Orbital Stability of Solitary Waves for Generalized Symmetric Regularized-Long-Wave Equations with Two Nonlinear Terms

Abstract

This paper investigates the orbital stability of solitary waves for the generalized symmetric regularized-long-wave equations with two nonlinear terms and analyzes the influence of the interaction between two nonlinear terms on the orbital stability. Since 𝐽 is not onto, Grillakis-Shatah-Strauss theory cannot be applied on the system directly. We overcome this difficulty and obtain the general conclusion on orbital stability of solitary waves in this paper. Then, according to two exact solitary waves of the equations, we deduce the explicit expression of discrimination 𝑑^′′(𝑐) and give several sufficient conditions which can be used to judge the orbital stability and instability for the two solitary waves. Furthermore, we analyze the influence of the interaction between two nonlinear terms of the equations on the wave speed interval which makes the solitary waves stable.

1. Introduction

Symmetric regularized-long-wave equations (SRLWE) 𝑢_𝑥𝑥𝑡−𝑢_𝑡=1𝑣+2𝑢²_𝑥,𝑣_𝑡+𝑢_𝑥=0, (1) which are the mathematical models describing the propagation of weakly nonlinear ion acoustic waves [1] and the typical equations in the field of nonlinear science, arise in many other areas of nonlinear mathematical physics [2]. References [1, 2] studied the solitary wave solutions, conservation laws, and interaction among the solitary wave solutions of (1). Moreover, [3–5] discussed the global solution and numerical solution of (1).

Many authors have studied some extended forms of (1).

Guo [3] studied the periodic initial value problem for generalized nonlinear wave equations including (1) 𝑢_𝑡−𝑢_𝑥𝑥𝑡+𝜌_𝑥+𝑓(𝑢)_𝑥=𝑔𝑢,𝜌,𝑢_𝑥,𝜌_𝑡+𝑢_𝑥=ℎ𝜌, (2) by spectral method, then proved the existence and uniqueness of the global generalized solution and classical solution, and gave the convergence and error estimates for the approximate solution in 1987. Zhang [6] obtained the exact solitary wave solutions for a class of the generalized SRLWE with high-order nonlinear terms in 2003.

In terms of the orbital stability of solitary wave solutions, Chen [7] studied it in 1998 for the following generalized SRLWE: 𝑢_𝑡𝑡−𝑢_𝑥𝑥+𝑓(𝑢)_𝑥𝑡−𝑢_{𝑥𝑥𝑡𝑡}=0, (3) where 𝑓(𝑢) is a 𝐶¹ function, satisfying 𝑓(𝑠)>0 if 𝑠>0 and |𝑓(𝑠)|=𝑜(|𝑠|^𝑝); |𝑓^′(𝑠)|=𝑜(|𝑠|^𝑝−1) as 𝑠→0 for 𝑝>1. In particular, 𝜑_𝑐>0 in the solitary wave solution (𝜑_𝑐,𝜓_𝑐)^𝑇 (𝑇 represents transposition) in Assumption 1 of [7]. Moreover, 𝐻_𝑐 only has a simple negative eigenvalue, whose kernel is spanned by 𝜑^′_𝑐. In addition, the rest of its eigenvalues are positive and bounded away from zero.

In this paper, we will consider the orbital stability and instability of solitary wave solutions for the following generalized SRLWE with two nonlinear terms: 𝑢_𝑥𝑥𝑡−𝑢_𝑡=𝑣+𝑏₂𝑢²+𝑏₃𝑢³_𝑥,𝑏_𝑖=constant,𝑏₃𝑣≥0,𝑖=2,3,_𝑡+𝑢_𝑥=0. (4) Our purpose is to investigate the influence of the interaction of the nonlinear terms on the orbital stability.

