Nonlinear coupled wave propagation in a n-dimensional layer

Applied Mathematics and Computation 294 (2017) 146–156

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Nonlinear coupled wave propagation in a n-dimensional layer Yury G. Smirnov, Dmitry V. Valovik∗ Department of Mathematics and Supercomputing, Penza State University, Krasnaya Str. 40, Penza 440026, Russia

a r t i c l e

i n f o

MSC: 47J10 35P15 78A60 Keywords: Nonlinear multi-parameter eigenvalue problem Linear multi-parameter eigenvalue problem Eigentuples Maxwell’s equations Nonlinear guided wave Dispersion equation

a b s t r a c t The paper focuses on the generalization of a nonlinear multi-parameter eigenvalue problem for a system of nonlinear differential equations. The problem is reduced to a system of nonlinear integral equations on a segment. The notion of eigentuple is introduced, the existence of a finite number of isolated eigentuples is proved, and their distribution is described. The corresponding linear multi-parameter eigenvalue problem is studied as well; it is proved that the linear problem has an infinite number of isolated eigentuples. Applications to nonlinear electromagnetic wave propagation theory are demonstrated. © 2016 Elsevier Inc. All rights reserved.

1. Introduction In this paper we investigate a nonlinear multi-parameter eigenvalue problem for a system of nonlinear ordinary differential equations. Such a problem arises, for instance, in mathematical models of coupled electromagnetic wave propagation in nonlinear media at different frequencies [1–4]. This problem is a generalization of one- and two-parameter eigenvalue problems arising in the theory of nonlinear dielectric waveguides [1–8] to the case of an n-dimensional layer filled with a nonlinear medium [9]. From the electromagnetic standpoint, the eigenvalue problem is formulated in unbounded domains, in particular on the real axis, and with transmission type conditions as well as conditions at infinity that contain the spectral parameters. However, then it can be reduced to a problem on a segment with boundary conditions of the third type [3,4] or more complicated ones [1,2]. Having different formulations, nonlinear multi-parameter eigenvalue problems have attracted considerable interest for many years (see [10–13] and the bibliography therein). In the most cases these problems are of pure mathematical nature without the outside applications. The problem under consideration gives an example of an applied nonlinear multiparameter eigenvalue problem [3,4]. It is worth mentioning that the theory of linear multi-parameter eigenvalue problems is quite well developed [14]. However, to the knowledge of the authors there is no comparable understanding of the nonlinear situation. In this paper we use the method of small parameter; this approach is justified by physical motivation. The method of solution employs the transition to nonlinear integral equations using the Green functions of the linear differential operators. After this, the eigenvalue problems are replaced by the determination of characteristic numbers of the integral operatorvalued functions which are nonlinear both with respect to the solution and to the spectral parameters. The latter problems

∗

Corresponding author. E-mail addresses: [email protected] (Y.G. Smirnov), [email protected] (D.V. Valovik).

http://dx.doi.org/10.1016/j.amc.2016.09.011 0 096-30 03/© 2016 Elsevier Inc. All rights reserved.

Y.G. Smirnov, D.V. Valovik / Applied Mathematics and Computation 294 (2017) 146–156

147

are reduced to functional dispersion equations, and their roots give the desired eigentuples. The existence and distribution of the roots are verified. The objective of the present work is to develop an appropriate technique to study specific nonlinear multi-parameter eigenvalue problems arising in the applied science and to prove the existence of eigentuples (coupled eigenvalues) for a generalization of the inhomogeneous nonlinear waveguide problem, to obtain the eigentuples – also as functions of the problem parameters (first of all, of the nonlinearity parameter). The outline of the paper is as follows: in Section 2 the problem is formulated, in Section 3 the eigenvalue problem is reduced to a system of integral equations and system the dispersion equations is derived, in Section 4 the system of integral equations is studied, in Section 5 the existence of eigentuples is proved both for the nonlinear problem and for the corresponding linear one, in Section 6 the found results are discussed and additional comments are given. 2. Statement of the problem We consider the system of differential equations









pi (x )ui (x ) + qi (x ) − λi ri (x ) + ai fi (|u|2 ) ui (x ) = 0,



i = 1, n,

(1)



 where x ∈ [−h, h] and u(x ) := u1 (x ), . . . , un (x ) , |u|2 := ni=1 u2i . Here (·) is the transposition operation. In this system the functions pi (x), qi (x), ri (x) are known and such that

pi (x ), qi (x ), ri (x ) ∈ C[−h, h], the constants ai > 0 are also known, whereas the vector λ := (λ1 , . . . , λn ) ∈ Rn and the vector-field u : [−h, h] → Rn are unknown. We assume that for every i

pi ( s ) > 0,

ri ( s ) > 0

for

s ∈ [−h, h],

f i (s ) > 0

for

s > 0,

and

f i ( 0 ) = 0.

