# Nonlinear Stability of Periodic Traveling Wave Solutions for(n+1)-Dimensional Coupled Nonlinear Klein-Gordon Equations

Authors: Cong Sun; Bo Jiang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

We study the existence and orbital stability of smooth periodic traveling waves solutions of the(n+1)-dimensional coupled nonlinear Klein-Gordon equations. Such a system occurs in quantum mechanics, fluid mechanics, and optical fiber communication. Inspired by Angulo Pava’s results (2007), and by applying the stability theory established by Grillakis et al. (1987), we prove the existence of periodic traveling waves solutions and obtain the orbital stability of the solutions to this system.


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## Body

## 1. Introduction

This paper pays close attention to(n+1)-dimensional system of coupled nonlinear Klein-Gordon equations(1)ϕtt-Δϕ+m1ϕ=a11ϕ2+a12ψ2ϕ,ψtt-Δψ+m2ψ=a21ϕ2+a22ψ2ψ,where Δ=∂2/∂x12+∂2/∂x22+⋯+∂2/∂xn2 is Laplace operator, x∈Rn, m1<0, m2<0, aij∈R (i,j=1,2), a11≠a21, and ϕ(x,t) and ψ(x,t) are complex functions.Equation (1) frequently describes physical questions, for example, crystal growths and dislocations. In recent decades, lots of methods have been put forward, such as the direct integral method, the hyperbolic function expansion method [1, 2], mixing exponential method, and Jacobi elliptic function expansion method [3, 4], to obtain the exact solutions of nonlinear evolution equations.In this paper, we are interested in the existence of a smooth periodic solution and the orbital stability of solution of (1). A comprehensive development of stability results for this type has been acquired by Grillakis et al. [5, 6], Benjamin [7], and Weinstein [8]. Moreover, many results of orbital stability of Klein-Gordon equation have been attained. For example, Shatah [9] has given sufficient conditions of orbital stability in such a case:(2)utt-Δu+u+guarg⁡u=0,t∈R,x∈Rn,and it is showed that the standing wave solutions eictϕ(x) of this equation have orbital stability when guarg⁡u=-|u|p-1u, with p>1 as an integer. In addition, Ohta and Todorova [10] have established the instability of (2) when 1+4/n<p<1+4/n-2. In this case, we say that this wave eictϕ(x) is unstable for (2); suppose for any ɛ>0, there exists (u0,u1)∈X=H1(R)×L2(R) such that (u0,u1)-(ϕ,icϕ)X<ɛ and the solution u(t)=(u,ut) of (2) with u(0)=(u0,u1) satisfies limt→∞u(t)X=∞. Moreover, Angulo Pava [11] has studied the orbital stability of cnoidal waves solution of the following system:(3)ut-auxxx+6uxu=2bvvx,vt+vxxx+3uvx=0,where b>0, and he obtained the orbital stability of the cnoidal solutions in the energy space Hper1(0,L)×Hper1([0,L]) with regard to the periodic flow of (3) when a≥1/20.In our paper, we work on the existence and orbital stability of periodic traveling waves solution of system (1). The plan of the paper is as follows: in Section 2, we obtain the existence of periodic traveling waves solution of system (1). In Section 3, the spectrum of operator L (see (26)) is studied in detail. In Section 4, the orbital stability of system (1) is established. In Section 5, we give a brief discussion and conclusion to state the application of system (1) in the field of engineering.Notation. Let Lp(Ω) denote Lebesgue measurable space for the open subset Ω in R, the norm fLpp=∫Ω|f|pdx, and p≥1. X=H1(R)×H1(R)×H1(R)×H1(R) denotes the complex space with real inner product (u1,u2)=Re∫(u1u2¯+u1xu2x¯). The dual space of X is X∗=H-1(R)×H-1(R)×H-1(R)×H-1(R). There exists a natural isomorphism I:X→X∗ denoted by 〈Iu,v〉=(u,v). Here, (4)u,v=∫Ruv¯dx,I=-∂2x2+1111.


