# Nonlinear Stability of the Periodic Traveling Wave Solution for a Class of Coupled KdV Equations

Authors: Cong Sun
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this paper, by applying the Jacobian ellipse function method, we obtain a group of periodic traveling wave solution of coupled KdV equations. Furthermore, by the implicit function theorem, the relation between some wave velocity and the solution’s period is researched. Lastly, we show that this type of solution is orbitally stable by periodic perturbations of the same wavelength as the underlying wave.


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## Body

## 1. Introduction

In fluid mechanics of the density stratification, the mechanism of propagation of nonlinear long wave is being researched by physicists and mathematicians. This wave usually appears in the lakes, fjords, and temperature jump layer in the coastal waters. Generally, the KdV equation is often used to describe the wave of this type when the depth of fluid is much shorter than the length of it. Moreover, weak interactions occur in the internal of the nonlinear long waves when wave phase speeds are unequal. In this case, each wave satisfies a KdV equation, and the interaction among waves could be depicted by the phase shift. However, strong interactions occur when these wave phase speeds are nearly equal although the waves belong to different modes. This case is described by a coupled Korteweg-de Vries equations, which has the following form:(1)ut+uxxx+a3vxxx+uux+a1vvx=0,b1vt+rvx+vvx+b2a3uxxx+vxxx+b2a1uvx=0,where a1, a2, a3, b1, b2, and r are real constants with positive b1 and b2. Here, u=ux,t and v=vx,t are real-valued functions, x∈R, t∈R+.System (1) was derived from Gear and Grimshaw [1]. Due to its significant physical meaning, many scholars at home and abroad carried out relevant research about these equations and obtained some results. As in [2], Marshall et al. have proved that (1) is globally well-posed when the initial data is rough. Moreover, Bona et al. also considered the similar question (for details, please see [3]).Recently, a number of researchers are interested in the theory of stability of the Korteweg-de Vries equation. For example, in [4], Linares and Pazotob considered the exponential stabilization with an initial boundary value problem for following the Korteweg-de Vries equation, which has the following form:
(2)ut+ux+uxxx+uux+axu=0,x,t∈R+,u0,t=0,t>0,ux,0=u0x,x>0.When ax>0, system (2) was shown to be exponentially stable. In [5], Russell and Zhang studied the KdV equation with the periodic boundary conditions, as follows:
(3)ut+ux+uux+uxxx=0,ux,0=φx,u0,t=uπ,t,uxx0,t=uxxπ,t,uxπ,t=αux0,t.Here,α∈−1,1. It was shown that if α≠−1/2, then (3) was locally exponentially stable.Furthermore, in [6], Bona et al. researched this type KdV equation with an initial boundary problem which posed in a quarter plane and appended a damping term, as follows:
(4)ut+ux+uux+uux+uxxx+αu=0,x,t≥0,ux,0=φx,u0,t=ht.They obtain such a result that the time-periodic solutionu∗x,t of (4) is either locally or globally exponentially stable in Hs(R+) when s∈3/4,1 or s>1, respectively.In recent decades, an extensive development of stability theory about solitary waves solutions has been obtained by Benjamin, Bona, Weinstein, Grillakis, and Strauss [7–12].In [13], Pava considered a modified Korteweg-de Vries equation, which has the following form:
(5)ut+3u2ux+uxxx=0,where x∈R, t>0. By applying the Lyapunov stability method ([10]), he gave an important conclusion that the periodic traveling solution η1dnη1ξ/2,k of (5) is orbitally stable in energy space Hper10,L.In this paper, I am interested in the existence of a smooth periodic solution and the orbital stability of the solution of a coupled Korteweg-de Vries (equation (1)).Our paper is organized as follows. In Section2, we applied the Jacobian ellipse function method to obtain a class of smooth cnoidal wave solution for system (1). Section 3 is devoted to studying the relation in a neighborhood of the relevant wave velocity between the period of above cnoidal wave solution and wave velocity. In Section 4, we present the orbital stability theory for the cnoidal wave solutions of system (1).


