intervals x∗,x1, x1,x2,…,xn2,h1+h2.
Using the same reasoning as in the proof of Statement2, one obtains the expression(56)∫-∞ηx∗dsw2s;γ+n2∫-∞+∞dsw2s;γ=h2.
Formula (56) shows that the solution to the Cauchy problem for (8) with initial conditions (47) exists and is defined globally at any segment x∗,h1+h2. The uniqueness of this solution and its continuity with respect to γ follows from smoothness of the right-hand side of (8) with respect to y2 and γ [37].Passing to the limitx∗→h1+0 in (56), one gets(57)Φ2γ;n2,p≡∫-∞pdsw2s;γ+n2∫-∞+∞dsw2s;γ=h2,where n2=0,1,2,….The following theorem takes place.Theorem 1 (of equivalence).
The valueγ^ is a PC of the problem Pβ1,β2 if and only if there are integers n1=n^1⩾0 and n2=n^2⩾0 such that γ=γ^ is a solution to the DE:(58)Φ1γ;n1,p=h1,Φ2γ;n2,p=h2with certain p=p^.Proof.
It follows from the derivation of expressions (46) and (57) that if γ=γ^ is an eigenvalue of the problem Pβ1,β2, then it is a solution to system (58) with n1=n^1, n2=n^2, and p=p^. Let us prove that each solution γ=γ^ to system (58) is an eigenvalue.
Let system (58) have a solution γ=γ^ with n1=n^1, n2=n^2, p=p^, and y10=A.
Consider the Cauchy problem for (7) with initial data (30), where κs=γ^2-k02εs. In accordance with Statement 2, its solution y1≡y1x,γ^ exists, is unique, and is defined for x∈0,h1. At this step, we do not claim that y1′h1/y1h1=p^. Using the found solution y1 and formula (33), one determines the functions τ1 and η1. It is clear that τ10=A2 and η10=κs.
Assuming thatη1h1=p1<p^ and using the found τ1 and η1, one obtains the expression (59)∫p1κsdsw1s;γ^+n^1∫-∞+∞dsw1s;γ^=h1,which corresponds to the first line in (58). We rewrite it in the form(60)∫p1p^dsw1s;γ^+∫p^κsdsw1s;γ^+n^1∫-∞+∞dsw1s;γ^=h1.
Sinceγ=γ^ satisfies the first line in (58) with p=p^, then in the first line in (58) and in (60) the integrands coincide. Subtracting one from another, one obtains(61)∫p1p^dsw1s;γ^=0.
Due to the obvious estimates∫-∞+∞ds/w1s;γ^>∫p1p^ds/w1s;γ^>0, one obtains that (61) is fulfilled only if p1=p^. Therefore, the condition ηh1=p1<p^ is false. In the same way, it can be shown that the condition ηh1=p1>p^ is also false. Thus, p1=p^.
Now let us pass to the Cauchy problem for (8) with initial data (47), where B≡Bγ^. In accordance with Statement 3, its solution y2≡y2x,γ^ exists, is unique, and is defined globally for x∈h1,h1+h2. In this case, using the continuity of Ey and Ey′, first integral (48), and (49), one finds that the quantity B is determined from the equation (62)B2=y1′2+k02ε2-γ^2y12+12β2y14>0,where y1 and y1′ are calculated at x=h1 with γ=γ^ (y1 and y1′ are already found).
At this step, we do not claim thaty2′h1/y2h1=p^. Using the found solution y2 and formula (50), one determines the functions τ2 and η2.
We assume thatη2h1=p2>p^. Using the found τ2 and η2, one obtains the expression (63)∫-∞p2dsw2s;γ^+n^2∫-∞+∞dsw2s;γ^=h2,which corresponds to the second line in (58). We rewrite it in the following form:(64)∫-∞p^dsw2s;γ^+∫p^p2dsw2s;γ^+m2∫-∞+∞dsw2s;γ^=h2.
Sinceγ=γ^ satisfies the second line in (58) with p=p^, then in the second line in (58) and in (64) the integrands coincide. Subtracting one from another, one obtains(65)∫p^p2dsw2s;γ^=0.Due to the obvious estimates ∫-∞+∞(ds/w2s;γ^)>∫p^p2(ds/w2s;γ^)>0, one obtains that (65) is fulfilled only if p2=p^. Thus, the condition η2h1=p2>p^ is false. In the same way, it can be shown that the condition ηh1=p2<p^ is also false. Thus, p2=p^.
In other words, we have shown that the functionsy1 and y2 satisfy (7) and (8) and conditions (10) and (11), and, therefore, the function (66)yx;γ^=y1x;γ^,x∈0,h1;y2x;γ^,x∈h1,h1+h2 is an eigenfunction of the problem Pβ1,β2 corresponding to the eigenvalue γ=γ^.


## 5. Solvability of the Nonlinear Problem

Theorem1 is derived for the general case, that is, β1,β2>0. The DE (58) can be studied theoretically and numerically. However, the existence of infinitely many PCs for the general case can be proven only if a special restriction is imposed on B≡Bγ. This restriction establishes the behaviour of B for big γ. To be more precise, Bγ must not decrease “too” rapid. As a matter of fact, such a restriction does not result from the physical formulation of the problem. For this reason below we study two simplified problems, where either β1 or β2 vanishes.In this section, we use the following notation for the eigenvaluesγ^ of problems P0,β2 and Pβ1,0: γ^i means that all the eigenvalues are arranged in the ascending order; γ^m means that this eigenvalue is a solution of (57) with n2=m [for the problem P0,β2] and (46) with n1=m [for the problem Pβ1,0].