Equation (4) is the generalization of (1). If (4) is converted into (3), then 𝑓(𝑢)=𝑏₂𝑢²+𝑏₃𝑢³, where 𝑓(𝑢) has two nonlinear terms and the symbols of 𝑏₂,𝑏₃ are unfixed. Indeed, 𝑓(𝑢) is not always positive when 𝑢>0, so the problem studied in this paper is not included by [7]. In the other hand, according to Theorem 1 in this paper, (4) has two bell-profile solitary wave solutions (𝜑_𝑖,𝜓_𝑖)^𝑇,𝑖=1,2, where 𝜑₁(𝜉)>0 and 𝜑₂(𝜉)<0. But the orbital stability of the solitary wave solution (𝜑₂(𝜉),𝜓₂(𝜉))^𝑇 is not considered in [7]. In this paper, we will consider it as well. So the content of this study is new. More significantly, we will study the influence of the interaction between nonlinear terms 𝑏₂𝑢² and 𝑏₃𝑢³ on the orbital stability. It is meaningful for the stability in the application of the practical problems and the selection of the models.

The paper is organized as follows. In Section 2, we will present two exact bell-profile solitary wave solutions of (4) and local existence for the solution of Cauchy problem. In Section 3, we will verify that (4) and its solitary wave solutions meet the requirements of the orbital stability theory of Grillakis-Shatah-Strauss and give the general conclusion. In Section 4, according to two exact solitary waves of the equations obtained in Section 2, we deduce the explicit expression of discrimination 𝑑^′′(𝑐) and give several sufficient conditions which can be used to judge the orbital stability and instability for the two solitary waves. Moreover, we will analyze the influence of two nonlinear terms on the orbital stability. In Section 5, we will focus on studying the orbital instability of solitary wave solutions for (4). Since the skew symmetric operator 𝐽 is not onto, we will define a new conservational functional 𝐼(^→𝑢)=∫_𝑅^→𝑢𝑑𝑥 and estimate solutions of the initial value problem. We will construct a formal Lyapunov function and present the sufficient condition on orbital instability of solitary wave solutions.

2. The Bell-Profile Solitary Wave Solutions and Local Existence for the Solution of Cauchy Problem

According to [6], the solitary wave solution of (4) satisfies 𝑢()(𝜉)()𝑣()(𝜉)()𝑥,𝑡=𝑢=𝑢𝑥−𝑐𝑡𝑥,𝑡=𝑣=𝑣𝑥−𝑐𝑡−𝑐𝑢^′′′(𝜉)+𝑐𝑢^′(𝜉)=𝑣^′(𝜉)+𝑏₂𝑢²(𝜉)+𝑏₃𝑢³(𝜉)_𝜉,𝑣^′(𝜉)=1𝑐𝑢^′(𝜉), (5) where 𝑢^′(𝜉),𝑢^′′(𝜉)→0,|𝜉|→∞. Their exact expressions are given by the following theorem.

Theorem 1. Suppose that 𝑐²−1>0.(1)If 𝑏₃𝑐>0, or 𝑏₃=0 and 𝑏₂𝑐>0, then (4) has a bell-profile solitary wave solution𝑢(𝑥,𝑡)=𝜑₁(𝜉)=𝜑₁(),𝑣()𝑥−𝑐𝑡𝑥,𝑡=𝜓₁(𝜉)=1𝑐𝜑₁(𝑥−𝑐𝑡), (6a) where 𝜑₁(𝜉)=𝐴₁secℎ²𝛼₁/2𝜉2+𝐵₁secℎ²𝛼₁𝜉,/2𝛼₁=√𝑐²−1𝑐,𝐴₁=3√2𝑐²−1𝑐2𝑏²₂𝑐+9𝑏₃𝑐²,𝐵−1₁𝑏=−1+²𝐴₁3𝛼²₁𝑐. (6b)(6c) (2)If 𝑏₃𝑐>0, or 𝑏₃=0 and 𝑏₂𝑐<0, then (4) has another bell-profile solitary wave solution𝑢(𝑥,𝑡)=𝜑₂(𝜉)=𝜑₂(),𝑣()𝑥−𝑐𝑡𝑥,𝑡=𝜓₂(𝜉)=1𝑐𝜑₂(𝑥−𝑐𝑡), (7a) where 𝜑₂(𝜉)=𝐴₂secℎ²𝛼₂/2𝜉2+𝐵₂secℎ²𝛼₂𝜉,/2𝛼₂=𝛼₁,𝐴₂=−𝐴₁,𝐵₂𝑏=−1+²𝐴₂3𝛼²₂𝑐=−2−𝐵₁. (7b)(7c)Next, we study the local existence for the solution of Cauchy problem for (4) by semigroup theory. Firstly, we give two lemmas (see [8, 9]).