## 2. The Periodic Solution for the Coupled Korteweg-de Vries Equations

In this section, by implying the Jacobian ellipse function method, we will show the existence of a class of smooth cnoidal wave solution for system (1).Firstly, we suppose that system (1) possesses solitary wave solution of the following form:
(6)ux,t=φξ,vx,t=ψξ,ξ=x−ct,where c is a traveling wave speed.By substituting (6) into (1), we get
(7)cφ′+φφ′φ′″+a3ψ′″+a1ψψ′+a2φψ′=0,cb1ψ′+rψ′+ψψ′+b2a3φ′″+ψ′″+b2a2φφ′+b2a1φψ′=0.Next, we study the form the solutionux,t,vx,t=φξ,0 of system (1).Letb2=0, so that we have
(8)cφ′+φφ′+φ′″=0.Integrating (8),
(9)cφ+φ22φ″=h1,where h1 is a positive integration constant.Multiply (9) by φ′ and integrate its equation, and we obtain
(10)cφ22+φ36+φ′22=h1φ+h2.Here,h2 is an integration constant.By further calculating, (10) converts
(11)φ′2=13φ−α1φ−α2α3−φ.Besides,α1, α2, α3, and c satisfy the following relation:
(12)α1α2α3+=6h2,α1α3+α2α3+α1α2=−6h1,α1+α2+α3=−3c,α2>c.Letζ=φ/α3, η1=α1/α3, and η2=α2/α3; hence, (11) can be written as
(13)ζ′2=α33ζ−η1ζ−η21−ζ,where η1=α1/α3 and η2=α2/α3.Furthermore, we define a new functionχ=χξ, satisfying the relation
(14)ζ=1+η2−1sin2χ.Substituting (14) into (13), through a tedious calculation, we get that
(15)χ′2=α3121−η11−k2sin2χ.Here,k2=1−η2/1−η1,0<k2<1.By (15) and the definition of [14], we obtain that
(16)ζ=1+1−η2cn2α31−η212ξ,k.So that(17)φ=α3+α3−α2cn2α3−α212ξ,k.Furthermore,cn has fundamental period 2Kk, i.e., cnx+2Kk=cnx and Kk is the complete elliptic integral of the first kind, so cn2x+2Kk=cn2x and this traveling wave solution φ have a fundamental period T, as follows:
(18)T=Tα2,α3=43Kkα3−α2.System (1) has the following exact periodic solution:
(19)ux,t=φξ=α3+α3−α2cn2α3−α212ξ,k,vx,t=ψξ=0.


