On the Instability of a Class of Periodic Travelling Wave Solutions of the Modified Boussinesq Equation

Abstract

This paper is concerned with instability of periodic travelling wave solutions of the modified Boussinesq equation. Periodic travelling wave solutions with a fixed fundamental period L will be constructed by using Jacobi's elliptic functions. It will be shown that these solutions, called dnoidal waves, are nonlinearly unstable in the energy space for a range of their speeds of propagation and periods.

1. Introduction

The original Boussinesq equations are among the classical models for the propagation of small amplitude, planar long waves on the surface of water [1, 2]. These equations possess special travelling wave solutions known as Scott Russel’s solitary waves or solitons [3, 4], cnoidal waves [5], and dnoidal waves ([6], Section 3 below). The cnoidal and dnoidal wave solutions are periodic travelling waves written in terms of the Jacobi elliptic functions.

Our purpose is to investigate the nonlinear stability of periodic travelling wave solutions 𝜙(𝑥−𝑐𝑡) of the modified Boussinesq equation𝑢_𝑡𝑡−𝑢_𝑥𝑥+(𝑢³+𝑢_𝑥𝑥)_𝑥𝑥=0. (1.1) The above equation (1.1), has the following equivalent form as a Hamiltonian system𝑢_𝑡=𝑣_𝑥,𝑣_𝑡=(𝑢−𝑢_𝑥𝑥−𝑢³)_𝑥 (1.2) for 𝑥∈ℝ, 𝑡>0. Here subscripts 𝑡 and 𝑥 denote partial differentiation with respect to 𝑡 and 𝑥.

The above equation conserves energy, namely, the integral1𝐻(𝑢,𝑣)=2^𝐿₀(𝑢²+𝑣²+𝑢²_𝑥−𝑢⁴2)𝑑𝑥 (1.3) does not depend on the time 𝑡. Another conservation law is the momentum𝐼(𝑢,𝑣)=^𝐿₀𝑢𝑣𝑑𝑥 (1.4) which turns out to be a relevant quantity in the investigation of stability properties of travelling waves.

To make precise the notion of stability we use, let 𝜏_𝑠 be the translation by 𝑠, 𝜏_𝑠𝜙(𝑥)=𝜙(𝑥+𝑠) for 𝑥∈ℝ and let 𝜙_𝑐=(𝜙_𝑐(𝑥−𝑐𝑡),𝜓_𝑐(𝑥−𝑐𝑡)) be an 𝐿-periodic travelling wave solution to system (1.2), where 𝜙_𝑐∶ℝ→ℝ, 𝜓_𝑐∶ℝ→ℝ, 𝐿>0 is the period of 𝜙_𝑐 and 𝜓_𝑐, and 𝑐 is the wave’s speed of propagation. If we define the 𝜙_𝑐-orbit to be the set Ω_𝜙𝑐={𝜙_𝑐(⋅+𝑠),𝑠∈ℝ}, 𝜙_𝑐 is called orbitally stable if profiles near its orbit remain near the orbit for as long as it exists.

So, we have the following definition. Let 𝑋 be a Hilbert space.

Definition 1.1 (Orbital Stability). Let 𝜙_𝑐=(𝜙_𝑐(𝑥−𝑐𝑡),𝜓_𝑐(𝑥−𝑐𝑡))∈𝑋 be an 𝐿-periodic travelling wave solution to system (1.2). We say that the orbit Ω_𝜙𝑐 is stable in the 𝑋-sense by the flow of system (1.2) if for each 𝜖>0 there exists 𝛿=𝛿(𝜖)>0 such that if 𝑢₀∈𝑋 and inf_𝑠∈ℝ‖𝑢₀−𝜏_𝑠(𝜙_𝑐)‖_𝑋<𝛿, then the solution 𝑢(𝑡) of (1.2) with 𝑢(0)=𝑢₀ satisfies, for all 𝑡 for which 𝑢=(𝑢,𝑣) exists, inf_𝑠∈ℝ‖𝑢(𝑡)−𝜏_𝑠(𝜙_𝑐)‖_𝑋<𝜖. (1.5) Otherwise, we say that Ω_𝜙𝑐 is 𝑋-unstable.

Here, 𝑋∶=𝐻¹_per([0,𝐿])×𝐿²_per([0,𝐿]). (The choice of norm in (1.5) is dictated by the form of the Hessian or “linearized Hamiltonian” 𝐻^′′(𝜙_𝑐)+𝑐𝐼^′′(𝜙_𝑐) and varies from problem to problem.)

Inserting the 𝐿-periodic travelling wave solution 𝜙_𝑐=(𝜙_𝑐(𝑥−𝑐𝑡),𝜓_𝑐(𝑥−𝑐𝑡)) in (1.2) leads to the system −𝑐𝜙^′_𝑐(𝜉)=𝜓^′_𝑐(𝜉),−𝑐𝜓^′_𝑐(𝜉)=(𝜙_𝑐−𝜙_𝑐^′′−𝜙³_𝑐)^′(𝜉), (1.6) where ′ connotes 𝑑/𝑑𝜉 and 𝜉=𝑥−𝑐𝑡. Integrating the latter system, we obtain the nonlinear system −𝑐𝜙(𝜉)=𝜓(𝜉)+𝐾₁,−𝑐𝜓(𝜉)=𝜙(𝜉)−𝜙^′′(𝜉)−𝜙³(𝜉)+𝐾₂, (1.7) where 𝐾₁,𝐾₂ are integration constants, which will be considered equal to zero here. Then, we obtain(𝐻^′+𝑐𝐼^′)(𝜙_𝑐)=0. (1.8) Next observe that relation (1.8) characterizes 𝜙_𝑐=(𝜙_𝑐,𝜓_𝑐) as a critical point of 𝐻 subject to the constraint 𝐼(𝑢,𝑣)=𝐼(𝜙_𝑐,𝜓_𝑐). In order to prove instability for 𝜙_𝑐, we will examine the relation between the concavity properties of the function𝑑(𝑐)=𝐻(𝜙_𝑐(⋅))+𝑐𝐼(𝜙_𝑐(⋅)), (1.9) and the properties of the functional 𝐻 near the critical point 𝜙_𝑐 under the constraint 𝐼=constant.

Bona and Sachs in [3] proved that the well-known solitary waves 𝜙=(𝜙_𝑐(𝑥−𝑐𝑡),𝜓_𝑐(𝑥−𝑐𝑡)) of the generalized Boussinesq equation𝑢_𝑡=𝑣_𝑥,𝑣_𝑡=(𝑢−𝑢_𝑥𝑥−𝑢^𝑝)_𝑥 (1.10) are stable in the 𝐻¹(ℝ)×𝐿²(ℝ) norm for speeds 𝑐 such that (𝑝−1)/4<𝑐²<1 if 𝑑 in (1.9) is a convex function of 𝑐. The aim of this paper is to prove that the solutions given by Theorem 3.2 below are unstable if 𝑑(𝑐) is concave. The proof follows the main ideas of Liu [4] (see also Bona et al. in [7]). Differently from the solitary wave solutions case, we do not know explicit periodic travelling wave solutions in the 𝑥-variable for the system (1.10) for every 𝑝. For this reason, we will treat here only the case 𝑝=3. Stability of dnoidal waves for this case is also treated by the author in a forthcoming paper [6]. Regarding the classical case 𝑝=2, in [5] the author proves nonlinear stability properties of a class of 𝐿-periodic travelling wave solutions, called cnoidal waves, in the energy space 𝐻¹_per([0,𝐿])×𝐿²_per([0,𝐿]), by periodic disturbances with period 𝐿.

In this paper, we first show the existence of a smooth curve 𝑐↦𝜙_𝑐=(𝜙_𝑐,𝜓_𝑐) of dnoidal wave solutions to system (1.2), with a fixed period 𝐿 (Theorem 3.2 below). Then, a proof of orbital instability of these solutions is established in 𝑋 for a certain range of their speeds of propagation and periods, based on a modification of the general procedure of [8]. More precisely, our main result regarding stability of the dnoidal waves 𝜙_𝑐, given by Theorem 3.2 below, is the following.

Theorem 1.2 (Instability Theorem). Let 𝑐∈(−1,1) and 𝐿>𝜋√2. Then the orbit Ω_𝜙𝑐 is 𝑋-unstable with respect to the flow of the modified Boussinesq equation, provided 𝑐²<1/2 and 1−𝑐²>2𝜋²/𝐿².