In addition, we assume that each fi satisfies the Lipschitz condition and is bounded in any bounded domain. To be more precise, let



Br := u ∈ C[−h, h] : uC  r



be a ball of radius r > 0 with its origin at zero, where

C[−h, h] = C[−h, h] × . . . × C[−h, h]







n times

is the space of continuous vector-functions with the norm u2C = i, the following inequality

n

i=1

| f i ( x1 ) − f i ( x2 )|  Lr |x1 − x2 |

ui C2 ; here ui C = maxx∈[−h,h] |ui (x )|. Then for each (2)

is fulfilled, where x1 , x2 ∈ [0, r]; Lr is a Lipschitz constant that, in general, depends on r. Let v = (v1 , . . . , vn ) be a real vector and V = [vi j ], where i, j = 1, n, be a real matrix. We shall use the vector norm  v22 = ni=1 v2i and the consistent matrix norm V22 = max1in μV(i) V , where μD(i) denotes an eigenvalue of the matrix D. For the simplicity, below we shall omit the subscript 2 in the vector and matrix norms. We denote Ar := maxu∈Br f(|u|2 )2 , where f = diag { f1 , . . . fn } is n × n diagonal matrix. The boundary conditions for system (1) are



ui (−h ) − αi ui (−h ) = 0, ui (h ) + βi ui (h ) = 0,

(3)

for i = 1, n with the coefficients

αi = αi (λ ), βi = βi (λ ), which are real-valued, continuous and satisfy the conditions

αi (λ )  0, βi (λ )  0, αi (λ ) + βi (λ ) > 0 for all λi ∈ i with certain sets i specified below. In addition we need to consider the scaling conditions

u ( h ) = c+ ,

(4)

where c+ := (c1+ , . . . , cn+ ) ∈ Rn is prescribed and such that ci+ = 0 for all i (without loss of generality ci+ > 0 for all i). We also denote c− := u(−h ), where c− := (c1− , . . . , cn− ) ∈ Rn , is unknown.

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Definition 1. Vector λ = λ is called a (scaled) eigentuple of the problem (1)–(4) if there exist nontrivial functions ui ∈ C 1 [−h, h] ∩ C 2 (−h, h ), i = 1, n, satisfying system (1), boundary conditions (3), and scaling conditions (4). The components λi of λ are also called coupled eigenvalues and the functions ui are called the eigenfunctions of the problem (1)–(4). Remark 1. In contrast to the linear case (i.e. all ai = 0 in (1)), where the eigenfunctions are unique up to multiplicative constants, here we have to fix the numbers ci+ . Both the eigenvalues and the eigenfunctions depend on these numbers. It can be shown that the property ci+ = 0 for some index i implies that ui = 0, since the boundary condition (3) at x = h leads to a Cauchy problem with vanishing initial conditions. Therefore we have to assume in (4) that ci+ = 0 for i = 1, n. The main task of this work is to prove the existence of eigentuples satisfying Definition 1. Statement 1. If the problem (1)–(4) has a solution, then λ ∈ Rn and u : [−h, h] → Rn . Proof. Multiplying the ith equation of (1) by ui (x ) and then integrating, we get

− − −

βi (λ ) pi (h )|ui (h )|2 − αi (λ ) pi (−h )|ui (−h )|2

h −h

λi

pi (x )|ui (x )|2 dx +

h

−h

h

−h

ri (x )|ui (x )|2 dx + ai

qi (x )|ui (x )|2 dx

h

−h

fi (|u|2 )|ui (x )|2 dx = 0.

(5)

Separating the imaginary part of (5) yields

−(Im λi )

h −h

ri (x )|ui (x )|2 dx = 0.

Hence Im λi = 0 if ui is a nontrivial function. Then it follows from the ith equation of (1) and (3), (4) that ui is real.



This statement explains why it is sufficient to consider only the case of Rn -valued λ and u. 3. Nonlinear integral equations Since we carry out our considerations for each index i, then for the brevity we will omit the phrase ‘for i = 1, n’ in every formula. Let us introduce n linear differential operators



Li :=

d d pi ( x ) dx dx



+ qi (x ) − λi ri (x ).

The operators Li correspond to the linear parts of equations (1). For these operators we consider the following boundary value problems

Li Gi = −δ (x − s ),

∂x Gi (x, s )|x=±h = 0.

(6)

It is known [15] that if λi is not an eigenvalue of Li , then the Green functions Gi exists for such λi . It is also known that the problem

Li ϕi = 0, with boundary conditions ϕi (±h ) = 0 has an infinite number of eigenvalues and eigenfunctions. Let {λi, j , ϕi, j }∞ be a comj=1





plete set of all eigenvalues and normalized [in the space L2 −h, h; ri (x ) with weight ri (x)] eigenfunctions for the aboveconsidered boundary value problem, see [15,16]. In [15] was shown that all the eigenvalues are real and simple (of multiplicity 1). Moreover, for each i there exists only a finite number (or does not exist at all) of positive eigenvalues and an infinite number of negative ones, such that λi, j → −∞ as j → ∞. In addition, the following asymptotics takes place λi, j = O∗ ( j2 ) as j → ∞. Arrange the eigenvalues in the descending order: . . . < λi, j+1 < λi, j < . . . < λi,2 < λi,1 . In [15] was also shown that the system of all eigenfunctions {ϕi, j }∞ forms an orthonormalized basis in L2 −h, h; ri (x ) . j=1 In the vicinity of an eigenvalue λi = λi, j the Green function Gi can be written in the form [15]

Gi (x, s; λi ) =

φi, j (x )φi, j (s )  + Gi (x, s; λi ), λi − λi, j

(7)

 (x, s ) is regular in the vicinity of λ . In addition, the Green function G is expanded into a series where G i i, j i

Gi (x, s; λi ) =

∞  j=1

φi, j (x )φi, j (s ) , λi − λi, j

(8)

with continuous terms. This series converges absolutely and uniformly for any (x, s, λ ) ∈ [−h, h] × [−h, h] × , where  is a segment that does not contain poles of the Green function [15].