### 5.1. Caseβ1=0 and β2≠0

Ifβ1=0 (or α1=0), solutions to (7) are found elementarily. This allows one to explicitly compute the quantity p, which is(67)p=κ1κscos⁡κ1h1-κ1sin⁡κ1h1κssin⁡κ1h1+κ1cos⁡κ1h1=κ1κs-κ1tan⁡κ1h1κ1+κstan⁡κ1h1.This expression (due to its analytical dependence on γ) can be used for κ12⩾0 as well as for κ12⩽0. Note that p is a real value for all γ2>k02εs. Indeed, for γ2>k02ε1, one has κ1=iκ~1, where κ~12=γ2-k02ε1, and then cos⁡ix=cosh⁡x and sin⁡ix=isinh⁡x; then (68)p=κ~1κscosh⁡κ~1h1+κ~1sinh⁡κ~1h1κssinh⁡κ~1h1+κ~1cosh⁡κ~1h1=κ1κs+κ~1tanh⁡κ~1h1κ~1+κstanh⁡κ~1h1.It can be checked that the substitution of the explicit expression forp into the first equation of system (58) leads to the identity. Thus, in this case, system (58) reduces to the only equation(69)Φ2γ;n2,p=h2,where Φ2 and p are given by (57) and (67), respectively; n2=0,1,2,….The solvability of the problemP0,β2 is established by the following.Theorem 2.
Forβ1=0, any β2>0, and any fixed A≠0, the problem P0,β2 has an infinite number of PCs γ^ii=1,2,… with the following properties:(1)
Ifγ^i is the solution to P0,β2, then γ^i2∈k02εs,+∞ and limi→∞⁡γ^i2=+∞.(2)
If the linear problemP0,0 has q solutions γ~1<γ~2<⋯<γ~q, then there exists a constant β0>0 such that, for any β2=β2′<β0, it is true that γ^i2∈k02εs,k02ε2 and limβ1′→0⁡γ^i=γ~ii=1,q¯, where γ^1,γ^2,…,γ^q are first q solutions to P0,β2′.(3)
Ifγ^i→∞, then maxx∈h1,h1+h2⁡y2x,γ^i→∞.(4)
For bigγ^ and arbitrary small Δ>0, the asymptotic two-sided inequality(70)1-Δ1h1ln⁡mπh2Aβ2⩽γ^m⩽1+Δ1h1ln⁡2m+1πh2Aβ2is valid.Proof.
Now let us study the behaviour ofC2≡C2γ with respect to γ. Explicit solution to (7) with initial data (30) has the form y1x=A(κs/κ1)sin⁡κ1x+cos⁡κ1x. At the point x=h1, one gets(71)y1h1=Aκsκ1sin⁡κ1h1+cos⁡κ1h1,y1′h1=Aκ1κsκ1cos⁡κ1h1-sin⁡κ1h1.Substituting (71) into (48), one obtains(72)C2=κ12κsκ1cos⁡κ1h1-sin⁡κ1h12A2+κ22κsκ1sin⁡κ1h1+cos⁡κ1h12A2+12β2κsκ1sin⁡κ1h1+cos⁡κ1h14A4.
Ifk02εs<γ2⩽k02ε1, then κ12⩾0 and κ22⩾0 and, therefore, C2>0.
Ifk02ε1<γ2⩽k02ε2, then κ12<0 and κ22⩾0. As before, we denote κ1=iκ~1, where κ~12=γ2-k02ε1>0. Then(73)C2=κ~12κsκ~1cosh⁡κ~1h1+sinh⁡κ~1h12A2+κ22κsκ~1sinh⁡κ~1h1+cosh⁡κ~1h12A2+12β2κsκ~1sinh⁡κ1h1+cosh⁡κ~1h14A4>0.
Ifγ2>k02ε2, then κ12<0 and κ22<0. As before, we denote κ1=iκ~1 and κ2=iκ~2, where κ~12=γ2-k02ε1>0 and κ~22=γ2-k02ε2>0. Then(74)C2=κ~12κsκ~1cosh⁡κ~1h1+sinh⁡κ~1h12A2-κ~22κsκ~1sinh⁡κ~1h1+cosh⁡κ~1h12A2+12β2κsκ~1sinh⁡κ1h1+cosh⁡κ~1h14A4.
Grouping terms in the last expression forC2, one gets (75)κsκ~1cosh⁡κ~1h1+sinh⁡κ~1h1=12κsκ~1eκ~1h1+e-κ~1h1+eκ~1h1-e-κ~1h1=12a+eκ~1h1+a-e-κ~1h1,κsκ~1sinh⁡κ~1h1+cosh⁡κ~1h1=12κsκ~1eκ~1h1-e-κ~1h1+eκ~1h1+e-κ~1h1=12a+eκ~1h1-a-e-κ~1h1,where a+≔κs/κ~1+1 and a-≔κs/κ~1-1.
Then(76)C2=132β2A4a+4e4κ~1h1+14A2k02ε2-k02ε1-12β2a+a-A2a+2e2κ~1h1+12A2κ~12+κ~22+38β2a+a-A2a+a-+14A2k02ε2-k02ε1-12β2a+a-A2a-2e-2κ~1h1+132β2A4a-4e-4κ~1h1.Since a+>1 and a->0 are bounded above, then it is clear that for sufficiently small β2>0 the constant C2>0 for all γ2>k02ε2. It is also clear that for any fixed β2>0 there exists γ∗ such that C2>0 for all γ>γ∗.
Using the expansion1+xα=1+Ox, where x<1 and -1<α<1, for sufficiently large γ, one gets the asymptotics(77)C2=β2A412+Oγ-2e4h1γ.
Since both integrands in (69) are positive, the estimate(78)n2T⩽Φ2γ;n2,p⩽n2+1Ttakes place, where n2⩾0 and T=∫-∞+∞(ds/w2). Thus we need to evaluate T. For further analysis, we use the easy checked inequalities 1/a+b⩽1/a2+b2⩽2/(a+b), where a⩾0 and b>0. These inequalities give(79)T∗⩽T⩽2T∗,where T∗=∫-∞+∞ds/s2+κ22+2β2C2. Integral T∗ is calculated explicitly.
Forγ2<k02ε2, the value η2+κ22>0 and, therefore, one has (80)T∗=2∫0+∞dss2+κ22+2β2C2=2∫0+∞dss2+κ22+2β2C2=πκ22+2β2C2.
Letγ2⩾k02ε2; then(81)T∗=2∫0+∞dss2+κ22+2β2C2=2∫0κ2ds-s2-κ22+2β2C2+2∫κ2+∞dss2+κ22+2β2C2=2∫0κ2dsκ22+2β2C2-s2+2∫κ2+∞dss2+2β2C2-κ22=2I1+2I2.Obviously, the denominator in I1 is always positive.
CalculatingI1, one gets (82)I1=∫0κ2dsκ22+2β2C2-s2=-12κ22+2β2C2ln⁡2β2C2κ2+κ22+2β2C22.
Two cases are possible forI2. If κ22-2β2C2⩾0, that is, γ2⩾k02ε2+2β2C2, then (83)I2=∫κ2+∞dss2-κ22-2β2C2=-12κ22-2β2C2ln⁡2β2C2κ2+κ22-2β2C22.
If-κ22+2β2C2>0, that is, k02ε2⩽γ2<k02ε2+2β2C2, then (84)I2=∫κ2+∞dss2-κ22+2β2C2=1-κ22+2β2C2π2-arctan⁡κ2-κ22+2β2C2.
Combining the found results, one arrives at the formula(85)T∗=πθ,γ2<k02ε2,-1θln⁡2β2C2κ2+θ2+2θ1π2-arctan⁡κ2θ1,k02ε2⩽γ2⩽k02ε2+2β2C2,-1θln⁡2β2C2κ2+θ2-1θ2ln⁡2β2C2κ2+θ22,γ2⩾k02ε2+2β2C2,where θ=κ22+2β2C2, θ1=-κ22+2β2C2, and θ2=κ22-2β2C2.
Taking into account formula (77), it is clear that the inequality γ2⩾k02ε2+2β2C2 with fixed β2 holds only if γ belongs to a finite interval.
Taking into account the fact that for largeγ the value C2 is asymptotically equal to 1/2β2A4e4h1γ, one finds that the second formula in (85) must be used for sufficiently big γ.
Further, we need the following expansions:ln⁡1+x=x+Ox2 and arctan⁡x=x+Ox3, which are valid for x<1. Using these expansions for the second line in (85), one finds(86)T∗=-12β2C241+κ222β2C2-1/2ln⁡2β2C2-2ln⁡κ2+θ+22β2C241-κ222β2C2-1/2π2-κ2θ1+Oκ2θ12=12β2C24π+Oκ222β2C24.
Taking into account the fact that(87)2β2C2-1/4=2β2·12β2A41+Oγ-2e4h1γ-1/4=e-h1γAβ21+Oγ-2,one finally obtains(88)T∗=πe-h1γAβ21+Oγ-2for sufficiently large γ.
It follows from (88) that limγ2→∞⁡T∗=0. This means that for any h1,h2>0 there is an integer n∗⩾0 such that, for any integer n2⩾n∗, the DE (69) has at least one solution. This implies the existence of infinitely many γ^i with an accumulation point at infinity.
Passing to the limitβ2→0 in the first line of (85), one obtains either (20) or (19) depending on k02εs<γ2<k02ε1 or k02ε1<γ2<k02ε2. This implies item (2) of the theorem.
Multiplying (8) by y2 and integrating over h1,h1+h2, one gets the following for γ2>k02ε2: (89)∫h1h1+h2y2y2′′dx=-κ22∫h1h1+h2y22dx-β2∫h1h1+h2y24dx.This gives(90)y2y2′h1h1+h2-∫h1h1+h2y2′2dx=-κ22∫h1h1+h2y22dx-β2∫h1h1+h2y24dx.Taking into account the fact that (91)y2h1y2′h1=A2κ~1κssinh⁡κ~1h1+κ~1cosh⁡κ~1h1κscosh⁡κ~1h1+κ~1sinh⁡κ~1h1>0,where κ~1h1>0 and y2h1+h2=0, formula (90) results in(92)β2∫h1h1+h2y24dx=A2κ~1κssinh⁡κ~1h1+κ~1cosh⁡κ~1h1κscosh⁡κ~1h1+κ~1sinh⁡κ~1h1+∫h1h1+h2y2′2dx+γ2-k02ε2∫h1h1+h2y22dx. If γ→∞, then, obviously, the right-hand side of this equality tends to ∞ and so does the left-hand side. This implies that limγ2→∞⁡maxx∈h1,h1+h2⁡y2x;γ=∞.
Formaxx∈h1,h1+h2⁡y2x;γ using (48), one can derive an exact formula provided that y2x;γ has at least two zeros inside the segment.
In order to derive the asymptotic estimates forγ^i, let us get back to formula (88). It follows from (78) and (79) that (93)n2T∗⩽n2T⩽Φ2γ;n2,p⩽n2+1T⩽2n2+1T∗.
Sincen2T∗ bounds h2 from below and 2n2+1T∗ bounds h2 from above, then, solving the equations h2=n2πe-h1γ/Aβ2 and h2=2n2+1πe-h1γ/Aβ2, with respect to γ, one finds (70).