The plan of this paper is as follows. A discussion of the evolution equation (1.1) and its natural invariants is given in Section 2. In Section 3 we introduce a smooth family {𝜙_𝑐}_𝑐 on the parameter 𝑐, of positive dnoidal wave solutions to system (1.2), with a fixed period 𝐿 (Theorem 3.2 below) and in Section 4 we present a complete study of the spectrum of the operator ℒ_𝑐. The existence of the smooth curve 𝑐↦𝜙_𝑐 will allow us to differentiate the function 𝑑(𝑐). Then, in section 5, we prove that 𝑑(𝑐) is indeed concave, for a certain range of speeds and periods of 𝜙_𝑐, which will imply our result. In Section 6 the Lyapunov functional [8, 9] is constructed and the instability result is proved. In Appendix, we give a review of those results about Jacobian elliptic functions which we use throughout the paper.

We remark that orbital instability of 𝜙 is established with respect to perturbations of periodic functions of the same period 𝐿 in 𝑋.

The following notation will be used: ⟨𝑓,𝑔⟩₀=⟨𝑓,𝑔⟩_{𝐿2per([0,𝐿])}=^𝐿₀𝑓𝑔𝑑𝑥,⟨𝑓,𝑔⟩₁=⟨𝑓,𝑔⟩_{𝐻1per([0,𝐿])}=^𝐿₀𝑓𝑔𝑑𝑥+^𝐿₀𝑓^′𝑔^′𝑑𝑥,‖𝑓‖₀=‖𝑓‖_{𝐿2per([0,𝐿])}=(^𝐿₀𝑓²𝑑𝑥)^1/2,‖𝑓‖₁=‖𝑓‖_{𝐻1per([0,𝐿])}=(^𝐿₀𝑓²𝑑𝑥+^𝐿₀𝑓^′2𝑑𝑥)^1/2,⟨(𝑓,𝑔),(𝑢,𝑣)⟩=⟨(𝑓,𝑔),(𝑢,𝑣)⟩_{𝐿2per([0,𝐿])×𝐿2per([0,𝐿])}=^𝐿₀𝑓𝑢𝑑𝑥+^𝐿₀𝑔𝑣𝑑𝑥,‖(𝑓,𝑔)‖=‖(𝑓,𝑔)‖_{𝐿2per([0,𝐿])×𝐿2per([0,𝐿])}=(^𝐿₀𝑓²𝑑𝑥+^𝐿₀𝑔²𝑑𝑥)^1/2,‖(𝑓,𝑔)‖_𝑋=‖(𝑓,𝑔)‖_{𝐻1per([0,𝐿])×𝐿2per([0,𝐿])}=(^𝐿₀𝑓²𝑑𝑥+^𝐿₀𝑓^′2𝑑𝑥+^𝐿₀𝑔²𝑑𝑥)^1/2. (1.11)

2. The Evolution Equation

The next lemma is the periodic version of a particular case of [4, Lemma 1.1].

Lemma 2.1. Let 𝑢₀=(𝑢₀,𝑣₀)∈𝑋≡𝐻¹_per([0,𝐿])×𝐿²_per([0,𝐿]). Then there exist 𝑇>0 and a uniquely weak solution 𝑢=(𝑢,𝑣) of (1.2) with 𝑢(0)=𝑢₀.

Proof. In order to obtain the existence of weak solutions for the system (1.2), we consider the approximate problem 𝑢_𝑡𝑢+𝐴𝑢=𝐹(𝑢),𝑢(0)=^𝑛₀, (2.1) with 𝑢^𝑛₀∈𝐷(𝐴)=𝐻³_per([0,𝐿])×𝐻¹_per([0,𝐿]) and 𝑢^𝑛₀→𝑢₀ in 𝑋, where 𝐴=0−𝜕_𝑥−𝜕_𝑥+𝜕³_𝑥0 (2.2) and −𝐴 is the infinitesimal generator of a 𝐶⁰ group of unitary operators in 𝑋 and 𝐹=𝐹(𝑡,𝑢,𝑣)=(⁰_{−𝜕𝑥(𝑢}³₎). Since 𝐹∈𝐶^∞, the map (𝑢,𝑣)→(0,𝜕_𝑥(𝑢³)) is locally Lipschitz on 𝑋. But then for all 𝑢^𝑛₀∈𝐷(𝐴), there exists a 𝑇_𝑛>0 such that the initial value problem (2.1) has a unique solution 𝑢^𝑛∈𝐶([0,𝑇_𝑛);𝐷(𝐴))∩𝐶¹([0,𝑇_𝑛);𝑋). Moreover, if 𝑇_𝑛<∞, then lim_{𝑡→𝑇𝑛}‖𝑢^𝑛(𝑡)‖_𝑋=∞, (2.3) by the semigroup theory [10]. By (2.1), we estimate on [0,𝑇_𝑛)12𝑑‖𝑢𝑑𝑡^𝑛(𝑡)‖²_𝑋=⟨𝑢^𝑛(𝑡),𝜕_𝑡𝑢^𝑛(𝑡)⟩_𝑋=⟨𝑢^𝑛,−𝐴𝑢^𝑛+𝐹(𝑢^𝑛)⟩_𝑋≤|^𝐿₀𝜕_𝑥(𝑢^𝑛)³(𝑡)⋅𝑣^𝑛(𝑡)𝑑𝑥|≤‖𝜕_𝑥(𝑢^𝑛)³(𝑡)‖₀‖𝑣^𝑛(𝑡)‖₀=3(^𝐿₀|(𝑢^𝑛(𝑡))²(𝑢^𝑛(𝑡))_𝑥|²𝑑𝑥)^1/2‖𝑣^𝑛(𝑡)‖₀≤3‖𝑢^𝑛(𝑡)‖²_∞(^𝐿₀(𝑢^𝑛_𝑥(𝑡))²𝑑𝑥)^1/2‖𝑣^𝑛(𝑡)‖₀≤3𝑆(𝐿)‖𝑢^𝑛(𝑡)‖²₁‖𝑢^𝑛_𝑥‖₀‖𝑣^𝑛(𝑡)‖₀≤3𝑆(𝐿)‖𝑢^𝑛(𝑡)‖²_𝑋‖𝑢^𝑛‖²_𝑋, (2.4) where we used in the first equality above that −𝐴𝑢^𝑛∈𝑋.
Consider now 𝑓(𝑠)=3𝑆(𝐿)𝑠, which is a continuous, positive and increasing function on ℝ⁺. Then by Gronwall’s inequality, it follows that ‖𝑢^𝑛(𝑡)‖²_𝑋≤‖𝑢^𝑛₀‖²_𝑋exp[^𝑡₀𝑓(‖𝑢^𝑛(𝜏)‖²_𝑋)𝑑𝜏],on[0,𝑇_𝑛). (2.5)
We compare ‖𝑢^𝑛(𝑡)‖²_𝑋 with the maximal solution 𝑦(𝑡)𝑦(𝑡)≡1(sup_𝑛‖𝑢^𝑛₀‖_𝑋)²1−3𝑆𝑡(sup_𝑛‖𝑢^𝑛₀‖_𝑋)²,𝑡∈[0,𝑇₀1)≡[0,3𝑆(sup_𝑛‖𝑢^𝑛₀‖_𝑋)²) (2.6) of the scalar Cauchy problem 𝑑𝑦=𝑑𝑡𝑓(𝑦)𝑦,𝑦(0)=𝑦₀=sup_𝑛‖𝑢^𝑛₀‖²_𝑋. (2.7) It follows that ‖𝑢^𝑛(𝑡)‖²_𝑋≤𝑦(𝑡),on[0,𝑇_𝑛)∩[0,𝑇₀). (2.8)
Let 𝑇<𝑇₀. Then 𝑢^𝑛 is defined on [0,𝑇] for all 𝑛. Moreover, ‖𝑢^𝑛(𝑡)‖²_𝑋≤𝐶₀𝑦₀=𝐾² (2.9) on [0,𝑇], where 𝐾 is a constant independent of 𝑛, since by (2.5), (2.8), and the fact that 𝑦(𝑡) is bounded on [0,𝑇], we have the following inequality on [0,𝑇]: ‖𝑢^𝑛(𝑡)‖²_𝑋≤‖𝑢^𝑛₀‖²_𝑋exp[^𝑡₀𝑓(‖𝑢^𝑛(𝜏)‖²_𝑋𝑢)𝑑𝜏]≤‖^𝑛₀‖²_𝑋exp[^𝑇₀𝑓(𝑦(𝜏))𝑑𝜏]≤𝐶₀(𝑇)‖𝑢^𝑛₀‖²_𝑋≤𝐶₀(𝑇)𝑦₀. (2.10)
Finally, from (2.9) and standard weak limit arguments, we have the existence of a unique solution 𝑢(𝑡)∈𝐶([0,𝑇];𝑋).