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For the convenience introduce the notation

αˆ i (λ ) := pi (−h )αi (λ ), βˆi (λ ) := pi (h )βi (λ ). Using the second Green formula, we find

h −h

(Gi Li ui − ui Li Gi )dx = pi (x )ui (x )Gi (x, s )|xx==h−h .

(9)

Using boundary conditions (3) and scaling conditions (4), Eq. (9) transforms

h −h

(Gi Li ui − ui Li Gi )dx = −βˆi ci+ Gi (h, s ) − αˆ i ci− Gi (−h, s ).

(10)

On the other hand, using (1), (6), we obtain

h −h

(Gi Li ui − ui Li Gi )dx = −ai

h −h

Gi (x, s ) fi (|u|2 )ui (x )dx + ui (s ).

(11)

Now using (10) and (11), we get the integral representation of solutions ui (s) of system (1) for s ∈ [−h, h]

ui ( s ) = ai

h −h

Gi (x, s ) fi (|u|2 )ui (x )dx − βˆi ci+ Gi (h, s ) − αˆ i ci− Gi (−h, s ).

(12)

For further study it is necessary to eliminate unknowns ci− from system (12). It is convenient to denote

Gi (±h, ±h ) := lim Gi (±h, s ). s→±h

Substituting consequently s = −h s = h into (12), we get

ci− = ai ci+ = ai

h

−h

h −h

Gi (x, −h ) fi (|u|2 )ui (x )dx − βˆi ci+ Gi (h, −h ) − αˆ i ci− Gi (−h, −h ),

(13)

Gi (x, h ) fi (|u|2 )ui (x )dx − βˆi ci+ Gi (h, h ) − αˆ i ci− Gi (−h, h ).

(14)

From (13), (14) we find

ci− = ai ci− = ai

h

−h

h

−h

Gi (x, −h ) βˆi Gi (h, −h ) f (|u|2 )ui (x )dx − ci+ , 1 + αˆ i Gi (−h, −h ) 1 + αˆ i Gi (−h, −h ) Gi (x, h ) 1 + βˆi Gi (h, h ) f (|u|2 )ui (x )dx − ci+ . αˆ i Gi (−h, h ) αˆ i Gi (−h, h )

(15) (16)

Then substituting (16) into (12), we get the system

ui ( s ) = ai

h −h





+

ci+



Gi (x, h )Gi (−h, s ) Gi (x, s ) − f (|u|2 )ui (x )dx Gi (−h, h )



1 + βˆi Gi (h, h ) Gi (−h, s ) − βˆi Gi (h, s ) . Gi (−h, h )

(17)

Equating (15) to (16), we obtain the system of dispersion equations

ci+ gi (λ ) = ai i (λ ),

(18)

where

gi (λ ) = −1 − αˆ i Gi (−h, −h ) − βˆi G(h, h )





− αˆ i βˆi Gi (−h, −h )Gi (h, h ) − Gi (−h, h )Gi (h, −h ) ,

h   

i (λ ) = αˆ i Gi (−h, h )Gi (x, −h ) − 1 + αˆ i Gi (−h, −h ) Gi (x, h ) f (|u|2 )ui (x )dx. −h

Solutions of the system (18) are the eigentuples of the problem (1)–(4). In order to study the solvability of the system (18), we are going to study system of nonlinear integral equations (17). For this reason let us rewrite the system (17) in an operator form. Introduce the linear matrix integral operator

Kg =

h

−h

K(x, s )g(x )dx,

(19)

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Y.G. Smirnov, D.V. Valovik / Applied Mathematics and Computation 294 (2017) 146–156

where g = (g1 , . . . , gn ) , the kernel K(x, s) has the form of a n × n diagonal matrix

K(x, s ) = diag {K11 (x, s ), . . . , Knn (x, s )}

(20)

and

Kii (x, s ) = Gi (x, s ) −

Gi (x, h )Gi (−h, s ) . Gi (−h, h )

Now system (17) can be written in the form





u = aK f ( |u|2 )u + h,

(21)

where a = diag {a1 , . . . , an }, f = diag { f1 , . . . , fn } are n × n diagonal matrices, and the vector h = (h1 , . . . , hn ) has the components



hi ( s ) =

Ci+



1 + βˆi Gi (h, h ) Gi (−h, s ) − βˆi Gi (h, s ) . Gi (−h, h )

(22)

We shall study Eq. (21) in the space C[−h, h] with the norm u2C . 4. Investigation of the operator equation In order to formulate a theorem about the existence of eigentuples it is necessary to derive some auxiliary statements for Eq. (21).