## 2. Existence of the Periodic Traveling Wave Solution for the KG Equations

In this section, motivated by Angulo Pava’s work [12], we will study the existence of a smooth curve of periodic traveling waves solutions to the coupled nonlinear Klein-Gordon equations (1) with the form(5)ϕx,t=eiη1uξ,ψx,t=eiη2φξ,η1=∑j=1nαjxj-ct,η2=∑j=1nβjxj-ct,where ξ=∑j=1nkjxj, kj,αj,βj∈R, j=1,2,…,n, and x=(x1,x2,…,xn). The profile (u(ξ),φ(ξ)) satisfies the boundary conditions |u(ξ)|→0 and |φ(ξ)|→0 as |ξ|→∞ and the constant c denotes the speed of traveling waves.By substituting (5) in (1), we obtain(6)-c2u-∑i=1nki2u′′+u∑j=1nαj2+m1u=a11u2+a12φ2u,-c2φ-∑i=1nki2φ′′+φ∑j=1nβj2+m2φ=a21u2+a22φ2φ,where α=∑j=1nαj2 and β=∑j=1nβj2.Now, we consider the the special solutions(u(ξ),φ(ξ))=(aφ(ξ),φ(ξ)) of (6) where a∈R-. Then, (6) is transformed into(7)φ′′-B0φ-φ3-a11a2-a12∑i=1nki2=0,φ′′-B0′φ-φ3-a21a2-a22∑i=1nki2=0,where B0=α+m1-c2/∑i=1nki2 and B0′=β+m2-c2/∑i=1nki2. Moreover, we assume that -m1+c2-α>0 and -m2+c2-β>0.Notice the relationu=aφ; hence, the corresponding parameters satisfy the relations a2=a22-a12/a11-a21 and B0=B0′.Next, to study the solutions of (7), we only need to consider the first equation of (7):(8)φ′′-B0φ-mφ3=0.Firstly, we suppose that B0>0 and m=(-a11a2-a12)/∑i=1nki2<0.Multiplying (8) by φ′ and integrating once, we obtain(9)φ′2=-m2P12-φ2φ2-Q12,where h is a constant of integration, P12=-1/m[B0+B02-4hm], Q12=1/m[-B0+B02-4hm], and h∈(-B02/4m,+∞). Without loss of generality, we suppose that P1>Q1>0. Define φ/P1=t and k2=P12-Q12/P12; hence, (9) becomes (10)t′2=-mP1221-t2t2-1+k2.Next, we continue to define a new variabler, which satisfies r≥0 and r(0)=0. By relation t2=1-k2sin2r, we obtain (11)r′2=-mP1221-k2sin2⁡r.So, we get fors=-m/2P1 that (12)∫0γξdt1-k2sin2⁡2=sξ.Next, inspired by Angulo Pava’s work [12], by applying the theory of Jacobian elliptic function [13], we obtain the dnoidal wave solution of (8),(13)φξ=P1dn-m2P1ξ,P12-Q12P1,with(14)k2=P12-Q12P12,P12+Q12=-2B0m.Because dn has fundamental period2K; that is, dn(x+2K;k)=dn(x,k). Here K(k) is the complete elliptic integral of first kind. Hence, we know that solution (13) with fundamental period is as follows:(15)T=2Kk-m/2P1.Next, expression (15) can be converted into a simple form according to (14). In fact, if we fix variables m and B0, we can find (15) as a function with only one variable Q1: (16)TQ1=2KkQ1B0+m/2Q12,k2Q1=-2B0/m-2Q12-2B0/m-Q12.Furthermore, ifQ1→0, k(Q1)→1, and K(k(Q1))→+∞, T(Q1)→+∞. If Q1→-B0/m, k(Q1)→0, and k(Q1)→π/2, therefore, T(Q1)→π/B0.Then, we will establish a smooth dn wave solution for (8) with fixed period L. For any given L>π/B0, B0>π2/L2, and m<0, we can know that the mapping Q1∈(0,-B0/m)→T(Q1) is strictly decreasing function. In what follows, we will show that there is a unique Q1=Q1(B0)∈(0,-B0/m) such that TQ1=T(Q1(B0)) is a fundamental period of (13).Theorem 1.
For any givenL>π/B0, consider B0,0>π2/L2; then there is a unique Q1,0∈(0,-B0,0/m) such that T(B0,0)=L. Then,(1)
there exist an intervalI(B0,0) around B0,0, an interval C(Q1,0) around Q1,0, and a smooth and unique function Υ:I(B0,0)→C(Q1,0), such that Υ(B0,0)=Q1,0 and (17)TQ1=2KkB0+m/2Q12.Here,B0∈I(B0,0) and Q1=Υ(B0).(2)
The solutionφ(ξ)=P1dn(-m/2P1ξ,P12-Q12/P1) is determined by P1=P1(B0) and Q1=Q1(B0), has fundamental period L, and satisfies (13). In addition, the mapping B0∈I(B0,0)→φB0∈Hper1([0,L]) is a smooth function.Proof.
The proof is via the implicit function theorem, the method is based on Angulo Pava’s work [12], and for more details, one may see Theorem 1 in [12].Theorem 2.
LetL>0. Then, there is a smooth curve of periodic traveling wave solutions for (8) with B0∈(0,+∞)→φB0∈Hper1([0,L]). Here, n∈N, φ(ξ)=P1dn(-m/2P1ξ,P12-Q12/P1).