Proposition 2.2. The unique solution 𝑢(𝑡) of (1.2) with initial data 𝑢(0)=𝑢₀, which is given by Lemma 2.1, satisfies 𝐻(𝑢(𝑡))=𝐻(𝑢,𝑣)=constant,0,𝐼(𝑢(𝑡))=𝐼(𝑢,𝑣)=constant,0. (2.11)

The proof is elementary.

3. Existence of a Smooth Curve of Dnoidal Wave Solutions with a Fixed Period 𝐿 for the System (1.2)

This section is devoted to establish the existence of a smooth curve of periodic travelling wave solutions for the system (1.2), which are solutions of the form𝑢(𝑥,𝑡)=(𝑢(𝑥,𝑡),𝑣(𝑥,𝑡))=(𝜙(𝑥−𝑐𝑡),𝜓(𝑥−𝑐𝑡)). (3.1)

Substituting (3.1) in (1.2) leads to the system−𝑐𝜙^′(𝜉)=𝜓^′(𝜉),−𝑐𝜓^′(𝜉)=(𝜙−𝜙^′′−𝜙³)^′(𝜉), (3.2) where ′ denotes 𝑑/𝑑𝜉 and 𝜉=𝑥−𝑐𝑡. Integrating (3.2), we obtain the nonlinear system−𝑐𝜙(𝜉)=𝜓(𝜉)+𝐾₁,−𝑐𝜓(𝜉)=𝜙(𝜉)−𝜙^′′(𝜉)−𝜙³(𝜉)+𝐾₂, (3.3) where 𝐾₁,𝐾₂ are integration constants, which will be considered equal to zero here. Then, 𝜙 must satisfy𝜙^′′−𝑤𝜙+𝜙³=0, (3.4) where 𝑤=𝑤(𝑐)=1−𝑐² will be considered positive.

Next, we show how to construct a smooth curve of solutions for (3.4) with a fixed fundamental period 𝐿, and depending on the parameter 𝑐. In order to do this, we first observe from (3.4) that 𝜙 satisfies the first-order equation (𝜙^′)²=12[−𝜙⁴+2𝑤𝜙²+4𝐵_𝜙1]=2(𝜂²₁−𝜙²)(𝜙²−𝜂²₂), (3.5) where 𝐵_𝜙 is an integration constant and −𝜂₁, 𝜂₁, −𝜂₂, 𝜂₂ are the real zeros of the polynomial 𝑝_𝜙(𝑡)=−𝑡⁴+2𝑤𝑡²+4𝐵_𝜙, which satisfy the relations2𝑤=𝜂²₁+𝜂²₂,4𝐵_𝜙=−𝜂²₁𝜂²₂. (3.6) Moreover, we assume without lost of generality that 𝜂₁>𝜂₂>0 and we obtain from (3.5) that 𝜂₂≤𝜙≤𝜂₁. By defining 𝜑=𝜙/𝜂₁ and 𝑘²=(𝜂²₁−𝜂²₂)/𝜂²₁, (3.5) becomes (𝜑^′)²=(𝜂²₁/2)(1−𝜑²)(𝜑²−1+𝑘²). We also impose the crest of the wave to be at 𝜉=0, that is, 𝜙(0)=1. Now, we define a further variable 𝜓 via the relation 𝜑²=1−𝑘²sin²𝜓 and so we get that (𝜓^′)²=(𝜂²₁/2)(1−𝑘²sin²𝜓). Then we obtain for 𝑙=𝜂₁/√2 that ∫₀^𝜓(𝜉)(𝑑𝑡/1−𝑘²sin²𝑡)=𝑙𝜉. Therefore, from the definition of the Jacobian elliptic function 𝑦=sn(𝑢;𝑘) (see in the appendix or in Byrd and Friedman [11]), we can write the last equality as sin𝜓=sn(𝑙𝜉;𝑘) and hence 𝜑(𝜉)=1−𝑘²sn²(𝑙𝜉;𝑘)=dn(𝑙𝜉;𝑘). Returning to the initial variable, we obtain the called dnoidal wave solution associated to (3.4),𝜙(𝜉)≡𝜙(𝜉;𝜂₁,𝜂₂)=𝜂₁𝜂dn(¹√2𝜉;𝑘) (3.7) with𝑘²=𝜂²₁−𝜂²₂𝜂²₁,𝜂²₁+𝜂²₂=2𝑤,0<𝜂₂<𝜂₁. (3.8) Next, dn has fundamental period 2𝐾, dn(𝑢+2𝐾;𝑘)=dn(𝑢;𝑘), where 𝐾=𝐾(𝑘) represents the complete elliptic integral of the first kind (see appendix); it follows that the dnoidal wave 𝜙 in (3.7) has fundamental period, 𝑇_𝜙, given by𝑇_𝜙≡2√2𝜂₁𝐾(𝑘). (3.9) Now, we show that 𝑇_𝜙>√2𝜋/√𝑤. First, we express 𝑇_𝜙 as a function of 𝜂₂ and 𝑤. In fact, for every 𝜂₂∈(0,√𝑤), there is a unique 𝜂₁∈(√𝑤,√2𝑤) satisfying the first relation in (3.6), namely, 𝜂₁=2𝑤−𝜂²₂. So, from (3.9) we obtain𝑇_𝜙(𝜂₂2√,𝑤)=22𝑤−𝜂²₂𝐾(𝑘(𝜂₂)),with𝑘²(𝜂₂,𝑤)=2𝑤−2𝜂²₂2𝑤−𝜂²₂. (3.10)

Then, by fixing 𝑤>0, we have that 𝑇_𝜙→+∞ as 𝜂₂→0 and 𝑇_𝜙(𝜂₂)→(𝜋√2/√𝑤) as 𝜂₂→√𝑤. So, since the mapping 𝜂₂↦𝑇_𝜙𝑤(𝜂₂) is strictly decreasing (see proof of Proposition 3.1), it follows that 𝑇_𝜙>√2𝜋/√𝑤.

Now, we obtain a dnoidal wave solution with period 𝐿. For 𝑤₀>2𝜋²/𝐿², there is a unique 𝜂_2,0∈(0,√𝑤₀) such that 𝑇_𝜙(𝜂_2,0,𝑤₀)=𝐿. So, for 𝜂_1,0 such that 𝜂²_1,0+𝜂²_2,0=2𝑤₀, the dnoidal wave 𝜙(⋅)=𝜙(⋅,𝜂_1,0,𝜂_2,0) has a fundamental period 𝐿 and satisfies (3.4) with 𝑤=𝑤₀.

By the above analysis the dnoidal wave 𝜙(⋅,𝜂₁,𝜂₂) in (3.7) is completely determined by 𝑤 and 𝜂₂ and will be denoted by 𝜙_𝑤(⋅;𝜂₂) or 𝜙_𝑤.

The next result, which corresponds to Theorem 2.1 and Corollary 2.2 in [12], is concerned with the existence of a smooth curve of dnoidal wave solutions for (3.4).

Proposition 3.1. Let 𝐿>0 be arbitrary but fixed. Consider 𝑤₀>2𝜋²/𝐿² and the unique 𝜂_2,0=𝜂₂(𝑤₀)∈(0,√𝑤₀) such that 𝑇_𝜙𝑤0=𝐿. Then, (1)there exist an interval ℐ(𝑤₀) around 𝑤₀, an interval 𝐽(𝜂_2,0) around 𝜂_2,0, and a unique smooth function Λ∶ℐ(𝑤₀)→𝐽(𝜂_2,0) such that Λ(𝑤₀)=𝜂_2,0 and 2√22𝑤−𝜂²₂𝐾(𝑘)=𝐿, (3.11) where 𝑤∈ℐ(𝑤₀), 𝜂₂=Λ(𝑤), and 𝑘²=𝑘²(𝑤)∈(0,1) is defined by (3.10); (2)the positive dnoidal wave solution in (3.7), 𝜙_𝑤(⋅;𝜂₁,𝜂₂), determined by 𝜂₁≡𝜂₁(𝑤)=2𝑤−𝜂²₂, 𝜂₂≡𝜂₂(𝑤), has fundamental period 𝐿 and satisfies (3.4). Moreover, the mapping 𝑤∈ℐ(𝑤₀)→𝜙_𝑤∈𝐻¹_per([0,𝐿])is a smooth function; (3)ℐ(𝑤₀) can be chosen as (2𝜋²/𝐿²,+∞); (4)the mapping Λ∶(2𝜋²/𝐿²,+∞)→𝐽(𝜂_2,0) is strictly decreasing.

Proof. From this result we conclude the following existence theorem.