## 3. The Periodic Property of the Exact Periodic Solution of the Coupled KdV Equations

In this section, by the implicit function theorem, the function relation between the periodT of the periodic solution of system (1) and wave velocity is obtained.Fromη1,η2, and α1+α2+α3=−3c, k2 can be rewritten as
(20)k2=α3−α22α3+α2−3c.Therefore,(21)Tα2,α3=43Kkα3−α2.From (12), we can obtain that
(22)α2=−α3+2c+α3+2c2−4α3+3α3c−6h12=gα3,c.so that T=Tα2,α3=Tgα3,c,c.If it exists a fixedL>0,c0>2h1+1/12,α3,0>4c0+23c02−2h1 and satisfies Tc0,α3,0=L, there is a function relation T=Tgα3,c,c=Tc in neighborhoods Uc0 of c0 and Uα3,0 of α3,0.Theorem 1.
If a fixedL>0,c0>2h1+1/12,α3,0>4c0+23c02−2h1 and satisfies Tc0,α3,0=L, there is a neighborhood Uc0 of c0, a neighborhood Uα3,0 of α3,0, and a unique smooth function Γ:Uc0⟶Uα3,0, α3=Tc, T=Tα3,c=Tc, where c∈Uc0,α3∈Uα3,0.Proof.
The idea and method are derived from [13]. Let
(23)Ω=c,α3c>2h1+112,α3>4c+23c2−2h1.
Now we defineΦ:Ω−⟶R by
(24)Φ=Φa3,c=43Kkα3−gα3,c−L.
Moreover,Φα3,0,c0=0,Φ,∂Φ/∂α3,∂Φ/∂c are continuous in Ω. Then, we will prove that ∂Φ/∂α3<0.
Obviously,(25)∂Φ∂α3=232∂K/∂k∂k/∂α3α3−ga3,c+1−∂g/∂α3α3−gα3,c3/2.
By [15], (12), and (20), we can obtain that
(26)∂K∂k>0,∂k∂α3=32kα2−c2α3+α2−3c2>0,∂g∂α3=−1−a3−gα3,c−1/2α3−4c−12<0.
Hence,∂Φ/∂α3<0inΩ. By the implicit function theorem, we can obtain that neighborhood Uc0 of c0, neighborhood Uα3,0,0 of α3,0, and smooth function Γ:Uc0⟶Uα3,0, s.t. α3=Tc, so that T=Tα3,c=Tc.
This completes the proof.


## 4. The Stability of the Periodic Traveling Wave Solution of the Coupled KdV Equations

In this section, we devote to researching the nonlinear stability of the periodic traveling wave solutionU of system (1), which is as follows:
(27)U=u,vt=α3+α3−α2cn2α3−α212ξ,k,0t.