### 5.2. Caseβ1≠0 and β2=0

Ifβ2=0 (or α2=0), solutions to (8) are found elementarily. This allows one to explicitly compute the quantity p, which is(94)p=κ2cot⁡κ2h2.This expression (due to its analytical dependence on γ) can be used for κ22⩾0 as well as for κ22⩽0. Note that p is a real value for all γ2>k02εs. Indeed, for γ2>k02ε1, one has κ2=iκ~2, where κ~22=γ2-k02ε2 and then cos⁡ix=cosh⁡x and sin⁡ix=isinh⁡x; then (95)p=κ2cos⁡κ2h2sin⁡κ2h2=iκ~2cosh⁡κ~2h2isinh⁡κ~2h2=κ~2cosh⁡κ~2h2sinh⁡κ~2h2.It is clear that the substitution of the explicit expression forp into the second equation of system (58) leads to the identity. Thus, in this case, system (58) reduces to the only equation(96)Φ1γ;n1,p=h1,where Φ1 and p are given by (46) and (94), respectively; n1=0,1,2,….The solvability of the problemPβ1,0 is established by the following.Theorem 3.
Forβ2=0, any β1>0, and any fixed A≠0, the problem Pβ1,0 has an infinite number of PCs γ^ii=1,2,… with the following properties:(1)
Ifγ^i is the solution to Pβ1,0, then γ^i2∈k02εs,+∞ and limi→∞⁡γ^i2=∞.(2)
If the linear problemP0,0 has q solutions γ~1<γ~2<⋯<γ~q, then there exists a constant β0>0 such that, for any β1=β1′<β0, it is true that γ^i2∈k02εs,k02ε2 and limβ1′→0⁡γ^i=γ~ii=1,q¯, where γ^1,γ^2,…,γ^q are first q solutions to Pβ1′,0.(3)
Ifs>q, then limβ1′→+0⁡γ^s=+∞.(4)
Ifγ^i→∞, then maxx∈0,h1⁡y2x,γ^i→∞.(5)
For bigγ^ and arbitrary small Δ>0, the asymptotic two-sided inequality (97)1-Δγ•m⩽γ^m⩽1+Δγ•2m+1is valid, where γ•2m=ε1+f-1h/4m2 and f-1 is the inversion of ft=t-1ln⁡t.Proof.
The constantC1 in (96) is defined explicitly by (32).
Using (16) at the point x=h1, one obtains (98)y2h1=-Bsin⁡κ2h2,y2′h1=Bκ2cos⁡κ2h2.Substituting these values into (31) and using (32), one obtains the biquadratic equation (99)B4+2κ12sin2⁡κ2h2+κ22cos2⁡κ2h2β1sin4⁡κ2h2B2-2k02ε1-εs+β1A2β1sin4⁡κ2h2A2=0.
Using the same estimations as in the proof of Theorem2 and evaluating the integral contained in (96), one finds(100)n1T∗⩽n1T⩽Φ1γ;n1,p⩽n1+1T⩽2n1+1T∗,where T∗=∫-∞+∞dη/η2+κ12+2β1C1.
IntegralT∗ is computed in the same way as in the proof of Theorem 2; the result has the form(101)T∗=πθ,γ2<k02ε1,-1θln⁡2β1C1κ1+θ2+2θ1π2-arctan⁡κ1θ1,k02ε1⩽γ2⩽k02ε1+2β1C1,-1θln⁡2β1C1κ1+θ2-1θ2ln⁡2β1C1κ1+θ22,γ2⩾k02ε1+2β1C1,where θ=κ12+2β1C1, θ1=-κ12+2β1C1, and θ2=κ12-2β1C1.
In order to derive the asymptotic estimates for the eigenvaluesγ^i, let us get back to formula (101). Since C1 is a (fixed) constant [it does not depend on γ; see formula (32)], the third line in (101) corresponds to sufficiently big γ. Thus, for sufficiently big γ, the third line of (101) gives the asymptotics(102)T∗=4ln⁡γγ+Oγ-1.
It follows from (102) that limγ2→∞⁡T∗=0. This implies that for any prescribed h1,h2>0 there is an integer n∗⩾0 such that (96) has at least one solution for any integer n1⩾n∗. Thus, there exists an infinite number of positive eigenvalues γ^i with accumulation point at infinity.
Passing to the limitβ1→+0 in the first line of (101) gives either (20) or (19) depending on k02εs<γ2<k02ε1 or k02ε1<γ2<k02ε2. This implies item (2) of the theorem.
Item(3) of the theorem results from the behaviour of the second and third lines in (101) as β1→+0.
Multiplying (7) by y1 and integrating over 0,h1, one obtains(103)y2h1y2′h1-κsA2+∫0h1y1′2dx=κ12∫0h1y12dx+β1∫0h1y14dx.
It follows from (103) that ∫0h1y14dx→∞ as γ→∞. This implies item (4) of the theorem.
The asymptotic estimates in the theorem result from formula (102).