Theorem 3.2. Let 𝐿>𝜋√2. Then there exists a smooth curve of dnoidal wave solutions for the system (1.2) in 𝐻^𝑛_per([0,𝐿])×𝐻^𝑚_per([0,𝐿]), 𝑛,𝑚≥0 which satisfy the system (3.3) with integration constants 𝐾₁=𝐾₂=0; this curve is given, for 𝑤(𝑐)=1−𝑐², by 𝑐∈(−1−2𝜋²𝐿²,1−2𝜋²𝐿²)→(𝜙_𝑤(𝑐),𝜓_𝑤(𝑐)). (3.12)
Moreover, 𝜙_𝑐(𝜉)∶=𝜙_𝑤(𝑐)(𝜉)=2𝑤−𝜂²₂dn[(2𝑤−𝜂²₂/√2)𝜉;𝑘],𝜓_𝑤(𝑐)=−𝑐𝜙_𝑤(𝑐), where the smooth function 𝜂₂≡𝜂₂(𝑤(𝑐)) is given by Proposition 3.1 and 𝑘=𝑘(𝑤(𝑐)) by (3.10).

Remark 3.3. 𝜕𝜙_𝑐/𝜕𝑐 is in 𝐻^∞_per([0,𝐿])×𝐻^∞_per([0,𝐿]) as soon as in 𝐻¹_per([0,𝐿])×𝐻¹_per([0,𝐿]). This follows from the equation and a bootstrap argument.

4. Spectral Analysis

In this section, we study the spectral properties associated to the linear operator ℒ_𝑐=𝐻^′′(𝜙_𝑤(𝑐),𝜓_𝑤(𝑐))+𝑐𝐼^′′(𝜙_𝑤(𝑐),𝜓_𝑤(𝑐)) (4.1) determined by the periodic solutions (𝜙_𝑤(𝑐),𝜓_𝑤(𝑐)) found in Theorem 3.2. We compute the Hessian operator ℒ_𝑐 by calculating the associated quadratic form, which is denoted by 𝒬_𝑐. By definition, 𝒬_𝑐(𝑔,ℎ) is the coefficient of 𝜖² in 𝐻(𝜙_𝑤(𝑐)+𝜖𝑔,𝜓_𝑤(𝑐)+𝜖ℎ)+𝑐𝐼(𝜙_𝑤(𝑐)+𝜖𝑔,𝜓_𝑤(𝑐)+𝜖ℎ), (4.2) and so is given by 𝒬_𝑐(𝑔,ℎ)=^𝐿₀{12(𝑔²+𝑔²_𝑥+ℎ²3)−2𝜙²_𝑤(𝑐)𝑔²=+𝑐𝑔ℎ}𝑑𝑥_𝐿0{12[(1−𝑐²)𝑔²+𝑔^′2−3𝜙²_𝑤(𝑐)𝑔²1]+2(ℎ+𝑐𝑔)²𝑑𝑥}∶=𝒬¹_𝑐1(𝑔)+2‖ℎ+𝑐𝑔‖²₀. (4.3) Note that 𝒬_𝑐 is the sum of the quadratic form 𝒬¹_𝑐 associated to the operator −(𝑑²/𝑑𝑥²)+1−𝑐²−3𝜙²_𝑤 and the nonnegative term (1/2)‖ℎ+𝑐𝑔‖²₀. From (3.2) for the dnoidal wave (𝜙_𝑤(𝑐),𝜓_𝑤(𝑐)), it follows that 𝑔=𝜙^′_𝑤(𝑐) and ℎ=𝜓^′_𝑤(𝑐) satisfy ℒ_𝑐(𝑔,ℎ)=0. To see that this is the only eigenfunction corresponding to the eigenvalue zero and the other expected properties of the operator ℒ_𝑐, we will first consider the following result about the periodic eigenvalue problem: ℒ_dn𝑑𝜉∶=(−²𝑑𝑥²+𝑤−3𝜙²_𝑤)𝜉=𝜆,𝜉(0)=𝜉(𝐿),𝜉^′(0)=𝜉^′(𝐿), (4.4) where 𝜙_𝑤 is given by Proposition 3.1.

The following result is a consequence of the Floquet theory (Magnus and Winkler [13]) and can be found in [12].

Theorem 4.1. Let ℒ_dn be the linear operator defined on 𝐻²_per([0,𝐿]) by (4.4). Then the first three eigenvalues 𝛽₁, 𝛽₂, and 𝛽₃ of ℒ_dn are simple, and satisfy 𝛽₁<0=𝛽₂<𝛽₃, and 𝜙^′ is the eigenfunction of 𝛽₂. Moreover, the rest of the spectrum consists of a discrete set of eigenvalues which are double.

To prove that the kernel of ℒ_𝑐 is spanned by (𝑑/𝑑𝑥)(𝜙_𝑤(𝑐),𝜓_𝑤(𝑐)), consider the quadratic form 𝒬_𝑐(𝑔,ℎ) as the pairing of (𝑔,ℎ) against (𝑔,ℎ) in the 𝐻¹_per([0,𝐿])×𝐿²_per([0,𝐿])−𝐻⁻¹_per([0,𝐿])×𝐿²_per([0,𝐿]) duality, where (𝑔,ℎ)^𝑡 is the unbounded operator ℒ_𝑐∶=1−𝜕_𝑥𝑥−3𝜙²_𝑤(𝑐)𝑐𝑐1 (4.5) applied to (𝑔,ℎ)^𝑡. Then ℒ_𝑐(𝑔,ℎ)^𝑡=0 implies −𝑔^′′+(1−𝑐²)𝑔−3𝜙²_𝑤(𝑐)𝑔=0,ℎ=−𝑐𝑔. (4.6) Now, from the properties of the operator ℒ_dn=−𝜕²_𝑥+𝑤−3𝜙²_𝑤 established in Theorem 4.1, it follows that 𝑔=𝜆𝜙^′_𝑤(𝑐) and ℎ=−𝑐𝑔=−𝑐𝜆𝜙^′_𝑤(𝑐)=𝜆𝜓^′_𝑤(𝑐), where 𝜆≠0∈ℝ.

To show that there is a single, simple, negative eigenvalue, consider 𝒬¹_𝑐 defined in (4.3) above. By Theorem 4.1, the operator ℒ_dn has exactly one negative eigenvalue which is simple, say 𝜆₀, with associated eigenfunction 𝜁>0. Thus, 𝒬¹_𝑐 achieves a negative value and so does 𝒬_𝑐. In fact, considering 𝜁=(𝜁,−𝑐𝜁), we have ℒ_𝑐(𝜁)=𝑄¹_𝑐(𝜁)+(1/2)‖𝑐𝜁−𝑐𝜁‖²=𝑄¹_𝑐(𝜁)=(1/2)𝜆₀<0. Denoting by 𝛽₀ the lowest eigenvalue of ℒ_𝑐, we will show that the next eigenvalue 𝛽₁ is 0, which is known to be simple, and consequently 𝛽₂ is in fact strictly positive. These results are proved using the (min-max) Rayley-Ritz characterization of eigenvalues (see [14, 15]), namely, 𝛽₁=max_{(𝜙1,𝜓1)∈𝑋}min_{(𝑔,ℎ)∈𝑋⧵{0}⟨𝑔,𝜙1⟩1+⟨ℎ,𝜓1⟩0=0}𝑄_𝑐(𝑔,ℎ)‖𝑔‖²₁+‖ℎ‖²₀. (4.7) Choosing 𝜙₁=𝜁, 𝜓₁=0, we obtain the lower estimate 𝛽₁≥min_{(𝑔,ℎ)∈𝑋⧵{0}⟨𝑔,𝜁⟩1=0}𝑄_𝑐(𝑔,ℎ)‖𝑔‖²₁+‖ℎ‖²₀. (4.8) The right-hand side of (4.8) is nonnegative on the subspace {(𝑔,ℎ)∈𝑋⧵{0};(𝑔,𝜁)₁=0}, since 𝑄¹_𝑐(𝑔)≥0 by Theorem 4.1. Thus, 𝛽₁=0 and, from earlier considerations, 𝛽₁ is simple and 𝛽₂>0.

The above analysis can be summarized in the form of the following theorem.

Theorem 4.2. Let ℒ_𝑐 be the linear operator defined on 𝐻²_per([0,𝐿])×𝐻¹_per([0,𝐿]) by (4.1). Then the first two eigenvalues 𝛽₀ and 𝛽₁ of ℒ_𝑐 are simple and satisfy 𝛽₀<𝛽₁=0; 𝜁_𝑐=(𝜁_1,𝑐,𝜁_2,𝑐), with 𝜁_1,𝑐>0 and 𝜙^′_𝑐 being the eigenfunctions of 𝛽₀ and 𝛽₁, respectively. Moreover, the rest of the spectrum consists of a discrete set of eigenvalues and the mapping 𝑐→𝜁_𝑐 is continuous with values in 𝐻²_per([0,𝐿])×𝐻¹_per([0,𝐿]).