### 4.1. Definition (Orbital Stability)

A periodic traveling wave solutionU of system (1) is orbitally stable. If every ε>0 and there exists a δ>0, then the following assumption holds: If Vt,x∈C0,T;H2 is a solution of system (1) with V0,x−U0,xH2<δ, every t∈0,T satisfies
(28)infγ∈RVx,t−U⋅−γ≤ε.Otherwise, the solution is unstable.We will apply the method which was established and developed by Angulo and Natali in [16] to show the orbital stability of the above periodic traveling wave solution for the coupled KdV equation (1).Initially, system (1) can be expressed as an abstract Hamiltonian system form
(29)dUdt=JE′U,with
(30)U=u,vt,J=∂x00−∂x,and
(31)EU=∫0T−ux22−α3uvxx−α1uv22−α2u22v+rv32b1+v36b1+b2α3b1vuxx+1b1vxx+b2α2b1u2v2+b2α1b1uv22dxis conserved quantity.Moreover, the coupled KdV equation (1) possesses the conserved quantity
(32)FU=14∫0Tu2+v2dx.E′U and F′U denote the variational derivative of EU and FU, respectively.
(33)E′U=−uxx−α3vxx−u22−α1v22rb1v+v22b1+b2b3b1uxx+vxxb1+b2α1b1uv,F′U=uv.Next, we introduce an operatorLu,0 and a scalar function dc, respectively.
(34)Lu,0=E″u,0+cF″u,0=−∂xx−u+u2c00∂xxb1+α1b2b1u,dc=EU+cFU=Eu,0+cFu,0.To be more precise, we have the following set of conditions which guarantees the stability of the periodic traveling solutionU of system (1):
(i)
It exists a nontrivial smooth curve of periodic traveling solutionU for system (1). Moreover, there are functionals EU and FU which are conserved for system (1)
(ii)
E′U+cF′U=0
(iii)
The operatorLu,0 has a unique negative eigenvalue, which is simple. Furthermore, The remainder of the spectrum consists of a discrete and positive set of eigenvalues
(iv)
d″c is positive in a neighborhood of c.Next, we focus on verifying these conditions hold. Firstly, in Section2, we have obtained that system (1) exists a periodic traveling solution (19) and conserves quantities EU and FU. Therefore, condition (i) holds. Moreover, substituting (19) into E′U+cF′U, we can easily get that
(35)E′U+cF′U=0.Hence, condition (ii) holds.Next, we study the spectral properties associated to the linear operatorLu,0=E″u,0+cF″u,0 determined by the periodic solution U. We will show that the spectrum of Lu,0 has simple negative eigenvalue, zero is simple eigenvalue, and the rest of the spectrum are away from zero.In the first place, we calculate the quadratic form of the operatorLu,0, for u,0∈H10,L×H10,L, by
(36)QL,2=Lu,0,u,0=∫0L−∂x2u−u3+u3c2dx.It represents the quadratic form of the operator(37)L1=−∂x2−u21−c2.From (8), we can get that L1φ′=0, so that zero is an eigenvalue whose eigenfunction is φ′. We will give the behavior of the eigenvalues related to operator L1 in the following theorem.Theorem 2.
Letφ be the cnoidal periodic traveling wave solution given by (17), then operator L1 defined in Lper20,L with domain Hper20,L has one simple negative eigenvalue, zero is simple eigenvalue, and the rest of spectrum are away from zero.Proof.
The idea and method are derived from [13, 17].
In fact,L1 is a self-adjoint operator. Since Hper20,L is compactly embedded in Lper20,L, operator L1 has a compact resolvent, so that the essential spectrum of L1 is an empty set, and the discrete spectrum of operator L1 consists of isolated eigenvalues with finite algebraic multiplicities. Consequently, operator L1 has only point spectrum. Because L1φ′=0, by the oscillation theorem in [18], the eigenvalue zero is the second or third eigenvalue of L1. Next, we will prove that zero is the second one.
The periodic eigenvalue problem associated with (37) is given by
(38)L1ν=λν,ν0=νL,ν′0=ν′L.
LetΨx=v,u2=2−4k2sn2x;k/2−c. We get the equivalent of problem (38):
(39)d2Ψdx2+1−2k2sn2x,kΨ=λ~Ψ,Ψ0=ΨL,Ψ′0=Ψ′L.
Here,λ~=−λ. Moreover, (39) with the Jacobian form is the Lam’e differential equation. According to the Floquet theory, we can get that (39) has two intervals of instability, so that the first three eigenvalues λ1~,λ2~,λ3~ of (39) are simple.
By some tedious computation, we obtain thatλ1~=k2−1 is a negative eigenvalue; λ2~ is a zero eigenvalue, λ3~=1−k2 is a positive eigenvalue.
This completes the proof.Furthermore, let us consider the eigenvalue equation:(40)Lu,0u0=θiu0,where i=1,2,3. It is clear that the smallest eigenvalue θ1 associated with operator Lu,0 is negative. Now, we will show that the next eigenvalue of Lu,0 is θ2(θ2=0, which is simple. The third eigenvalue, θ3, is strictly positive. By min−max characterization in [19], we have that
(41)θ2=maxψ1,ψ2∈Zming,h∈ZQL,2g2+h2,where Z=Hper20,L×Hper20,L, g,ψ1+h,ψ2=0. If we consider ψ1=v,ψ2=0, we can get that
(42)θ2≥ming,h∈ZQL,2g2+h2≥0.so that θ2=0.Next, we prove thatθ3>0. Through taking the two-dimensional subspace spanned by (v,0) and (φ′ 0), since g⊥v,g⊥φ′, therefore,
(43)θ3≥minv,0,,φ′,0∈ZQL,2v2+φ′2>0,where θ3 the third eigenvalue related to Lu,0 is obviously positive.So, we deduce that operatorLu,0 has one negative eigenvalue and one zero eigenvalue, and the rest of the spectrum is positive and bounded away from zero, which proves the condition (iii) holds.In the following, we will proved″c>0. In fact,
(44)d′c=dUdcE′U+cF′U+FU=FU.Furthermore,(45)d″c=F′UdUdc=14∫0Tudξ⋅dudc=14α3T+23α3−α2∫02Kcn2x;kdxdudc=16K21+k2+k2T2c4Tk2+4KTE−k′Kdudc,where
(46)dudc=−32K21+k2kTdTdc−96KT3dTdc⋅cn2α3−α212ξ,k−2α3−α2cnα3−α212ξ,k⋅snα3−α212ξ,k⋅dnα3−α212ξ,k⋅2dTdc−tα3−α212+1.In Section3, we have calculated dT/dc<0; obviously, du/dc<0, so that condition (iv) holds.
(47)d″c>0.So, an application of the orbital stability theorem in [16] gives the following theorem:Theorem 3.
If conditions (i), (ii), (iii), and (iv) hold, the periodic solution (19) of system (1) is orbitally stable.