## 3. Spectral Analysis

Firstly, (1) can be rewritten as the following form: (18)ϕt=ϕ~,ϕt~=Δϕ-m1ϕ+a11ϕ2+a12ψ2ϕ,ψt=ψ~,ψt~=Δψ-m2ψ+a21ϕ2+a22ψ2ψ.Moreover, let U→=(ϕ,ϕ~,ψ,ψ~)T. So, (1) can be rewritten as the following Hamiltonian system: (19)dU→dt=JE′U→,where J is the skew-symmetric linear operator: (20)J=0-1001000000-10010,EU→=12∫0Lϕ~2-∇·ϕ2-m1ϕ2+12a11ϕ4+a12ψ2ϕ2+ψ~2-∇·ψ2-m2ψ2+12a22ψ4dx.LetT be a one-parameter group of unitary operator on X defined by (21)TsU→·=ϕ·eis,ϕ~·eis,ψ·eis,ψ~eis,where U→(·)∈X and s∈R.Obviously,(22)T′0=i0000i0000i0000i.SinceJB=T′0, we can deduce that (23)B=0i00-i000000-i00i0.Next, we define the functional(24)QU→=12∫0Lϕϕ~+ψ~ψdx,HU→=EU→-cQU→.According to [5], it can be verified that E(U→) and Q(U→) are conserved quantities, and the solutions of system (1) are critical points of H(U→); namely, (25)H′U→=0.Next, we define an operatorL from X to X∗:(26)L=H′′U→=E′′U→-cQ′′U→.Then, for any ζ→=(ζ1,ζ2,ζ3,ζ4)T, (27)Lζ→=Δζ1-m1ζ1+a11ϕ2ζ1+2a12ϕ2ζ1+a12ψ2ζ1-cζ2+2a12ψϕζ3-cζ1+ζ2Δζ3-m2ζ3+2a12ϕψζ1+a12ϕ2ζ3+a22ψ2ζ3+2a22ψ2ζ3-cζ4-cζ3+ζ4.By a simple computation, we can deduce thatL is a self-adjoint operator. It is shown that I-1L is a bounded self-adjoint operator. The spectrum of the L consists of the real numbers λ such that λI-L is not invertible, and λ=0 belongs to the spectrum of L.Moreover, we can also prove that(28)LT′0U→=0.Let(29)Z=k∈R∣kT′0U→.From (28), we can know that Z is contained in the kernel of L.By [14], we have the following theorem.Theorem 3.
The spaceX is decomposed as a direct sum; X=Z+N+P. Here, Z is defined by (29), N is a finite-dimensional subspace, such that (30)Lu→,u→<0,for0≠u→∈N,and P is a closed subspace, such that (31)Lu→,u→≥σu→X2,foru→∈P,where σ is a positive number and independent of u→.Proof.