5. Concavity of 𝑑(𝑐)

Lemma 5.1. Let 𝑐∈(−1,1) and 𝐿>𝜋√2. Then the function 𝑑(𝑐) is concave, provided 𝑐²<1/2 and 1−𝑐²>2𝜋²/𝐿².

Remark 5.2. Relation (1.8) implies that 𝑑^′′(𝑐)<0 is equivalent to the condition 𝑑𝑑𝑐𝐼(𝜙_𝑤(𝑐),𝜓_𝑤(𝑐))<0. (5.1)

Proof of Lemma 5.1. Note that 𝑑𝑑𝑐𝐼(𝜙_𝑤,𝜓_𝑤𝑑)=𝑑𝑐^𝐿₀𝜙_𝑤𝜓_𝑤=−^𝐿₀𝜙²_𝑤𝑑𝑑𝑥−𝑐[𝑑𝑤^𝐿₀𝜙²_𝑤𝑑𝑥]𝑑𝑤𝑑𝑐=−^𝐿₀𝜙²_𝑤𝑑𝑥+2𝑐²𝑑[𝑑𝑤^𝐿₀𝜙²_𝑤𝑑𝑥]. (5.2) Now, 𝑑1𝑑𝑤2^𝐿₀𝜙²_𝑤4𝑑𝑥=𝐿𝑑𝑑𝑘[𝐾(𝑘)𝐸(𝑘)]𝑑𝑘𝑑𝑤>0. (5.3) Indeed, we observe from (3.7), (3.8), and (3.11) that ‖𝜙_𝑤‖²=√2𝜂₁^(𝜂1/√₀^2)𝐿dn²(𝑥;𝑘)𝑑𝑥=8𝐾(𝑘)𝐿^𝐾₀dn²(𝑥;𝑘)𝑑𝑥, (5.4) where we used the fact that the Jacobi elliptic function dn has fundamental period 2𝐾 and is an even function. Now, by using that ∫^𝐾₀cn²(𝑥;𝑘)𝑑𝑥=(1/𝑘²)[𝐸(𝑘)−(𝑘^′)²𝐾(𝑘)] and dn²(𝑥;𝑘)=1−𝑘²+𝑘²cn²(𝑥;𝑘), it follows from (5.4) that 12^𝐿₀𝜙²_𝑤4𝑑𝑥=𝐿𝐾(𝑘)𝐸(𝑘). (5.5) Now, Proposition 3.1 and Theorem 3.2 imply that the map 𝑤→Λ(𝑤)≡𝜂₂(𝑤) is strictly decreasing and from (3.10), with 𝜂₂=𝜂₂(𝑤), we have that 𝑑𝑘=1𝑑𝑤[2𝑘2𝜂²₂−4𝑤𝜂₂𝜂^′₂(2𝑤−𝜂²₂)²]>0. (5.6) Thus, since 𝑘∈(0,1)→𝐾(𝑘)𝐸(𝑘) is strictly increasing (see Appendix), the claim (5.3) follows from (5.5) and (5.6).
So, from (5.2), (5.3), and (5.5), we get 𝑑𝑑𝑐𝐼(𝜙_𝑤,𝜓_𝑤8)=−𝐿𝐾(𝑘)𝐸(𝑘)+16𝑐²𝐿𝑑𝑑𝑘[𝐾(𝑘)𝐸(𝑘)]𝑑𝑘𝑑𝑤. (5.7) Now, considering the function Ψ defined by (2.12) in [12] and using (5.6), we obtain 𝜕Ψ=2√𝜕𝑤22𝑤−𝜂²₂(𝑑𝐾/𝑑𝑤)(𝑘(𝜂₂,𝑤))−2√2𝐾(𝑘(𝜂₂,𝑤))(2𝑤−𝜂²₂)^−(1/2)(2𝑤−𝜂²₂)=2√22𝑤−𝜂²₂(𝑑𝐾/𝑑𝑘)(𝑘(𝜂₂,𝑤))(𝜂²₂/𝑘(2𝑤−𝜂²₂)²)(2𝑤−𝜂²₂)−2√2𝐾(𝑘(𝜂₂,𝑤))(2𝑤−𝜂²₂)^−(1/2)(2𝑤−𝜂²₂), (5.8) hence 𝑑𝑘=1𝑑𝑤2𝑘(2𝑤−𝜂²₂)²{2𝜂²₂−4𝑤[𝑘(2𝑤−𝜂²₂)𝐾−𝜂²₂(𝑑𝐾/𝑑𝑘)𝑘(2𝑤−𝜂²₂)𝐾−2𝑤(𝑑𝐾/𝑑𝑘)]}>0. (5.9)
From (5.7), (5.9), and using that 2𝑤−𝜂²₂=𝜂²₂/(𝑘^′)², we obtain 𝜃(𝐿/8)(𝑑𝐼(𝜙_𝑤,𝜓_𝑤)/𝑑𝑐)=𝜃{−𝐸𝐾+2𝑐²((𝐸²−𝑘^′2𝐾²)/𝑘𝑘^′2)(𝑑𝑘/𝑑𝑤)}=−𝜃𝐸𝐾+2𝑐²(𝐸²−𝑘^′2𝐾²)𝐾(𝜂²₂−2𝑤)(𝑘^′)²=𝐾{−𝜂²₂𝐸(𝑘²𝜂²₂𝐾−2𝑤𝐸+2𝑤𝑘^′2𝐾)}+𝐾{2𝑐²(𝐸²−𝑘^′2𝐾²)(𝑘^′)²(𝜂²₂−2𝑤)}, or equivalently, 𝜃𝐿8𝐾𝑑𝐼(𝜙_𝑤,𝜓_𝑤)𝑑𝑐=−𝜂²₂𝐸(𝑘²𝜂²₂𝐾−2𝑤𝐸+2𝑤𝑘^′2𝐾)+2𝑐²(𝑘^′)²(𝐸²−𝑘^′2𝐾²)(𝜂²₂−2𝑤)=𝜂²₂(−2𝑤𝑘^′2−𝜂²₂𝑘²)𝐸𝐾+(2𝑐²(𝑘^′)²𝜂²₂−4𝑐²𝑤(𝑘^′)²+2𝑤𝜂²₂)𝐸²+2𝑐²(𝑘^′)²𝜂²₂𝐾², (5.10) where 𝜃=𝜂²₂(𝑘²𝜂²₂𝐾−2𝑤𝐸+2𝑤𝑘^′2𝐾)<0.
Now, given that (𝑘^′)²=𝜂²₂/(2𝑤−𝜂²₂), we rewrite the coefficient of 𝐸² in (5.10) as 2𝑐²(𝑘^′)²𝜂²₂−4𝑐²𝑤(𝑘^′)²+2𝑤𝜂²₂=2𝑐²𝜂⁴₂2𝑤−𝜂²₂−4𝑐²𝑤𝜂²₂2𝑤−𝜂²₂+2𝑤𝜂²₂=2𝑐²𝜂⁴₂−4𝑐²𝑤𝜂²₂+4𝑤²𝜂²₂−2𝑤𝜂⁴₂2𝑤−𝜂²₂=𝜂²₂(2𝑤−𝜂²₂)(2𝑤−2𝑐²)2𝑤−𝜂²₂=𝜂²₂(2𝑤−2𝑐²). (5.11) Also, the coefficient of 𝐸𝐾 can be rewritten as 𝜂²₂(−2𝑤𝑘^′2−𝜂²₂𝑘²)=2𝜂⁴₂. Thus, 𝐿8𝐾𝑑𝐼(𝜙_𝑤,𝜓_𝑤)=𝑑𝑐−2𝜂⁴₂𝐸𝐾+2𝜂²₂(𝑤−𝑐²)𝐸²+2𝑐²(𝑘^′)²𝜂²₂𝐾²𝜂²₂(𝑘²𝜂²₂𝐾−2𝑤𝐸+2𝑤𝑘^′2.𝐾) (5.12)
We remark that we can write 𝑤 as a function of complete elliptic integrals. In fact, by integrating (3.4) from 0 to 𝐿, we obtain ∫𝑤=𝑤(𝑐)=^𝐿0𝜙³_𝑤(𝜉)𝑑𝜉∫^𝐿₀𝜙_𝑤(𝜉)𝑑𝜉, (5.13) which is well defined, since the solution 𝜙_𝑤 is positive.
Now, using (3.7), the expression 314.01 in [11], and the fact that 𝐹(𝜋/2;𝑘)=𝐾(𝑘) (see Appendix), we obtain ^𝐿₀𝜙_𝑤(𝜉)𝑑𝜉=^𝐿₀𝜂₁𝜂dn(¹√2𝜉;𝑘)𝑑𝜉=√2^{𝜂1√𝐿/2}₀=√dn(𝑦;𝑘)𝑑𝑦2₀^2𝐾dn(𝑦;𝑘)𝑑𝑦=2√2^𝐾₀dn(𝑦;𝑘)𝑑𝑦=𝜋√2. (5.14) Similarly using (3.7), the expression 314.03 in [11], and the special values sn0=0, and sn𝐾=1, cn𝐾=0 (see Appendix), it follows that ^𝐿₀𝜙³_𝑤(𝜉)𝑑𝜉=^𝐿₀𝜂³₁dn³(𝜂₁√2𝜉;𝑘)𝑑𝜉=√2𝜂²₁₀^2𝐾dn³√(𝑦;𝑘)𝑑𝜉=22𝜂²₁^𝐾₀dn³(𝑦;𝑘)𝑑𝜉=16√2𝐾²𝐿²12[(1+(𝑘^′)²)𝜋2+𝑘²√sn𝐾cn𝐾]=4𝜋2(1+(𝑘^′)²)𝐾²𝐿². (5.15) Substituting (5.14) and (5.15) in (5.13), we deduce that 𝑤(𝑐)=1−𝑐²=4(1+(𝑘^′)²)𝐾²𝐿². (5.16) Using (5.16) and 𝜂²₂=2𝑤(𝑘^′)²/(1+(𝑘^′)²), the numerator of (5.12) will be positive if and only if 𝑐²(𝑘^′)²𝐾²>(𝑐²−𝑤)𝐸²+𝜂²₂𝐸𝐾⇔(1−𝑤)(𝑘^′)²𝐾²>(1−2𝑤)𝐸²+(2𝑤(𝑘^′)²/(1+(𝑘^′)²))𝐸𝐾⇔𝑤[2𝐸²−(𝑘^′)²𝐾²−(2(𝑘^′)²/(1+𝑘^′2))𝐸𝐾]>𝐸²−(𝑘^′)²𝐾²⇔𝑤>((𝐸²−(𝑘^′)²𝐾²)/((2𝐸²−(𝑘^′)²𝐾²)−(2(𝑘^′)²/(1+𝑘^′2))𝐸𝐾)).
Remark 5.3. 2𝐸²−(𝑘^′)²𝐾²−(2(𝑘^′)²/(1+𝑘^′2))𝐸𝐾>0 since the functions 𝐸𝐾 and 𝐸+𝐾 are strictly increasing (see Appendix).Claim 1. lim_𝑘→0𝐸²−(𝑘^′)²𝐾²2𝐸²−(𝑘^′)²𝐾²−(2(𝑘^′)²/(1+𝑘^′2=2))𝐸𝐾5. (5.17)Proof. Indeed, denoting by 𝑓(𝑘)∶=𝐸²−(𝑘^′)²𝐾² and by 𝑔(𝑘)∶=𝐸²−(2(𝑘^′)²/(1+𝑘^′2))𝐸𝐾, we use 𝐿^′Hospital’s rule to find the limit (5.17). Specifically, we show using (A.3) that lim_𝑘→0𝑓^(𝑗)(𝑘)=lim_𝑘→0𝑔^(𝑗)(𝑘)=0,(𝑗=0,1,2,3),lim_𝑘→0𝑓⁽⁴⁾(𝑘)=3𝜋²4,lim_𝑘→0𝑔⁽⁴⁾(𝑘)=9𝜋²8, (5.18) which implies our claim.
Note that, by lim_𝑘→1𝑘^′2𝐾²=0, we have that lim_𝑘→1𝐸²−(𝑘^′)²𝐾²𝐸²−(𝑘^′)²𝐾²+𝐸²−(2(𝑘^′)²/(1+𝑘^′2=1))𝐸𝐾2. (5.19) Moreover, 0<((𝐸²−(𝑘^′)²𝐾²)/(𝐸²−(𝑘^′)²𝐾²+𝐸²−(2(𝑘^′)²/(1+𝑘^′2))𝐸𝐾))<1/2  forall𝑘∈(0,1), since 𝐸²−(2𝑘^′2/(1+𝑘^′2))𝐾²>𝐸²−𝑘^′2𝐾²forall𝑘∈(0,1). In, addition we get (1+𝑘^′2)𝐾>2𝐸, since the function 𝑚(𝑘)∶=(1+𝑘^′2)𝐾−2𝐸 has the following properties: 𝑚(0)=0 and 𝑚^′(𝑘)>0forall𝑘∈(0,1). We conclude that the function 𝑓(𝑘)=𝐸𝑓(𝑘)+𝑔(𝑘)²−(𝑘^′)²𝐾²𝐸²−(𝑘^′)²𝐾²+𝐸²−(2(𝑘^′)²/(1+𝑘^′2))𝐸𝐾 (5.20) is strictly positive on [0,1]. Now, continuity plus (5.17) and (5.19) implies that 𝑐₀∶=max_0≤𝑘≤1(𝑓(𝑘)/(𝑓(𝑘)+𝑔(𝑘))) satisfies 0<𝑐₀≤1/2.
This concludes the lemma.