( p)

( p)

Approximate solutions u( p) (x ) = u1 (x ), . . . , un (x ) mapping method:





, x ∈ [−h, h] of Eq. (21) can be determined using the contracting



u( p+1) = aK f(|u( p) |2 )u( p) + h.

(23)

Since under certain conditions the right-hand side of Eq. (21) defines a contracting operator, then the sequence u(p) converges uniformly to a unique solution of Eq. (21). To be more precise, the following theorem takes place. Theorem 1. Let Br0 = {u ∈ C[−h, h] : uC  r0 } be a closed ball of radius r0 with the origin at zero. Assume that the two conditions

q := a · K(2r02 Lr0 + Ar0 ) < 1,

a · KAr0 r0 + hC  r0

are satisfied. Then there exists a unique solution u ∈ Br0 of Eq. (21) (or, equivalently, of system (17)). The sequence of approximate solutions u( p) ∈ Br0 which is defined by the iterative process (23) converges in C[−h, h] to a unique exact solution u ∈ Br0 of Eq. (21) (or system (17)) for any initial approximation u(0 ) ∈ Br0 with the geometric rate of order q. Remark 2. The quantities Lr0 and Ar0 depend on the radius r0 . Proof. Consider the equation u = A(u ) with the nonlinear operator





A ( u ) ≡ aK f ( |u|2 )u + h in the space C[−h, h]. Let u, v ∈ Br0 ; uC ≤ r0 , vC ≤ r0 , then

   A(u ) − A(v )C = aK f(|u|2 )u − f(|v|2 )v C  a · K(2r02 Lr0 + Ar0 )u − vC .

(24)

Let us prove estimation (24). Indeed,

      f ( | u | 2 ) u − f ( | v | 2 ) v  =  f ( | u | 2 ) u − f ( | v | 2 ) u + f ( | v | 2 ) u − f ( | v | 2 ) v  C    C  f(|u|2 )u − f(|v|2 )u + f(|v|2 )u − f(|v|2 )v C   C    f(|u|2 ) − f(|v|2 ) · uC + f(|v|2 ) · u − vC      L r 0 | u | 2 − | v | 2  ·  u  C + f ( | v | 2 )  ·  u − v  C        Lr0 |u| − |v| · |u| + |v| · uC + f(|v|2 ) · u − vC . C C

(25)

 Taking into account that |u|  |u − v| + |v|, |u| − |v|  |u − v| and |v|  |u − v| + |u|, |v| − |u|  |u − v|, one finds |u| −    |v|  |u − v|  u − vC . Thus |u| − |v|C  u − vC . Then



 







Lr0 |u| − |v| · |u| + |v| · uC + f(|v|2 ) · u − vC C

C

Y.G. Smirnov, D.V. Valovik / Applied Mathematics and Computation 294 (2017) 146–156

151

     Lr0 u − vC · |u| + |v| · uC + maxf(|v|2 ) · u − vC C v∈Br0

(

2r02 Lr0

+ Ar0 ) u − vC .

(26)

  We have found that f(|u|2 )u − f(|v|2 )v  (2r02 Lr0 + Ar0 )u − vC . This implies (24). C Since

     A ( u )  C = a K f ( | u | 2 ) u + h C   a  ·  K  A r 0 r 0 +  h  C ,

(27)

then the operator A maps the ball Br0 into itself. This means that A is contracting in the ball Br0 .



For further study we consider the auxiliary equation

a · K · f(r2 )r + h = r.

(28)

Introduce the function y(r ) := r − a · K · f(r 2 )r and consider the following three cases: (a) y is an unbounded and monotonically increasing function for r ∈ [0, +∞]; (b) y is a bounded and monotonically increasing function for r ∈ [0, +∞]; (c) y is a nonmonotonic function for r ≥ 0. Several applied two-parameter problems that fit the cases (a) and (c) are given in [4] and [1–3], respectively. If the parameters in [4] are specially chosen, then it can fit to the case (b). In the case (a), Eq. (28) has a unique positive solution r = r∗ for any positive matrix a and any vector h such that 0 < hC . In the case (b), Eq. (28) has a unique positive solution r = r∗ for any positive matrix a and any vector h which norm satisfies the inequality 0 < hC < ymax , where ymax := limr→+∞ y(r ). Obviously, the value ymax depends on matrix a. In the case (c), let us consider the equation y (r ) = 0. Let rm be its minimal positive root (it exists due to nonmonotonic behavior of f). So for r ∈ [0, rm ] the function y monotonically increases having maximum y∗max at the point r = rm . This means that equation (28) has a unique positive solution r = r∗ for any positive matrix a and any vector h which norm satisfies the inequality 0 < hC < y∗max . Obviously, the value y∗max depends on matrix a. Statement 2 (case (a)). Let Br∗ = {u : uC  r∗ } be a ball of radius r∗ with the origin at zero, where r∗ is an arbitrary 1 positive constant. If the matrix a is such that a < A, where A = , and the vector h satisfies the inequality 2 K ( 2r∗ Lr∗ +Ar∗ )

hC > 0, then Eq. (21) has a unique solution u ∈ C[−h, h] in the ball Br∗ and uC ≤ r∗ . Proof. Let u ∈ Br∗ , then the following estimate is true

    A(u )C = aK f (|u|2 )u + hC  a · KAr∗ r∗ + hC = r∗ .