## 5.1. Caseβ1=0 and β2≠0

Ifβ1=0 (or α1=0), solutions to (7) are found elementarily. This allows one to explicitly compute the quantity p, which is(67)p=κ1κscos⁡κ1h1-κ1sin⁡κ1h1κssin⁡κ1h1+κ1cos⁡κ1h1=κ1κs-κ1tan⁡κ1h1κ1+κstan⁡κ1h1.This expression (due to its analytical dependence on γ) can be used for κ12⩾0 as well as for κ12⩽0. Note that p is a real value for all γ2>k02εs. Indeed, for γ2>k02ε1, one has κ1=iκ~1, where κ~12=γ2-k02ε1, and then cos⁡ix=cosh⁡x and sin⁡ix=isinh⁡x; then (68)p=κ~1κscosh⁡κ~1h1+κ~1sinh⁡κ~1h1κssinh⁡κ~1h1+κ~1cosh⁡κ~1h1=κ1κs+κ~1tanh⁡κ~1h1κ~1+κstanh⁡κ~1h1.It can be checked that the substitution of the explicit expression forp into the first equation of system (58) leads to the identity. Thus, in this case, system (58) reduces to the only equation(69)Φ2γ;n2,p=h2,where Φ2 and p are given by (57) and (67), respectively; n2=0,1,2,….The solvability of the problemP0,β2 is established by the following.Theorem 2.
Forβ1=0, any β2>0, and any fixed A≠0, the problem P0,β2 has an infinite number of PCs γ^ii=1,2,… with the following properties:(1)
Ifγ^i is the solution to P0,β2, then γ^i2∈k02εs,+∞ and limi→∞⁡γ^i2=+∞.(2)
If the linear problemP0,0 has q solutions γ~1<γ~2<⋯<γ~q, then there exists a constant β0>0 such that, for any β2=β2′<β0, it is true that γ^i2∈k02εs,k02ε2 and limβ1′→0⁡γ^i=γ~ii=1,q¯, where γ^1,γ^2,…,γ^q are first q solutions to P0,β2′.(3)
Ifγ^i→∞, then maxx∈h1,h1+h2⁡y2x,γ^i→∞.(4)
For bigγ^ and arbitrary small Δ>0, the asymptotic two-sided inequality(70)1-Δ1h1ln⁡mπh2Aβ2⩽γ^m⩽1+Δ1h1ln⁡2m+1πh2Aβ2is valid.Proof.
Now let us study the behaviour ofC2≡C2γ with respect to γ. Explicit solution to (7) with initial data (30) has the form y1x=A(κs/κ1)sin⁡κ1x+cos⁡κ1x. At the point x=h1, one gets(71)y1h1=Aκsκ1sin⁡κ1h1+cos⁡κ1h1,y1′h1=Aκ1κsκ1cos⁡κ1h1-sin⁡κ1h1.Substituting (71) into (48), one obtains(72)C2=κ12κsκ1cos⁡κ1h1-sin⁡κ1h12A2+κ22κsκ1sin⁡κ1h1+cos⁡κ1h12A2+12β2κsκ1sin⁡κ1h1+cos⁡κ1h14A4.
Ifk02εs<γ2⩽k02ε1, then κ12⩾0 and κ22⩾0 and, therefore, C2>0.
Ifk02ε1<γ2⩽k02ε2, then κ12<0 and κ22⩾0. As before, we denote κ1=iκ~1, where κ~12=γ2-k02ε1>0. Then(73)C2=κ~12κsκ~1cosh⁡κ~1h1+sinh⁡κ~1h12A2+κ22κsκ~1sinh⁡κ~1h1+cosh⁡κ~1h12A2+12β2κsκ~1sinh⁡κ1h1+cosh⁡κ~1h14A4>0.
Ifγ2>k02ε2, then κ12<0 and κ22<0. As before, we denote κ1=iκ~1 and κ2=iκ~2, where κ~12=γ2-k02ε1>0 and κ~22=γ2-k02ε2>0. Then(74)C2=κ~12κsκ~1cosh⁡κ~1h1+sinh⁡κ~1h12A2-κ~22κsκ~1sinh⁡κ~1h1+cosh⁡κ~1h12A2+12β2κsκ~1sinh⁡κ1h1+cosh⁡κ~1h14A4.
Grouping terms in the last expression forC2, one gets (75)κsκ~1cosh⁡κ~1h1+sinh⁡κ~1h1=12κsκ~1eκ~1h1+e-κ~1h1+eκ~1h1-e-κ~1h1=12a+eκ~1h1+a-e-κ~1h1,κsκ~1sinh⁡κ~1h1+cosh⁡κ~1h1=12κsκ~1eκ~1h1-e-κ~1h1+eκ~1h1+e-κ~1h1=12a+eκ~1h1-a-e-κ~1h1,where a+≔κs/κ~1+1 and a-≔κs/κ~1-1.
Then(76)C2=132β2A4a+4e4κ~1h1+14A2k02ε2-k02ε1-12β2a+a-A2a+2e2κ~1h1+12A2κ~12+κ~22+38β2a+a-A2a+a-+14A2k02ε2-k02ε1-12β2a+a-A2a-2e-2κ~1h1+132β2A4a-4e-4κ~1h1.Since a+>1 and a->0 are bounded above, then it is clear that for sufficiently small β2>0 the constant C2>0 for all γ2>k02ε2. It is also clear that for any fixed β2>0 there exists γ∗ such that C2>0 for all γ>γ∗.
Using the expansion1+xα=1+Ox, where x<1 and -1<α<1, for sufficiently large γ, one gets the asymptotics(77)C2=β2A412+Oγ-2e4h1γ.