6. Instability

Consider the function 𝑑(𝑐) defined by (1.9). We now examine the relation between concavity properties of 𝑑 and the properties of the functional 𝐻 near the critical point 𝜙_𝑐 subject to the constraint 𝐼(𝑢)=𝐼(𝜙_𝑐).

Theorem 6.1. Let 𝑐≠0 be fixed. If 𝑑^′′(𝑐)<0, then there exists a curve 𝑤→Φ_𝑤 which satisfies 𝐼(Φ_𝑤)=𝐼(𝜙_𝑐), Φ_𝑐=𝜙_𝑐, and on which 𝐻(𝑢) has a strict local maximum at 𝑢=𝜙_𝑐.

Proof. We follow the ideas of [3, 8, 9]. Let 𝜁_𝑐 be the unique, negative eigenfunction of ℒ_𝑐. Define Φ_𝑤∶=𝜙_𝑤+𝑠(𝑤)𝜁_𝑐, for 𝑤 near 𝑐, where 𝑠(𝑤) satisfies 𝑠(𝑐)=0 and 𝐼(Φ_𝑤)=𝐼(𝜙_𝑐). The function 𝑠(𝑤) can be defined by the implicit function theorem, since 𝜕𝜙𝜕𝑠𝐼(_𝑤+𝑠𝜁_𝑐)|_{{𝑠=0,𝑤=𝑐}}=^𝐿₀(𝜙_𝑐𝜁_2,𝑐+𝜓_𝑐𝜁_1,𝑐)𝑑𝑥, (6.1) where 𝜁_𝑐=(𝜁_1,𝑐,𝜁_2,𝑐) with 𝜁_2,𝑐=(𝑐/(𝛽₀−1))𝜁_1,𝑐, and 𝛽₀ is the unique negative eigenvalue of ℒ_𝑐 and 𝜓_𝑐=−𝑐𝜙_𝑐, 𝜙_𝑐>0. Thus 𝜕𝜙𝜕𝑠𝐼(_𝑤+𝑠𝜁_𝑐)|_{{𝑠=0,𝑤=𝑐}}1=−𝑐(1+1−𝛽₀)^𝐿₀𝜙_𝑐𝜁_1,𝑐𝑑𝑥≠0. (6.2) It is easy to see that 𝑑²𝑑𝑤²𝐼(Φ_𝑤)|_{𝑤=𝑐}=⟨ℒ_𝑐𝑦,𝑦⟩, (6.3) where 𝑦=𝜕Φ_𝑤/𝜕𝑤|_{𝑤=𝑐}=𝜕𝜙_𝑐/𝜕𝑐+𝑠^′(𝑐)𝜁_𝑐. In fact, by some calculations, we have that 𝑑^′′(𝑐)=−𝑠^′(𝑐)⟨𝐼^′(𝜙_𝑐),𝜁_𝑐ℒ⟩,_𝑐𝑦=−𝐼^′(𝜙_𝑐)+𝑠^′(𝑐)ℒ_𝑐𝜁_𝑐, (6.4) so that ⟨ℒ_𝑐𝑦,𝑦⟩=−𝑠^′(𝑐)⟨𝐼^′(𝜙_𝑐),𝜁_𝑐⟩+𝑠^′(𝑐)²⟨𝜁_𝑐,ℒ_𝑐𝜁_𝑐⟩<0 (6.5) in view of the fact that 𝑑^′′(𝑐)<0.

To prove the instability, we need the following lemmas which are proved in [8] and as in the analogous case of [7]; therefore, we state them without proof.