(29)

Thus if u, v ∈ Br∗ , then estimate (24) is valid with r0 := r∗ . Since a < A, then the inequality a · K(2r∗2 Lr∗ + Ar∗ ) < 1 is valid. Thus the conditions of Theorem 1 are satisfied with q := a · K(2r∗2 Lr∗ + Ar∗ ). We conclude that A maps Br∗ into itself and is a contracting operator in Br∗ . This implies the existence of a unique solution to Eq. (21) in the ball Br∗ .  Statement 3 (case (b)). Let Br∗ = {u : uC  r∗ } be a ball of radius r∗ with the origin at zero, where r∗ is an arbitrary 1 positive constant. If the matrix a is such that a < A, where A = , and the vector h satisfies the inequality 0 2 K ( 2r∗ Lr∗ +Ar∗ )

< hC < ymax , then Eq. (21) has a unique solution u ∈ C[−h, h] in the ball Br∗ and uC ≤ r∗ . Statement 4 (case (c)). Let Br∗ = {u : uC  r∗ } be a ball of radius r∗ with the origin at zero, where r∗ = rm . If the matrix a 1 is such that a < A, where A = , and the vector h satisfies the inequality 0 < hC < y∗max , then Eq. (21) has 2 K ( 2r∗ Lr∗ +Ar∗ )

a unique solution u ∈ C[−h, h] in the ball Br∗ and uC ≤ r∗ . Statements 3 and 4 are proved in the same way as statement 2. Remark 3. In the cases (a), (b), (c) the constants Lr∗ and Ar∗ are depended on r∗ . The following step is to prove the continuous dependence of the solution u ≡ u(x; λ) on the parameter λ if the parameter belongs to a certain set in Rn . The theorem below guarantees the continuity. Theorem 2. Let the matrix integral operator K = K(λ ) and the vector h = h(λ ) of Eq. (21) depend continuously on the parameters λ ∈  for a certain closed real n-dimensional set  := 1 × . . . × n . Let also one of the statements 2, 3, or 4 takes place. Then the solution u ≡ u(x; λ) of Eq. (21) exists for x ∈ [−h, h], is unique and continuously depends on λ ∈ . Proof. Let us consider Eq. (21). It follows from Theorem 1 that there exists a unique nontrivial solution u(x, λ) to Eq. (21). Now we are going to prove that the solution depends continuously on parameter λ ∈ .

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Choosing r∗ , we define Lr∗ ≡ Lr∗ (λ ) and Ar∗ ≡ Ar∗ (λ ), which continuously depend on λ. We stress that in general the value r∗ depends on λ in the case (c); however we assume that after fixing the set , one chooses the maximal possible value r∗ (thus r∗ no longer depends on ). Denote Lmax := maxλ∈ Lr∗ (λ ) and Amax := maxλ∈ Ar∗ (λ ). It follows from the above inequality that

  f(|u|2 )u − f(|v|2 )v  (2r∗2 Lmax + Amax )u − vC . C

First, let us assume that

u(γ )C  u(γ + γ )C .

(30)

Then the following inequalities are valid

|u(s; λ + λ ) − u(s; λ )|  h    =  K(x, s; λ + λ ) f |u(x; λ + λ )|2 u(x; λ + λ )dx −h

−

h

−h



 



K(x, s; λ ) f |u(x; λ )|2 u(x; λ )dx + h(s; λ + λ ) − h(s; λ )

 h          K(x, s; λ + λ ) − K(x, s; λ ) f |u(x; λ + λ )|2 u(x; λ + λ )dx −h  h         2 2  + K(x, s; λ ) f |u(x; λ + λ )| u(x; λ + λ ) − f |u(x; λ )| u(x; λ ) dx −h

+ | h ( s ; λ + λ ) − h ( s ; λ ) | . It follows from the previous lines that

u(λ + λ ) − u(λ )C  a · K(λ + λ ) − K(λ )Amax r∗ + a · K(λ )(2r∗2 Lmax + Amax )u(λ + λ ) − u(λ )C + h(λ + λ ) − h(λ )C . Thus, we obtain

 u ( λ + λ ) − u ( λ )  C 

a · K(λ + λ ) − K(λ )Amax r∗ + h(λ + λ ) − h(λ )C , 1 − a(2r∗2 Lmax + Amax ) max K(λ )

(31)

λ∈ 

where the denominator in the right-hand side is strongly positive and does not depend on λ. Obviously, the numerator in the right-hand side tends to zero as λ → 0. Now, let u(λ )C  u(λ + λ )C . Then, all the preceding estimations remain valid if we replace λ by λ + λ and λ + λ by λ. Thus estimate (31) also remains valid.  5. Existence of eigentuples In order to prove the solvability of DEs (18), one should study the corresponding linear problem. 5.1. The linear problem Setting a = 0, we get from (18)

gi ( λ ) = 0 ,

(32)

where i = 1, n. Using (7), we obtain

φi,2 j (−h ) i (−h, −h; λi ), +G λi − λi, j φi,2 j (h ) i (h, h; λi ), Gi (h, h; λi ) = +G λi − λi, j

Gi (−h, −h; λi ) =

(33)

where i = 1, n. It is easy to show that φi, j (−h ) = 0 for i = 1, n and for all j. Indeed, if φi, j (−h ) = 0, then one finds from (3) that

  φi, j (−h ) = 0. So one has the Cauchy problem for the equation pi (x )vi + qi (x )vi − λi ri (x )vi = 0 with zero initial data. By virtue of the uniqueness theorem [16] this equation has only a trivial solution; this contradicts to the statement that an eigenfunction φ i, j (x) does not equal identically to zero on [−h, h]. In the same way one can prove that φ i, j (h) = 0.