Since both integrands in (69) are positive, the estimate(78)n2T⩽Φ2γ;n2,p⩽n2+1Ttakes place, where n2⩾0 and T=∫-∞+∞(ds/w2). Thus we need to evaluate T. For further analysis, we use the easy checked inequalities 1/a+b⩽1/a2+b2⩽2/(a+b), where a⩾0 and b>0. These inequalities give(79)T∗⩽T⩽2T∗,where T∗=∫-∞+∞ds/s2+κ22+2β2C2. Integral T∗ is calculated explicitly.
Forγ2<k02ε2, the value η2+κ22>0 and, therefore, one has (80)T∗=2∫0+∞dss2+κ22+2β2C2=2∫0+∞dss2+κ22+2β2C2=πκ22+2β2C2.
Letγ2⩾k02ε2; then(81)T∗=2∫0+∞dss2+κ22+2β2C2=2∫0κ2ds-s2-κ22+2β2C2+2∫κ2+∞dss2+κ22+2β2C2=2∫0κ2dsκ22+2β2C2-s2+2∫κ2+∞dss2+2β2C2-κ22=2I1+2I2.Obviously, the denominator in I1 is always positive.
CalculatingI1, one gets (82)I1=∫0κ2dsκ22+2β2C2-s2=-12κ22+2β2C2ln⁡2β2C2κ2+κ22+2β2C22.
Two cases are possible forI2. If κ22-2β2C2⩾0, that is, γ2⩾k02ε2+2β2C2, then (83)I2=∫κ2+∞dss2-κ22-2β2C2=-12κ22-2β2C2ln⁡2β2C2κ2+κ22-2β2C22.
If-κ22+2β2C2>0, that is, k02ε2⩽γ2<k02ε2+2β2C2, then (84)I2=∫κ2+∞dss2-κ22+2β2C2=1-κ22+2β2C2π2-arctan⁡κ2-κ22+2β2C2.
Combining the found results, one arrives at the formula(85)T∗=πθ,γ2<k02ε2,-1θln⁡2β2C2κ2+θ2+2θ1π2-arctan⁡κ2θ1,k02ε2⩽γ2⩽k02ε2+2β2C2,-1θln⁡2β2C2κ2+θ2-1θ2ln⁡2β2C2κ2+θ22,γ2⩾k02ε2+2β2C2,where θ=κ22+2β2C2, θ1=-κ22+2β2C2, and θ2=κ22-2β2C2.
Taking into account formula (77), it is clear that the inequality γ2⩾k02ε2+2β2C2 with fixed β2 holds only if γ belongs to a finite interval.
Taking into account the fact that for largeγ the value C2 is asymptotically equal to 1/2β2A4e4h1γ, one finds that the second formula in (85) must be used for sufficiently big γ.
Further, we need the following expansions:ln⁡1+x=x+Ox2 and arctan⁡x=x+Ox3, which are valid for x<1. Using these expansions for the second line in (85), one finds(86)T∗=-12β2C241+κ222β2C2-1/2ln⁡2β2C2-2ln⁡κ2+θ+22β2C241-κ222β2C2-1/2π2-κ2θ1+Oκ2θ12=12β2C24π+Oκ222β2C24.
Taking into account the fact that(87)2β2C2-1/4=2β2·12β2A41+Oγ-2e4h1γ-1/4=e-h1γAβ21+Oγ-2,one finally obtains(88)T∗=πe-h1γAβ21+Oγ-2for sufficiently large γ.
It follows from (88) that limγ2→∞⁡T∗=0. This means that for any h1,h2>0 there is an integer n∗⩾0 such that, for any integer n2⩾n∗, the DE (69) has at least one solution. This implies the existence of infinitely many γ^i with an accumulation point at infinity.
Passing to the limitβ2→0 in the first line of (85), one obtains either (20) or (19) depending on k02εs<γ2<k02ε1 or k02ε1<γ2<k02ε2. This implies item (2) of the theorem.
Multiplying (8) by y2 and integrating over h1,h1+h2, one gets the following for γ2>k02ε2: (89)∫h1h1+h2y2y2′′dx=-κ22∫h1h1+h2y22dx-β2∫h1h1+h2y24dx.This gives(90)y2y2′h1h1+h2-∫h1h1+h2y2′2dx=-κ22∫h1h1+h2y22dx-β2∫h1h1+h2y24dx.Taking into account the fact that (91)y2h1y2′h1=A2κ~1κssinh⁡κ~1h1+κ~1cosh⁡κ~1h1κscosh⁡κ~1h1+κ~1sinh⁡κ~1h1>0,where κ~1h1>0 and y2h1+h2=0, formula (90) results in(92)β2∫h1h1+h2y24dx=A2κ~1κssinh⁡κ~1h1+κ~1cosh⁡κ~1h1κscosh⁡κ~1h1+κ~1sinh⁡κ~1h1+∫h1h1+h2y2′2dx+γ2-k02ε2∫h1h1+h2y22dx. If γ→∞, then, obviously, the right-hand side of this equality tends to ∞ and so does the left-hand side. This implies that limγ2→∞⁡maxx∈h1,h1+h2⁡y2x;γ=∞.
Formaxx∈h1,h1+h2⁡y2x;γ using (48), one can derive an exact formula provided that y2x;γ has at least two zeros inside the segment.
In order to derive the asymptotic estimates forγ^i, let us get back to formula (88). It follows from (78) and (79) that (93)n2T∗⩽n2T⩽Φ2γ;n2,p⩽n2+1T⩽2n2+1T∗.
Sincen2T∗ bounds h2 from below and 2n2+1T∗ bounds h2 from above, then, solving the equations h2=n2πe-h1γ/Aβ2 and h2=2n2+1πe-h1γ/Aβ2, with respect to γ, one finds (70).