## 4.1. Definition (Orbital Stability)

A periodic traveling wave solutionU of system (1) is orbitally stable. If every ε>0 and there exists a δ>0, then the following assumption holds: If Vt,x∈C0,T;H2 is a solution of system (1) with V0,x−U0,xH2<δ, every t∈0,T satisfies
(28)infγ∈RVx,t−U⋅−γ≤ε.Otherwise, the solution is unstable.We will apply the method which was established and developed by Angulo and Natali in [16] to show the orbital stability of the above periodic traveling wave solution for the coupled KdV equation (1).Initially, system (1) can be expressed as an abstract Hamiltonian system form
(29)dUdt=JE′U,with
(30)U=u,vt,J=∂x00−∂x,and
(31)EU=∫0T−ux22−α3uvxx−α1uv22−α2u22v+rv32b1+v36b1+b2α3b1vuxx+1b1vxx+b2α2b1u2v2+b2α1b1uv22dxis conserved quantity.Moreover, the coupled KdV equation (1) possesses the conserved quantity
(32)FU=14∫0Tu2+v2dx.E′U and F′U denote the variational derivative of EU and FU, respectively.
(33)E′U=−uxx−α3vxx−u22−α1v22rb1v+v22b1+b2b3b1uxx+vxxb1+b2α1b1uv,F′U=uv.Next, we introduce an operatorLu,0 and a scalar function dc, respectively.
(34)Lu,0=E″u,0+cF″u,0=−∂xx−u+u2c00∂xxb1+α1b2b1u,dc=EU+cFU=Eu,0+cFu,0.To be more precise, we have the following set of conditions which guarantees the stability of the periodic traveling solutionU of system (1):
(i)
It exists a nontrivial smooth curve of periodic traveling solutionU for system (1). Moreover, there are functionals EU and FU which are conserved for system (1)
(ii)
E′U+cF′U=0
(iii)
The operatorLu,0 has a unique negative eigenvalue, which is simple. Furthermore, The remainder of the spectrum consists of a discrete and positive set of eigenvalues
(iv)
d″c is positive in a neighborhood of c.Next, we focus on verifying these conditions hold. Firstly, in Section2, we have obtained that system (1) exists a periodic traveling solution (19) and conserves quantities EU and FU. Therefore, condition (i) holds. Moreover, substituting (19) into E′U+cF′U, we can easily get that
(35)E′U+cF′U=0.Hence, condition (ii) holds.Next, we study the spectral properties associated to the linear operatorLu,0=E″u,0+cF″u,0 determined by the periodic solution U. We will show that the spectrum of Lu,0 has simple negative eigenvalue, zero is simple eigenvalue, and the rest of the spectrum are away from zero.In the first place, we calculate the quadratic form of the operatorLu,0, for u,0∈H10,L×H10,L, by
(36)QL,2=Lu,0,u,0=∫0L−∂x2u−u3+u3c2dx.It represents the quadratic form of the operator(37)L1=−∂x2−u21−c2.From (8), we can get that L1φ′=0, so that zero is an eigenvalue whose eigenfunction is φ′. We will give the behavior of the eigenvalues related to operator L1 in the following theorem.Theorem 2.
Letφ be the cnoidal periodic traveling wave solution given by (17), then operator L1 defined in Lper20,L with domain Hper20,L has one simple negative eigenvalue, zero is simple eigenvalue, and the rest of spectrum are away from zero.Proof.