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In addition, from (8) one finds

Gi (−h, −h )Gi (h, h ) − Gi (−h, h )Gi (h, −h ) =

∞ φ 2 (h ) ∞ ∞ 2  φi,k (−h )  φi, j (h )φi, j (h ) φi,k (−h )φi,k (h )  i, j − λ λ λ λi − λi, j i − λi,k i − λi, j i − λi,k j=1 j=1 k=1 k=1

∞ 

2 φi,k (−h )φi,2 j (h ) − φi,k (−h )φi,k (h )φi, j (h )φi, j (h ) (λi − λi,k )(λi − λi, j ) k=1 j=1  2 ∞  ∞  φi,k (−h )φi, j (h ) − φi,k (h )φi, j (−h ) = (λi − λi,k )(λi − λi, j ) k=1 j>k  2  ∞  ∞  φi,k (−h )φi, j (h ) − φi,k (h )φi, j (−h )  −1 1 = + , λi, j − λi,k λi − λi,k λi − λi, j k=1 j>k

=

∞  ∞ 

(34)

where i = 1, n. The transformations made above (the grouping and transmutation of the terms of series) are justified, because series (8) converge absolutely. Then, in the neighbourhood of λi, l , gi (λ) can be represented in the form

φi,l2 (−h ) φi,l2 (h ) − βˆi − wi ( λ ) λi − λi,l λi − λi,l    2  ∞  ∞  φi,k (−h )φi, j (h ) − φi,k (h )φi, j (−h )  −1 1 ˆ − αˆ i βi + , λi, j − λi,k λi − λi,k λi − λi, j k=1 j>k

gi (λ ) = −αˆ i

(35)

where i = 1, n and

i (−h, −h; λi ) + βˆi G i (h, h; λi ) wi (λ ) := 1 + αˆ i G is continuous in this neighbourhood. Multiplying gi (λ) by λi − λi,l , we get

(λi − λi,l )gi (λ ) = −αˆ i φi,l2 (−h ) − βˆi φi,l2 (h ) − (λi − λi,l )wi (λ )    2  l−1  ∞  φi,k (−h )φi, j (h ) − φi,k (h )φi, j (−h )  −1 1 ˆ − αˆ i βi (λi − λi,l ) + λi, j − λi,k λi − λi,k λi − λi, j k=1 j>k  2 ∞  φi,l (−h )φi, j (h ) − φi,l (h )φi, j (−h ) ˆ − αˆ i βi λi − λi, j j>l    2  ∞  ∞  φi,k (−h )φi, j (h ) − φi,k (h )φi, j (−h )  −1 1 − αˆ i βˆi (λi − λi,l ) + . λi, j − λi,k λi − λi,k λi − λi, j

(36)

k=l+1 j>k

2 (−h ) > 0, φ 2 (h ) > 0, and λ Since αi2 (λ ) + βi2 (λ ) = 0, αi2 (λ )  0, βi2 (λ )  0, p(−h ) > 0, p(h) > 0, φi,l i, l > λi, j for l < j, i,l we have





lim (λi − λi,l )gi (λ ) = −αˆ i φ (−h ) − βˆi φ (h ) − αˆ i βˆi

λi →λi,l

2 i,l

2 i,l

∞  j>l



φi,l (−h )φi, j (h ) − φi,l (h )φi, j (−h ) |λi,l − λi, j |

2

< 0.

(37)

Then,

lim

λi →λi,l ±0

gi (λ ) = ∓∞

and, taking into account that the function gi (λ) is continuous in the interval (λi,l+1 , λi,l ) for all l ≥ 1, we obtain that, between λ and λ , there is at least one root  λ of the equation g (λ ) = 0. Obviously, this is true for each i = 1, n. i,l+1

i, l

i,l

i

In other words, for each i = 1, n and any integer j ≥ 1, there exist  λi, j such that the function gi (λ) can be represented in the form

gi (λ ) = (λi −  λi, j ) gi ( λ ) , where  λi, j ≡  λi, j (λ1 , . . . , λi−1 , λi+1 , . . . , λn ). Thus one has proved the following

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Theorem 3. The problem (1)–(3) with a = 0 has an infinite set of isolated eigentuples

 λ

j1 ,..., jn

= ( λ1, j1 , . . . ,  λn, jn ) ,

where jk = 1, 2, . . . , k = 1, n. In addition, all the eigentuples are real-valued vectors and, for each i, the ith component of an eigentuple is a simple root of gi (λ ) = 0. Moreover, for each i, the sequence { λi, j }∞ contains only a finite number (or does not contain at all) of positive terms j=1 and an infinite number of negative ones, such that  λ → −∞ as j → ∞. The following asymptotics is valid  λ = O∗ ( j2 ) as j → i, j

∞.

i, j

5.2. The nonlinear problem The main result of this study consists in the following Theorem 4. Let  = 1 × · · · × n be a bounded set, where each of i is a segment on the real axis, such that Theorem 2 takes place, one of the statements 2, 3, or 4 holds, and the linear problem has p1 × . . . × pn eigentuples

 λ

j1 ,..., jn

= ( λ1, j1 , . . . ,  λn, jn )

inside  and each  λi, jk ∈ i , where j1 = 1, p1 , . . . , jn = 1, pn .