## 5.2. Caseβ1≠0 and β2=0

Ifβ2=0 (or α2=0), solutions to (8) are found elementarily. This allows one to explicitly compute the quantity p, which is(94)p=κ2cot⁡κ2h2.This expression (due to its analytical dependence on γ) can be used for κ22⩾0 as well as for κ22⩽0. Note that p is a real value for all γ2>k02εs. Indeed, for γ2>k02ε1, one has κ2=iκ~2, where κ~22=γ2-k02ε2 and then cos⁡ix=cosh⁡x and sin⁡ix=isinh⁡x; then (95)p=κ2cos⁡κ2h2sin⁡κ2h2=iκ~2cosh⁡κ~2h2isinh⁡κ~2h2=κ~2cosh⁡κ~2h2sinh⁡κ~2h2.It is clear that the substitution of the explicit expression forp into the second equation of system (58) leads to the identity. Thus, in this case, system (58) reduces to the only equation(96)Φ1γ;n1,p=h1,where Φ1 and p are given by (46) and (94), respectively; n1=0,1,2,….The solvability of the problemPβ1,0 is established by the following.Theorem 3.
Forβ2=0, any β1>0, and any fixed A≠0, the problem Pβ1,0 has an infinite number of PCs γ^ii=1,2,… with the following properties:(1)
Ifγ^i is the solution to Pβ1,0, then γ^i2∈k02εs,+∞ and limi→∞⁡γ^i2=∞.(2)
If the linear problemP0,0 has q solutions γ~1<γ~2<⋯<γ~q, then there exists a constant β0>0 such that, for any β1=β1′<β0, it is true that γ^i2∈k02εs,k02ε2 and limβ1′→0⁡γ^i=γ~ii=1,q¯, where γ^1,γ^2,…,γ^q are first q solutions to Pβ1′,0.(3)
Ifs>q, then limβ1′→+0⁡γ^s=+∞.(4)
Ifγ^i→∞, then maxx∈0,h1⁡y2x,γ^i→∞.(5)
For bigγ^ and arbitrary small Δ>0, the asymptotic two-sided inequality (97)1-Δγ•m⩽γ^m⩽1+Δγ•2m+1is valid, where γ•2m=ε1+f-1h/4m2 and f-1 is the inversion of ft=t-1ln⁡t.Proof.
The constantC1 in (96) is defined explicitly by (32).
Using (16) at the point x=h1, one obtains (98)y2h1=-Bsin⁡κ2h2,y2′h1=Bκ2cos⁡κ2h2.Substituting these values into (31) and using (32), one obtains the biquadratic equation (99)B4+2κ12sin2⁡κ2h2+κ22cos2⁡κ2h2β1sin4⁡κ2h2B2-2k02ε1-εs+β1A2β1sin4⁡κ2h2A2=0.
Using the same estimations as in the proof of Theorem2 and evaluating the integral contained in (96), one finds(100)n1T∗⩽n1T⩽Φ1γ;n1,p⩽n1+1T⩽2n1+1T∗,where T∗=∫-∞+∞dη/η2+κ12+2β1C1.
IntegralT∗ is computed in the same way as in the proof of Theorem 2; the result has the form(101)T∗=πθ,γ2<k02ε1,-1θln⁡2β1C1κ1+θ2+2θ1π2-arctan⁡κ1θ1,k02ε1⩽γ2⩽k02ε1+2β1C1,-1θln⁡2β1C1κ1+θ2-1θ2ln⁡2β1C1κ1+θ22,γ2⩾k02ε1+2β1C1,where θ=κ12+2β1C1, θ1=-κ12+2β1C1, and θ2=κ12-2β1C1.
In order to derive the asymptotic estimates for the eigenvaluesγ^i, let us get back to formula (101). Since C1 is a (fixed) constant [it does not depend on γ; see formula (32)], the third line in (101) corresponds to sufficiently big γ. Thus, for sufficiently big γ, the third line of (101) gives the asymptotics(102)T∗=4ln⁡γγ+Oγ-1.
It follows from (102) that limγ2→∞⁡T∗=0. This implies that for any prescribed h1,h2>0 there is an integer n∗⩾0 such that (96) has at least one solution for any integer n1⩾n∗. Thus, there exists an infinite number of positive eigenvalues γ^i with accumulation point at infinity.
Passing to the limitβ1→+0 in the first line of (101) gives either (20) or (19) depending on k02εs<γ2<k02ε1 or k02ε1<γ2<k02ε2. This implies item (2) of the theorem.
Item(3) of the theorem results from the behaviour of the second and third lines in (101) as β1→+0.
Multiplying (7) by y1 and integrating over 0,h1, one obtains(103)y2h1y2′h1-κsA2+∫0h1y1′2dx=κ12∫0h1y12dx+β1∫0h1y14dx.
It follows from (103) that ∫0h1y14dx→∞ as γ→∞. This implies item (4) of the theorem.
The asymptotic estimates in the theorem result from formula (102).