Lemma 6.2. There exist 𝜖>0 and a unique 𝐶¹  map 𝛼∶𝑈_𝜖→ℝ, such that, for any 𝑢=(𝑢,𝑣)∈𝑈_𝜖 and 𝑟∈ℝ, ⟨𝑢(⋅+𝛼(𝑢),𝜕_𝑥𝜙_𝑐𝛼)⟩=0,𝛼(𝑢(⋅+𝑟))=𝛼(𝑢)−𝑟,modulotheperiod,_′(𝜕𝑢)=^𝑥𝜙_𝑐(⋅−𝛼(𝑢))⟨𝑢,𝜕²_𝑥𝜙_𝑐,(⋅−𝛼(𝑢))⟩ (6.6) where 𝑈_𝜖 is the “tube” 𝑈_𝜖={𝑢∈𝑋∶inf_𝑠∈ℝ‖𝑢−𝜏_𝑠(𝜙_𝑐)‖_𝑋<𝜖}. (6.7)

Definition 6.3. For 𝑢∈𝑈_𝜖, define 𝐵(𝑢) by the formula 𝐵(𝑢)=𝑦(⋅−𝛼(𝑢))−𝐾𝜕_𝑥𝛼^′(𝑢), (6.8) where 𝐾=(⁰¹₁₀).

By Lemma 6.2, 𝐵 may also be expressed as 𝐵(𝑢)=𝑦(⋅−𝛼(𝑢))−⟨𝐾𝑦(⋅−𝛼(𝑢)),𝑢⟩⟨𝜕²_𝑥𝜙_𝑐(⋅−𝛼(𝑢)),𝑢⟩𝐾𝜕²_𝑥𝜙_𝑐(⋅−𝛼(𝑢)). (6.9)

Lemma 6.4. 𝐵 is a 𝐶¹ function from 𝑈_𝜖 into 𝑋, 𝐵 commutes with translations, 𝐵(𝜙_𝑐)=𝑦, and ⟨𝐵(𝑢),𝐾𝑢⟩=0, for any 𝑢∈𝑈_𝜖.

Lemma 6.5. There exists a 𝐶¹ functionΠ∶{𝑣∈𝑈_𝜖∶𝐼(𝑣)=𝐼(𝜙_𝑐)}→ℝ (6.10) which is invariant under translations, such that if 𝑣∈𝑈_𝜖 with 𝐼(𝑣)=𝐼(𝜙_𝑐) and 𝑣 not being a translation of 𝜙_𝑐, we have 𝐻(𝜙_𝑐)<𝐻(𝑣)+Π(𝑣)⟨𝐻^′(𝑣),𝐵(𝑣)⟩. (6.11)

Lemma 6.6. The curve 𝑤→Φ_𝑤 constructed in Theorem 6.1 satisfies 𝐻(Φ_𝑤)<𝐻(𝜙_𝑐) for 𝑤≠𝑐, and ⟨𝐻^′(Φ_𝑤),𝐵(Φ_𝑤)⟩ changes sign as 𝑤 passes through 𝑐, with 𝑐≠0.

6.1. Proof of Theorem 1.2

First, consider 𝑐≠0 with |𝑐|<1. Let 𝜖>0 be given small enough. By Lemma 6.6, we can choose 𝑢₀∈𝑋 arbitrarily close to 𝜙_𝑐 such that 𝐼(𝑢₀)=𝐼(𝜙_𝑐), 𝐻(𝑢₀)<𝐻(𝜙_𝑐) e ⟨𝐻^′(𝑢₀),and𝐵(𝑢₀)⟩>0. To prove the instability of 𝜙_𝑐, it suffices to show that there are some elements 𝑢₀∈𝑋 which are close to 𝜙_𝑐 but for which the solution 𝑢 of (1.2) with initial data 𝑢₀ exits from 𝑈_𝜖 in finite time. Let [0,𝑡₁) denote the maximal interval for which 𝑢(⋅,𝑡) lies continuously in 𝑈_𝜖. By Lemma 2.1, 𝑡₁>0. Let 𝑇 be the maximum existence time of solution 𝑢 in (1.1) with initial data 𝑢₀. If 𝑇 is finite, then we have the 𝑋-instability for 𝜙_𝑐 by definition. So we may assume that 𝑇=+∞ and it suffices to show that 𝑡₁<+∞.

In view of Lemma 2.1 and Proposition 2.2, 𝑢 has the following properties: 𝑢∈𝐶([0,𝑡₁];𝑋),𝑢(0,𝑥)=𝑢₀(𝑥), and 𝐼(𝑢(𝑡))and𝐻(𝑢(𝑡)) are constant for 𝑡∈[0,𝑡₁). Let 𝜌(𝑡)=𝛼(𝑢(𝑡)), where 𝛼 is defined by Lemma 6.2 and define 𝑑Φ𝑦(𝑧)=^𝑐=𝑑𝜙𝑑𝑐𝑑𝑐^𝑐+𝑠^′(𝑐)𝜒_𝑐,𝑌(𝑥)=^𝑥₀𝐾𝑦(𝑧)𝑑𝑧,𝐴(𝑡)=^𝐿₀𝑌(𝑥−𝜌(𝑡))⋅𝑢(𝑥,𝑡)𝑑𝑥,for0≤𝑡<𝑡₁,𝛾=^𝐿₀𝐾𝑦(𝑥)𝑑𝑥. (6.12) The function 𝐴(𝑡) serves as a Lyapunov function in our argument.

Now we estimate 𝑑𝐴/𝑑𝑡. By differentiation, 𝑑𝐴𝑑𝑡=−𝜌^′(𝑡)^𝐿₀𝐾𝑦(𝑥−𝜌(𝑡))⋅𝑢(𝑥,𝑡)𝑑𝑥+^𝐿₀𝜕𝑢𝑌(𝑥−𝜌(𝑡))⋅𝜕𝑡(𝑥,𝑡)𝑑𝑥, (6.13) where 𝜕𝑢/𝜕𝑡 is in distribution sense. Since 𝜌^′(𝑡)=𝑑𝛼(𝑢(𝑡))/𝑑𝑡=⟨𝛼^′(𝑢),(𝜕𝑢/𝜕𝑡)⟩, 𝑑𝐴𝑑𝑡=⟨−⟨𝐾𝑦(⋅−𝜌),𝑢⟩𝛼^′(𝜕𝑢𝑢)+𝑌(⋅−𝜌),𝜕𝑡⟩. (6.14)

Since 𝑑𝑢𝑑𝑡=𝐽𝐻^′(𝑢),where𝐽=𝜕_𝑥𝐾,𝜕_𝑥𝑌=𝐾𝑦, (6.15) it follows that 𝑑𝐴𝑑𝑡=⟨−⟨𝐾𝑦(⋅−𝜌(𝑡)),𝑢⟩𝜕_𝑥𝛼^′(𝑢)−𝐾𝑦(⋅−𝜌(𝑡)),𝐾𝐻^′(𝑢)⟩=⟨−⟨𝐾𝑦(⋅−𝜌(𝑡)),𝑢⟩𝐾𝜕_𝑥𝛼^′(𝑢)−𝑦(⋅−𝜌(𝑡)),𝐻^′(𝑢)⟩=−⟨𝐵(𝑢),𝐻^′(𝑢)⟩. (6.16) Since 0<𝐻(𝜙_𝑐)−𝐻(𝑢₀)=𝐻(𝜙_𝑐)−𝐻(𝑢(𝑡)), (6.17) Lemma 6.5 implies that 0<Π(𝑢(𝑡))⟨𝐵(𝑢(𝑡)),𝐻^′(𝑢(𝑡))⟩. (6.18) By ⟨𝐵(𝑢₀),𝐻^′(𝑢₀)⟩>0 and continuity, we obtain ⟨𝐵(𝑢(𝑡)),𝐻^′(𝑢(𝑡))⟩>0 for 0<𝑡<𝑡₁. Moreover, since 𝑢(𝑡)∈𝑈_𝜖 and Π(𝜙_𝑐)=0, we may assume that 0<Π(𝑢(𝑡))<1 for 0<𝑡<𝑡₁, by choosing 𝜖 smaller if necessary. Therefore, for all 𝑡∈[0,𝑡₁), by Lemma 6.5, ⟨𝐵(𝑢(𝑡)),𝐻^′(𝑢(𝑡))⟩≥Π(𝑢(𝑡))⟨𝐵(𝑢(𝑡)),𝐻^′(𝜙𝑢(𝑡))⟩>𝐻(_𝑐)−𝐻(𝑢(𝑡))=𝐻(𝜙_𝑐)−𝐻(𝑢₀)>0. (6.19) Hence, −𝑑𝐴𝜙𝑑𝑡≥𝐻(_𝑐)−𝐻(𝑢₀)>0. (6.20) So we conclude that 𝑡₁<+∞.