Then there exists a matrix a0 = diag {a01 , . . . , a0n }, where each a0i > 0, such that for any matrix a = diag {a1 , . . . , an }, where

ai  a0i , i = 1, n, the problem (1)–(4) has at least p1 × . . . × pn eigentuples

λ

j1 ,..., jn

= ( λ1, j1 , . . . , λn, jn ),

where each λi, jk ∈ i . Proof. Let  λ j1 ,..., jn = ( λ1, j1 , . . . ,  λn, jn ) be a solution to the linear problem. Then  λi, jk ∈ i for i = 1, n. Let positive sufficiently ± small values δi, j be such that k

[ λi, jk − δi,−j ,  λi, jk + δi,+j ] =: i, jk ⊂ i k

k

for i = 1, n. The quantities δi,±j are chosen in the way that for all possible i and jk the segments i, jk do not contain poles k

of the Green function Gi (x, s; λi ), see (6), (7). Introduce the notation

k1 ,...,kn := 1,k1 ∪ . . . ∪ n,kn , where (k1 , . . . , kn ) is a particular n-tuplet such that ki ∈ {1, . . . , pi }. 1 Let us consider A = . It is clear that A ≡ A(λ). Introduce the numbers 2 K ( 2r∗ Lr∗ +Ar∗ )

Ak1 ,...,kn :=

min

λ∈ k1 ,...,kn

A ( λ ).

Then choosing a0 such that a0  < Ak1 ,...,kn , in accordance with one of the statements 2,3, or 4, a unique solution u ≡ u(s, λ) to Eq. (21), defined globally on s ∈ [−h, h], exists for all λ ∈ k1 ,...,kn . This solution is continuous in s, λ and uC  r∗ = r∗ (λ ), where r∗ is defined in one of the statements 2,3, or 4. Let rk1 ,...,kn := maxλ∈ r∗ . Estimating i , one finds k1 ,...,kn

max

λ∈ k1 ,...,kn

| i (λ )|  ck1 ,...,kn rk1 ,...,kn · max Ar∗ , λ∈ k1 ,...,kn

where ck1 ,...,kn is a positive constant (ck1 ,...,kn does not depend on λ or any of its components). It follows from the proof of Theorem 3 that each of the functions gi (λ) is continuous and has different signs at λi =  λi, jk − δi,−j and λi =  λi, jk + δi,+j and (λ1 , . . . , λi−1 , λi+1 , . . . , λn ) ⊂ 1,k1 ∪ . . . ∪ i−1,ki−1 ∪ i+1,ki+1 ∪ . . . ∪ n,kn for all possible k

k

i and (n − 1 )-tuplet (k1 , . . . , ki−1 , ki+1 , . . . , kn ). Introduce the numbers

mk1 ,...,kn :=

max

λ∈ k1 ,...,kn

|gi ( λ ) |.

Now it is clear that if a0 is such that a0  



ci+ gi (λ

(1 )

mk ,...,kn 1

ck ,...,kn rk ,...,kn ·maxλ∈ Ar∗ 1 1 k1 ,...,kn

, then

   ) − ai (λ(1) ) · ci+ gi (λ(2) ) − ai (λ(2) ) < 0

for all ai  a0i and i = 0, n, where λ (2 )

(1 )

(1 )

(1 )

= (λ1 , . . . , λn ), λ

(2 )

(38) (2 )

(2 )

(1 )

= (λ1 , . . . , λn ); λi

is equal either  λi, jk − δi,−j or  λi, jk + δi,+j k

k

and λi is equal either  λi, jk + δi,+j or  λi, jk − δi,−j , respectively. Due to continuous dependence the left-hand side of (38) on k k each parameter, the latter inequality implies the existence of n-tuplet λ = ( λ1 , . . . , λn ) in each k1 ,...,kn such that

ci+ gi ( λ ) − ai ( λ ) = 0.