## 6. Numerical Results

Figures2–13 show results of numerical experiments. The following parameters are used: A=1 [E] and εs=1; ε1=2 and ε2=2.5 (Figures 2–9); ε1=5 and ε2=10 (Figures 10 and 11); ε1=5 and ε2=5.01 (Figures 12 and 13); other parameters are specified in the captions; ω=2πf/c, where c is the speed of light.Figure 2
DCs for the linear case:h1=1 and h2=4 [mm]; vertical dashed line f=45 [GHz]; γ~1≈1.0170 (green dot) and γ~2≈1.3709 (brown dot).Figure 3
EigenmodesEyx for the PCs marked in Figure 2: γ~1≈1.0170 (green) and γ~2≈1.3709 (brown).Figure 4
(a) DCs for nonlinear (blue) and linear (red) cases:α1=0.02 [E-2], α2=0, h1=5, h2=4 [mm], and Ey0=1 [E]; vertical dashed line f=36.1 [GHz]; nonlinear PCs: γ^1≈1.1509 (blue dot), γ^2≈1.2701 (green dot), and γ^3≈4.5003 (brown dot), respectively; linear PC: γ~2≈1.0860 (red dot). (b) The dependence γ versus A for nonlinear (blue) and linear (red) cases: α1=0.02 [E-2], α2=0, h1=5, h2=4 [mm], and f=36.1 [GHz]; vertical dashed line Ey0=1 [E]; nonlinear PCs: γ^1≈1.1509 (blue dot), γ^2≈1.2701 (green dot), and γ^3≈4.5003 (brown dot), respectively; linear PC: γ~2≈1.0860 (red dot).
(a)
(b)Figure 5
EigenmodesEyx for the PCs marked in Figure 4: γ~2≈1.0860 (red, linear case), γ^1≈1.1509 (blue), γ^2≈1.2701 (green), and γ^3≈4.5003 (brown).Figure 6
(a) DCs for nonlinear (blue) and linear (red) cases:α1=0, α2=0.03 [E-2], h1=1 [mm], h2=4 [mm], and Ey0=1 [E]; vertical dashed line f=45 [GHz]; nonlinear PCs: γ^1≈1.4625 (blue dot), γ^2≈1.7012 (green dot), and γ^3≈2.6038 (brown dot), respectively. The linear case coincides with one presented in Figure 2. (b) The dependence γ versus A for nonlinear (blue) and linear (red) cases: α1=0, α2=0.03 [E-2], h1=1 [mm], h2=4 [mm], and f=45 [GHz]; vertical dashed line Ey0=1 [E]; nonlinear PCs: γ^1≈1.4625 (blue dot), γ^2≈1.7012 (green dot), and γ^3≈2.6038 (brown dot), respectively; linear PC: γ~2≈1.3709 (red dot).
(a)
(b)Figure 7
EigenmodesEyx: γ~2≈1.3709 (red, linear case), γ^1≈1.4625 (blue), γ^2≈1.7012 (green), and γ^3≈2.6038 (brown).Figure 8
(a) DCs for nonlinear (blue) and linear (red) cases:α1=0.02, α2=0.03 [E-2], h1=1, h2=4 [mm], and Ey0=1 [E]; black dashed line f=45 [GHz]; nonlinear PCs: γ^1≈1.4548 (blue dot), γ^2≈1.7476 (green dot), and γ^3≈2.7896 (brown dot), respectively. The linear case coincides with one presented in Figure 2. (b) The dependence γ versus A for nonlinear (blue) and linear (red) cases: α1=0.02, α2=0.03 [E-2], h1=1, h2=4 [mm], and f=45 [GHz]; black dashed line Ey0=1 [E]; nonlinear PCs: γ^1≈1.4548 (blue dot), γ^2≈1.7476 (green dot), and γ^3≈2.7896 (brown dot), respectively; linear PC: γ~2≈1.3709 (red dot).
(a)
(b)Figure 9
EigenmodesEyx for the PCs marked in Figure 8: γ~2≈1.3709 (red, linear case), γ^1≈1.4548 (blue), γ^2≈1.7476 (green), and γ^3≈2.7896 (brown).Figure 10
(a) DCs for nonlinear (blue) and linear (red) cases:α1=0.02, α2=0.01 [E-2], h1=10, h2=5 [mm], and Ey0=1 [E]; black dashed line f=30.2 [GHz]; nonlinear PCs: γ^1≈0.8145 (blue dot) and γ^2≈1.0723 (green dot), respectively. (b) The dependence γ versus A for nonlinear (blue) and linear (red) cases: α1=0.02, α2=0.01 [E-2], h1=10, h2=5 [mm], and f=30.2 [GHz]; black dashed line Ey0=1 [E]; nonlinear PCs: γ^1≈0.8145 (blue dot) and γ^2≈1.0723 (green dot), respectively.
(a)
(b)Figure 11
EigenmodesEyx for the PCs marked in Figure 10: γ^1≈0.8145 (blue) and γ^2≈1.0723 (green).Figure 12
(a) DCs for nonlinear (blue) and linear (red) cases:α1=0.02, α2=0.01 [E-2], h1=10, h2=5 [mm], and Ey0=1 [E]; black dashed line f=33 [GHz]; nonlinear PCs: γ^1≈0.7189 (blue dot) and γ^2≈0.9937 (green dot), respectively. (b) The dependence γ versus A for nonlinear (blue) and linear (red) cases: α1=0.02, α2=0.01 [E-2], h1=10, h2=5 [mm], and Ey0=1 [E]; black dashed line f=33 [GHz]; nonlinear PCs: γ^1≈0.7189 (blue dot) and γ^2≈0.9937 (green dot), respectively.
(a)
(b)Figure 13
EigenmodesEyx for the PCs marked in Figure 12: γ^1≈0.7189 (blue) and γ^2≈0.9937 (green).In Figures2, 4, 6, 8, 10, and 12(a), the dispersion curves (DCs) are plotted (γ versus f). The DCs for the linear and nonlinear cases are given in red and blue colours, respectively. Dashed lines in Figure 2 correspond to the boundaries for γ. Points of intersections of the vertical dashed line with DCs are PCs.Dashed lines in Figures3, 5, 7, and 9 correspond to the boundary between the layers.Figures2 and 3 correspond to the linear problem (α1=α2=0). For chosen frequency, there are 2 PCs; eigenmodes for them are shown in Figure 3.The dependenceγ versus A, where A=Ey0, is shown in Figures 2, 4, 6, 8, 10, and 12(b). The linear and nonlinear cases are given in red and blue colours, respectively. Points of intersections of the vertical dashed line (corresponding to A=1) with the curves are PCs. Obviously, in the linear case, the PCs do not depend on A and therefore linear solutions are horizontal (red) lines.There are two types of PCs for the nonlinear case. The first type corresponds to the PCs, which are close to the corresponding linear solutions (in the linear limit, they reduce to the linear solutions; see item(2) in Theorems 2 and 3). The second type corresponds to the PCs, which present a novel guiding regime (they do not reduce to any linear solutions in the linear limit; see item (3) in Theorem 3). In the latter case, we call them “purely nonlinear” PCs; see also [23–25]. In Figures 4, 6, and 8, the PC γ^1 marked with a blue dot corresponds to the first type; the PCs γ^2 and γ^3 marked with green and brown dots, respectively, correspond to the second type.In Figures6 and 8, for any frequency, there are infinitely many PCs in the nonlinear cases (few of them are shown and marked in the figures) and only 2 PCs in the linear case.DCs for the linear cases (red DCs) presented in Figures6 and 8 coincide with the DCs for the linear case presented in Figure 2.Figures5, 7, and 9 allow one to compare linear and nonlinear modes. It can be seen from these pictures that, for a “linear” PC and for a “nonlinear” one (which reduces to the “linear” one in the linear limit), eigenmodes are also close. Obviously, for such a nonlinear PC, perturbation methods can be applied. However, two other nonlinear eigenmodes (shown in green and brown colours) are not close to any linear solution and they do not reduce to linear solutions in the linear limit; these very eigenmodes we call purely nonlinear.In Figures8–13, the nonlinear case for different sets of α1,α2 is shown, where both coefficients α1 and α2 are nonzero. In this case, we did not prove the existence of infinitely many PCs; however, these figures are similar to corresponding figures, where α1 or α2 equals zero.