Finally, consider the case 𝑐=0 and 𝑑^′′(0)<0.

Let 𝑁={𝑐∣thednoidalwave𝜙_𝑐is𝑋-unstable}. The curve 𝑐→𝜙_𝑐 is continuous in 𝑋. From the definition of stability, the set 𝑁 is closed. Since 𝑑^′′(𝑐)=⟨𝐼^′(𝜙_𝑐𝜕𝜙),^𝑐⟩𝜕𝑐 (6.21) by Theorem 3.2, 𝑑^′′(𝑐) is continuous on (−1−(2𝜋²/𝐿²),1−(2𝜋²/𝐿²))⊂(−1,1). For any nonzero sequence 𝑐_𝑛→0, we have 𝑑^′′(𝑐_𝑛)→𝑑^′′(0)<0. Hence 𝑑^′′(𝑐_𝑛)<0 for large 𝑛. Thus 𝑐_𝑛∈𝑁, for large 𝑛. It follows that 𝑐=0∈𝑁.

Appendix

In this Appendix, we recall some properties of the Jacobian elliptic integrals that have been used in this work (see [11]).

First, we define the normal elliptic integral of the first and second kinds, 𝐹(𝜑,𝑘)∶=^𝜑₀𝑑𝜃1−𝑘²sin²𝜃=^𝑦₀𝑑𝑡(1−𝑡²)(1−𝑘²𝑡²),𝐸(𝜑,𝑘)∶=^𝜑₀1−𝑘²sin²𝜃𝑑𝜃=^𝑦₀1−𝑘²𝑡²1−𝑡²𝑑𝑡, (A.1) respectively, where 𝑦=sin𝜑.

In their algebraic forms, these two integrals possess the following properties: the first is finite for all real (or complex) values of 𝑦, including infinity; the second has a simple pole of order 1 for 𝑦=∞. The number 𝑘 is called the modulus. This number may take any real or imaginary value. Here we wish to take 0<𝑘<1. The number 𝑘^′ is called the complementary modulus and is related to 𝑘 by 𝑘^′=1−𝑘². The variable 𝜑 is the argument of the normal elliptic integrals. When 𝑦=1, the integrals above are said to be complete. In this case, one writes 𝐹(𝜋/2,𝑘)≡𝐾(𝑘)≡𝐾 and 𝐸(𝜋/2,𝑘)≡𝐸(𝑘)≡𝐸.

Some special values of 𝐾 and 𝐸 are 𝐾(0)=𝐸(0)=𝜋/2, 𝐸(1)=1 and 𝐾(1)=+∞. For 𝑘∈(0,1), one has 𝐾^′(𝑘)>0, 𝐾^′′(𝑘)>0, 𝐸^′(𝑘)<0, 𝐸^′′(𝑘)<0, and 𝐸(𝑘)<𝐾(𝑘). Moreover, 𝐸(𝑘)+𝐾(𝑘) and 𝐸(𝑘)𝐾(𝑘) are strictly increasing functions in (0,1).

Now, we give some derivatives of the complete elliptic integrals 𝐾 and 𝐸 and some important limits involving these functions, that we used in this work (cf. [16] ou [11]):𝑑𝐾=𝑑𝑘𝐸−𝑘^′2𝐾𝑘𝑘^′2,𝑑𝐸=𝑑𝑘𝐸−𝐾𝑘,𝑑²𝐸𝑑𝑘²1=−𝑘𝑑𝐾𝑑𝑘=−𝐸−𝑘^′2𝐾𝑘²𝑘^′2,lim_𝑘→0𝐸−𝑘^′2K𝑘²=lim_𝑘→0𝐾−𝐸𝑘²=𝜋4,lim_𝑘→0𝑑𝐸𝑑𝑘=lim_𝑘→0(𝐸−𝐾)𝑘²𝑘=0,lim_𝑘→0𝑑𝐾𝑑𝑘=0,lim_𝑘→0𝑑²𝐸𝑑𝑘²=−lim_𝑘→01𝑘𝑑𝐾𝜋𝑑𝑘=−4,lim_𝑘→0𝑑²𝐾𝑑𝑘²=lim_𝑘→0[1𝑘^′2𝐾+(3𝑘²−1)𝑘𝑘^′2𝑑𝐾𝜋𝑑𝑘]=4,lim_𝑘→0𝑑³𝐸𝑑𝑘³=0,lim_𝑘→0𝑑³𝐾𝑑𝑘³=0,lim_𝑘→0𝑑⁴𝐸𝑑𝑘⁴=−9𝜋16,lim_𝑘→0𝑑⁴𝐾𝑑𝑘⁴=27𝜋,16lim_𝑘→01𝑘(𝑑𝐾)𝑑𝑘²=lim_𝑘→01𝑘^′4(𝐸−𝑘^′2𝐾)𝑘²𝑑𝐾𝑑𝑘=0. (A.2)(A.3)

Remark .7. To see that lim_𝑘→0(𝑑³𝐾/𝑑𝑘³)=0, we write 𝐾(𝑘)=(𝜋/2){∑^∞_𝑟=0((2𝑟)!(2𝑟)!/2^4𝑟(𝑟!)⁴)𝑘^2𝑟} (see [16, page 110]), and then we get 𝑑³𝐾𝑑𝑘³=𝜋2𝑘{_𝑟≥2(2𝑟)!(2𝑟)!2^4𝑟(𝑟!)⁴2𝑟(2𝑟−1)(2𝑟−2)𝑘^2𝑟−2}, (A.4) where the series converges absolutely. Actually, denoting by 𝑎_𝑟∶=((2𝑟)!(2𝑟)!/2^4𝑟(𝑟!)⁴)2𝑟(2𝑟−1)(2𝑟−2)𝑘^2𝑟−2, we have |𝑎_𝑟+1|/|𝑎_𝑟|=((2𝑟+2)²(2𝑟+1)³/2³(𝑟+1)³(2𝑟−1)(2𝑟−2))𝑘; then we get lim_𝑟→∞(|𝑎_𝑟+1|/|𝑎_𝑟|)<1. So, lim_𝑘→0(𝑑³𝐾/𝑑𝑘³)=0.
To see that lim_𝑘→0(𝑑³𝐸/𝑑𝑘³)=0, we write 𝐸(𝑘)=(𝜋/2){1−∑^∞_𝑟=1((2𝑟−2)!(2𝑟)!/2^4𝑟−1(𝑟−1)!(𝑟!)³)𝑘^2𝑟} (see [16, page 110]), from which we get 𝑑³𝐸𝑑𝑘³𝜋=−2{_𝑟≥2(2𝑟−2)!(2𝑟)!2^4𝑟−1(𝑟−1)!(𝑟!)³2𝑟(2𝑟−1)(2𝑟−2)𝑘^2𝑟−3}. (A.5) Proceeding as before, we obtain the desired limit.
To see that lim_𝑘→0(𝑑⁴𝐾/𝑑𝑘⁴)=27𝜋/16 and lim_𝑘→0(𝑑⁴𝐸/𝑑𝑘⁴)=−(9𝜋/16), differentiating again the series in (A.4) and (A.5), we get 𝑑⁴𝐾/𝑑𝑘⁴=(𝜋/2)∑^∞_𝑟=2𝑏_𝑟2𝑟(2𝑟−1)(2𝑟−2)(2𝑟−3)𝑘^2𝑟−4 and 𝑑⁴𝐸/𝑑𝑘⁴=−(𝜋/2)∑^∞_𝑟=2𝑐_𝑟2𝑟(2𝑟−1)(2𝑟−2)(2𝑟−3)𝑘^2𝑟−4, where 𝑏_𝑟∶=((2𝑟)!(2𝑟)!/2^4𝑟(𝑟!)⁴)2𝑟(2𝑟−1)(2𝑟−2)(2𝑟−3)𝑘^2𝑟−4 and 𝑐_𝑟∶=((2𝑟−2)!(2𝑟)!/2^4𝑟−1(𝑟−1)!(𝑟!)³)2𝑟(2𝑟−1)(2𝑟−2)(2𝑟−3)𝑘^2𝑟−4. It is easy to see that 𝑑⁴𝐾/𝑑𝑘⁴→(𝜋/2)𝑏₂4!=27𝜋/16 and 𝑑⁴𝐸/𝑑𝑘⁴→−(𝜋/2)𝑐₂4!=−(9𝜋/16), as 𝑘→0.

Acknowledgment

Work partially supported by the FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo-Brazil). Grant 2009/14795-0.