Y.G. Smirnov, D.V. Valovik / Applied Mathematics and Computation 294 (2017) 146–156

155

Thus the found λ is a solution to (1)–(4), that is, an eigentuple of the nonlinear problem. Then one can choose a0 in such a way that

⎧ ⎨

⎫ ⎬

mk1 ,...,kn a0   min Ak1 ,...,kn , . ck1 ,...,kn rk1 ,...,kn · max Ar∗ ⎭ ⎩ λ∈ k1 ,...,kn

Obviously, procedure described above is valid for each k1 ,...,kn . This implies the existence of at least one solution to the nonlinear problem in the vicinity of each solution to the linear problem in the fixed domain . Thus there exist at least p1 × . . . × pn eigentuples λ j1 ,..., jn = ( λ1, j1 , . . . , λn, jn ), where j1 = 1, p1 , . . . , jn = 1, pn . Carrying out the above made estimations for all n-tuplet k1 , . . . , kn , one derives the estimation for a0 , which is valid for "  := k1 ,...,kn k1 ,...,kn .  6. Results and discussion The proof of Theorem 4 employs the method of small parameters where the nonlinearity coefficients given by the matrix a are supposed to be small. This approach can be applied for the analysis of nonlinear eigenvalue problems because in many physical cases involving nonlinear eigenvalue problems of the type considered in this study, it is known that a is sufficiently small. Theorem 4 states the existence of eigentuples λ ∈ Rn . In the case of n = 1, 2 this theorem results in the existence of nonlinear polarized electromagnetic guided and coupled guided waves in a plane dielectric waveguide having a scalar permittivity ε = q1 (x ) + a1 f1 (|u|2 ) and a tensor permittivity of the form

⎛

q1 ( x ) + a1 f 1 ( |u|2 )

0

0

q2 ( x ) + a2 f 2 ( |u|2 )

ε=⎝

0

0

⎞

0

⎠, q3 ( x ) + a3 f 3 ( |u| ) 0

2

respectively, where u corresponds to the electric field vector. Results regarding to the scalar permittivity and constant q1 are presented in [8], regarding to the tensor permittivity and constant qi in [3,4]. In case of all ai are zeros one has an inhomogeneous (for nonconstant qi ) or homogeneous (if all qi are constants) waveguide. The case of nonconstant qi is very important both in linear and nonlinear waveguiding theory. The linear theory for inhomogeneous waveguides was developed in [17]; the nonlinear theory for nonconstant qi is given in this paper. We stress that in the electromagnetic case pi ≡ 1 and ri ≡ 1 for all i. The following issue seems to be very important for further development in this field. If all qi are constants and all fi are bounded (some restrictions are needed [4]), then the integral dispersion equation method (IDEM) can be effectively applied. However, in the case of nonconstant qi and/or unbounded nonlinearities (like, fi (|u|2 ) = |u|2 , see also case (c) above), the IDEM is not yet developed. It would be interesting to develop the IDEM or any other technique for these cases. The other important issue is that in some nonlinear eigenvalue problems that contain small nonlinearity coefficient, there exists an infinite number of solutions that do not reduce to any solution of the corresponding linear problems [8]. To be more precise, for the case n = 1, p1 ≡ 1, q1 ≡ const > 0, r1 ≡ 1, and f1 (s ) = s, the existence of infinitely many nonperturbative eigenvalues is proved in [8]. In other words, the small parameter method does not allow finding all possible results about eigenvalues and eigentuples in some intriguing nonlinear eigenvalue (including multi-parameter) problems. Acknowledgments The first author (Yu.S.) was supported by the Ministry of Education and Science of the Russian Federation (agreement no. 2.1102.2014K) and by the Mathematisches Forschungsinstitut Oberwolfach (under RiP 2014 project); the second author (D.V.) was supported by the Ministry of Education and Science of the Russian Federation (agreement no. 2.1102.2014K), the Russian Foundation for Basic Research (grant no. 15-01-00206), and the Russian Federation President Grant (grant no. MK-4684.2016.1). References [1] D.V. Valovik, On the problem of nonlinear coupled electromagnetic TE-TM wave propagation, J. Math. Phys. 54 (4) (2013) 042902(14). [2] Y.G. Smirnov, D.V. Valovik, Coupled electromagnetic TE-TM wave propagation in a cylindrical waveguide with Kerr nonlinearity, J. Math. Phys. 54 (4) (2013) 043506(22). [3] Y.G. Smirnov, D.V. Valovik, Problem of nonlinear coupled electromagnetic TE-TE wave propagation, J. Math. Phys. 54 (8) (2013) 083502(13). [4] D.V. Valovik, Nonlinear coupled electromagnetic wave propagation: saturable nonlinearities, Wave Motion 60 (2016) 166–180. [5] A.D. Boardman, P. Egan, F. Lederer, U. Langbein, D. Mihalache, H.-E. Ponath, G.I. Stegeman, Third-Order Nonlinear Electromagnetic TE and TM Guided Waves, Elsevier Sci. Publ., 1991. reprinted from Nonlinear Surface Electromagnetic Phenomena. [6] P.N. Eleonskii, L.G. Oganes’yants, V.P. Silin, Structure of three-component vector fields in self-focusing waveguides, Sov. Phys. JETP 36 (2) (1973) 282–285. [7] P.N. Eleonskii, L.G. Oganes’yants, V.P. Silin, Vector structure of electromagnetic field in self-focused waveguides, Sov. Phys. Usp. 15 (4) (1972) 524–525. [8] Y.G. Smirnov, D.V. Valovik, Guided electromagnetic waves propagating in a plane dielectric waveguide with nonlinear permittivity, Phys. Rev. A 91 (1) (2015) 013840-1–013840-6.

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