## 7. Conclusion

The paper focuses on the problem of wave propagation in a plane layered dielectric waveguide filled with Kerr medium. The existence of guided modes that have linear counterparts and guided modes that do not have linear counterparts is proven. The latter waves correspond to a novel guided regime. Since the Kerr nonlinearity is widely studied in nonlinear optics (see, e.g., [7–14]), the results found here can be interesting and important from both theoretical and applied points of view.It is worth noting that similar results have been found for some other cases. Indeed, it was proven lately that, even in a simpler case of a one-layer waveguide, the Kerr nonlinearity results in the existence of novel guided regimes as well [23–25]. Moreover, similar results have recently been found in the case of polynomial nonlinearity [39]. Thus, the existence of infinitely many nonperturbative PCs is a general feature of polynomial (nonlinear) permittivities with positive terms.One of the most known peculiarities of nonlinear guided waves is power-dependent PCs; this is clearly shown in Figures2, 4, 6, 8, 10, and 12(b). It is interesting that varying the value A, one strongly affects the corresponding PC and therefore the eigenmode. The fact that the power-dependent PCs can have some potential for optical signal processing is pointed out in many papers (see, e.g., [28–34]). For additional description of applications of nonlinear guided waves and Kerr effect, see [32, 34], where many useful references are also given.The result given in the paper clearly shows that nonlinear problems can have solutions that cannot be considered as perturbations of solutions of corresponding linear problems. Thus, it is necessary to be careful when one linearizes a nonlinear problem and considers the linearized problem without proving that there are no other solutions.Of course, it is impossible to expect the existence of purely nonlinear waves for infinitely many PCs. However, it is possible that purely nonlinear waves can be observed in an experiment for some first purely nonlinear PCs. Indeed, as is seen from Figures5, 7, and 9, max values of few first purely nonlinear eigenmodes are not too big (in comparison with the linear eigenmode). This probably gives an opportunity to observe such waves in an experiment. For the rest of the PCs, the value maxx∈0,h1+h2⁡Eyx;γ^i is so high that the Kerr law is no longer valid.As a matter of fact, different types of nonlinearities admit nonlinear solutions that become linear ones in the linear limit. For example, for a wide range of saturated nonlinearities, there exist only a finite number of PCs [40, 41]. Thus, there is a qualitative difference between saturated and unbounded (Kerr, qubic-quintic-septic, and, more generally, power and polynomial) nonlinearities. It seems that this difference can be used in order to understand what kinds of nonlinear permittivities are closer to real situations.If the purely nonlinear guided modes are confirmed by experiment, the theory of nonlinear guided wave propagation will definitely advance. If they are not observed in experiments, then well-known and widespread formulas for nonlinear permittivities must be changed so that mathematical analysis of these models can give results that better satisfy reality.In spite of the fact that in some papers the nonlinear eigenmodes are searched in an explicit form (see, e.g., [7, 16, 17, 20–22, 34]), this way is not an appropriate one. Indeed, on one hand, explicit solutions to nonlinear equations are often complicated special functions (if it is possible to find them at all) and, therefore, it is almost impossible to study such solutions. On the other hand, many properties/characteristics of eigenwaves can be calculated from original differential equations and boundary conditions, like it is done in this paper. In order to check calculations, one solves numerically the DE (58) with respect to γ in a prescribed segment; then, for each found PC, one solves the Cauchy problem for (7) and (8) with initial data at x=0 and transmission conditions (11) at x=h1. In this case, the second condition (10) is fulfilled indispensably (obviously, in numerical calculations, y2h1+h2 cannot be exactly zero). Thus, Theorems 1, 2, and 3 together with computations allow one to find all nonlinear eigenwaves for which the PCs are calculated.



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