For anyω∈X, let (32)ω=ω1,ω2,ω3,ω4T.Then, (33)Lω,ω=L1ω1,ω1+L2ω3,ω3+2Re⁡∫R-cω1¯ω2+2a12ψϕω3ω1¯-cω3¯ω4dx+ω2+ω4,ω2+ω4-c2ω1-αω1,ω1-c2ω3-βω3,ω3,where (34)L1=Δ-m1+a11ϕ2+a12ψ2+c2-α,L2=Δ-m2+a12ϕ2+2a22ψ2+c2-β.
As|x|→∞, ϕ,ψ→0. So that (35)σessL1=-m1+c2-α,+∞,σessL2=-m2+c2-β,+∞.
Moreover, we may obtain that(36)L1u=0,L2φ=0.
From (13) and [5, 15], they imply u has a simple zero. By Sturm-Liouville theorem, 0 is the second eigenvalue of L1, and L1 has exactly one strictly negative eigenvalue -σ12, with an eigenfunction χ; that is, (37)L1χ=-σ12χ.
Furthermore,0 is the first simple eigenvalue of L2. From [16], we have the following lemmas (Lemmas 4 and 5).
Lemma
4.
For any real function
z
1
∈
H
(
R
)
, if it satisfies
(38)
z
1
,
χ
=
z
1
,
u
=
0
.
There exists a number
σ
1
∈
R
+
independent of
z
1
, such that
(39)
L
1
z
1
,
z
1
≥
σ
1
z
1
H
1
2
.
Lemma
5.
For any real function
z
2
∈
H
(
R
)
, if it satisfies
(40)
z
2
,
φ
=
0
.
Then, there exits a number
σ
2
∈
R
+
independent of
z
2
, such that
(41)
L
2
z
2
,
z
2
≥
σ
2
z
2
H
1
2
.
Lety→=(χ1,0,φ,0)T∈X. By (3), we can obtain (42)Ly→,y→=-σ12y→,y→.
Notice that the kernel ofL is spanned by ν1→ and ν2→; then (43)Z=k1ν1→+k2ν2→∣k1,k2∈R.
Let(44)P→=p→∈X∣p→,y→=p→,ν1→=p→,ν2→=0,N=k2y→∣k2∈R.
For anyω∈X and p→∈P, we choose a1=p→,y→/y→,y→, a2=p→,ν1→/ν1→,ν1→, and a3=p→,ν2→/ν2→,ν2→. Then ω can be uniquely represented by (45)ω=a1y→+a2ν1→+a3ν2→+p→.This shows that the space X is decomposed as a direct sum; X=Z+N+P.
In the following, we will prove that(46)∀p→∈P,Lp→,p→≥σp→H12.
Letp→=(p1,p2,p3,p4)T. By Lemmas 4 and 5, we have (47)Lp→,p→≥δ1p1H12+δ2p3H12-cp1¯H12+p2¯H12-cp3¯H12+p4¯H12+p2¯H12+p4¯H12-c2-αp1H12-c2-βp3H12≥δp1H12+p2H12+p3H12+p4H12,where δ=min⁡{α+δ1-2c2,1-c,β+δ2-2c2}. Thus, we complete the proof of Theorem 3.