The idea and method are derived from [13, 17].
In fact,L1 is a self-adjoint operator. Since Hper20,L is compactly embedded in Lper20,L, operator L1 has a compact resolvent, so that the essential spectrum of L1 is an empty set, and the discrete spectrum of operator L1 consists of isolated eigenvalues with finite algebraic multiplicities. Consequently, operator L1 has only point spectrum. Because L1φ′=0, by the oscillation theorem in [18], the eigenvalue zero is the second or third eigenvalue of L1. Next, we will prove that zero is the second one.
The periodic eigenvalue problem associated with (37) is given by
(38)L1ν=λν,ν0=νL,ν′0=ν′L.
LetΨx=v,u2=2−4k2sn2x;k/2−c. We get the equivalent of problem (38):
(39)d2Ψdx2+1−2k2sn2x,kΨ=λ~Ψ,Ψ0=ΨL,Ψ′0=Ψ′L.
Here,λ~=−λ. Moreover, (39) with the Jacobian form is the Lam’e differential equation. According to the Floquet theory, we can get that (39) has two intervals of instability, so that the first three eigenvalues λ1~,λ2~,λ3~ of (39) are simple.
By some tedious computation, we obtain thatλ1~=k2−1 is a negative eigenvalue; λ2~ is a zero eigenvalue, λ3~=1−k2 is a positive eigenvalue.
This completes the proof.Furthermore, let us consider the eigenvalue equation:(40)Lu,0u0=θiu0,where i=1,2,3. It is clear that the smallest eigenvalue θ1 associated with operator Lu,0 is negative. Now, we will show that the next eigenvalue of Lu,0 is θ2(θ2=0, which is simple. The third eigenvalue, θ3, is strictly positive. By min−max characterization in [19], we have that
(41)θ2=maxψ1,ψ2∈Zming,h∈ZQL,2g2+h2,where Z=Hper20,L×Hper20,L, g,ψ1+h,ψ2=0. If we consider ψ1=v,ψ2=0, we can get that
(42)θ2≥ming,h∈ZQL,2g2+h2≥0.so that θ2=0.Next, we prove thatθ3>0. Through taking the two-dimensional subspace spanned by (v,0) and (φ′ 0), since g⊥v,g⊥φ′, therefore,
(43)θ3≥minv,0,,φ′,0∈ZQL,2v2+φ′2>0,where θ3 the third eigenvalue related to Lu,0 is obviously positive.So, we deduce that operatorLu,0 has one negative eigenvalue and one zero eigenvalue, and the rest of the spectrum is positive and bounded away from zero, which proves the condition (iii) holds.In the following, we will proved″c>0. In fact,
(44)d′c=dUdcE′U+cF′U+FU=FU.Furthermore,(45)d″c=F′UdUdc=14∫0Tudξ⋅dudc=14α3T+23α3−α2∫02Kcn2x;kdxdudc=16K21+k2+k2T2c4Tk2+4KTE−k′Kdudc,where
(46)dudc=−32K21+k2kTdTdc−96KT3dTdc⋅cn2α3−α212ξ,k−2α3−α2cnα3−α212ξ,k⋅snα3−α212ξ,k⋅dnα3−α212ξ,k⋅2dTdc−tα3−α212+1.In Section3, we have calculated dT/dc<0; obviously, du/dc<0, so that condition (iv) holds.
(47)d″c>0.So, an application of the orbital stability theorem in [16] gives the following theorem:Theorem 3.
If conditions (i), (ii), (iii), and (iv) hold, the periodic solution (19) of system (1) is orbitally stable.



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