## 4. Orbital Stability of the Periodic Wave Solutions for the(n+1)-Dimensional KG Equations

In this section, we shall apply the method established by Grillakis et al. [5] to give the orbital stability of the periodic traveling solutions of system (1):(48)Φ→x,t=ϕ,ψ=eiη1aφξ,eiη2φξ,where(49)φξ=P1dn-m2P1ξ,P12-Q12P1.By [16, 17] and by applying classical semigroups method, for any (50)Φ→0=ϕx,0,ψx,0∈Hper1Rn,0,L×Hper1Rn,0,L,there exists a solution Φ→(x,t)∈Hper1(Rn,[0,L])×Hper1(Rn,0,L) for system (1) and L=+∞ or L<+∞.In Section2, we have obtained that system (1) has periodic traveling waves solution of (48) in Hper1[0,L]×Hper1[0,L]. Moreover, in Section 3, we also obtained that operator L has only one negative simple eigenvalue, the kernel of L is spanned by T′0Φ, and the rest of its spectrum is positive and bounded away from zero. According to orbital stability theory of Grillakis et al. [6], we only need to prove that the function is strictly convex as follows: (51)dc=EU→-cQU→.Firstly, we state our definition of orbital stability.Definition 6.
The orbit ofΦ→(x,t)=(ϕ,ψ) is defined by (52)ΩΦ→≔TsΦ→≔Φ→eis,s∈R2.We say orbit ΩΦ→ is orbital stable in Banach space X, if for all ɛ>0 there exists a δ>0 such that if Φ→(x,0)=(ϕ(x,0),ψ(x,0))∈X=Hper1(Rn,[0,L])×Hper1(Rn,[0,L]) satisfies Φ→(0)-Φ→X<δ, and Φ→(x,t) is a solution of (1) in internal [0,t0) with the initial value Φ→x,0=Φ→(0), then Φ→(x,t) can be extended to a solution in 0≤t<+∞ has (53)supt>0⁡infs∈R⁡Φ→t-TsΦ→X<ɛ.Otherwise, we say that Φ→(x,t) is orbital unstable.Theorem 7.
Letac+1<0 and Φ→x,t=aeη1φξ,eη2φξ, φξ determined by (49), be a periodic traveling waves solution of system (1). Then, this solution is orbital stable in Hper1(Rn,[0,L])×Hper1(Rn,[0,L]) by periodic flow of system (1).Proof.
From (3), we know that (54)E′U→-cQ′U→=0.
Therefore(55)d′c=E′U→-cQ′U→-QU→=-QU→=-ac+1∫0Lφξ2dξ.Then (56)d′′c=-ddcac+1∫0Lφξ2dξ.Through (13), we obtain that (57)φ2=P12∫0Ldn2-m2P1ξ,kdξ=P12∫0-m/2P1Ldn2ξ,kdξ=P1-m/2∫02Kdn2ξ,kdξ=2K-m/2L∫0Kdn2ξ,kdξ.Now, by using that (see [13]), (58)∫0Kcn2ξ;kdξ=1k2Ek-1-k2Kk,(59)dn2ξ;k=1-k2+k2cn2ξ;k.Therefore, we can obtain that (60)∫0Lφ2ξdξ=4KEL1-m/2.So that(61)d′′c=-a∫0Lφ2ξdξ+ac+1ddc∫0Lφ2ξdξ=-a4KEL1-m/2+ac+1-1L12mddk4EKdB0dcdkdB0=-a4KEL1-m/2+ac+1-1L12mddk4EK-2cdkdB0.Indeed,(62)k2=P12-Q12P12=2B02-4hmB02+B02-4hm.So that(63)dkdB0=4hmB02-4hm+B02B02-4hm<0.Moreover, according to [13], E(k)K(k) is strictly increasing functions; hence, d(E(k)K(k))/dk>0. Hence, we can obtain d′′(c)>0. So, we complete the proof of this theorem.


## 5. Discussion and Conclusion

In this paper, inspired by Angulo Pava’s ideas [11, 12, 18, 19], and by using the stability theory established by Grillakis et al. [5], we obtain solution (48) of system (1) and prove that the solution is orbital stable. The method is helpful to look for periodic solution and obtain the orbital stability for a class of nonlinear equations, which is widely used in quantum mechanics, fluid mechanics, and optical fiber communication. Hence, it may contribute to solving these engineering problems.



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