Lemma 2 (Hille-Yosida-Phillips). A linear unbounded operator 𝐴 is the infinitesimal generator of a 𝐶₀ semigroup of {𝑇(𝑡)∶𝑡≥0} if and only if 𝐴 is a closed operator with dense domain and there exist real numbers 𝑀 and 𝜔, such that when 𝜆>𝜔, one has(1)𝜆∈𝜌(𝐴), (2)‖𝑅(𝜆;𝐴)^𝑛‖≤𝑀/(𝜆−𝜔)^𝑛, 𝑛=1,2,…,
where 𝜌(𝐴) is the resolvent set and 𝑅(𝜆;𝐴)^𝑛 is the resolvent of 𝐴.

Lemma 3. Consider the Cauchy problem of nonlinear equation 𝑑𝑢(𝑡)(𝑡)((𝑡𝑢(0)𝑑𝑡=𝐴𝑢+𝑓𝑡,𝑢)),𝑡>0,=𝑢₀,𝑢₀∈𝑋. (8) If the following two conditions hold:(1)𝐴 is the infinitesimal generator of a 𝐶₀ semigroup 𝑇(𝑡) in 𝑋;(2)𝑓∈𝐶(𝑅⁺×𝑋,𝑋) satisfies the Lipschitz manner, which means for any 𝑇>0,
there exists 𝐾=𝐾(𝑡), such that ‖𝑓(𝑡,𝑢)−𝑓(𝑡,𝑣)‖≤𝐾(𝑡)‖𝑢−𝑣‖, for all 𝑢,𝑣∈𝑋, 𝑡∈[0,𝑇], then the initial value problem (8) has a unique solution 𝑢(𝑡)=𝑇(𝑡)𝑢₀+^𝑡₀𝑇(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠 (9) in 𝑅⁺.
From Lemmas 2 and 3, we can prove the following Lemma 4, which describes the local existence for the solution of Cauchy problem for (4).

Lemma 4. For any ^→𝑢₀∈𝑋(𝐻¹(𝑅)×𝐿²(𝑅)), there exists 𝑡₀>0, which only depends on ‖^→𝑢₀‖_𝑋, such that (4) has a unique solution ^→𝑢∈𝐶([0,𝑡₀);𝐻¹(𝑅)×𝐿²(𝑅)) with ^→𝑢(0)=^→𝑢₀.

Proof. Firstly, (4) can be written as 𝑢_𝑡=(Δ−1)⁻¹𝑣+𝑏₂𝑢²+𝑏₃𝑢³_𝑥,𝑏_𝑖=constant,𝑏₃𝑣≥0,_𝑡=−𝑢_𝑥, (10) where Δ=𝜕²/𝜕𝑥². (Δ−1)⁻¹ is the pseudodifferential operator. The initial value problem of (10) is equal to 𝑑^→𝑢(𝑡)𝑑𝑡=𝐴^→𝑢(𝑡)+𝑓𝑡,^→𝑢(𝑡),𝑡>0,_→𝑢(0)=^→𝑢₀, (11) where ^→𝑢(𝑡)=𝑢(𝑡)𝑣(𝑡),^→𝑢₀=𝑢(0)𝑣(0),0()𝐴=Δ−1⁻¹𝜕_𝑥−𝜕_𝑥0,𝑓𝑡,^→𝑢(𝑡)=(Δ−1)⁻¹𝑏₂𝑢²+𝑏₃𝑢³_𝑥0. (12)
Since for any 𝑇>0, there exists 𝐾=𝐾(𝑡), such that for any ^→𝑢₁,^→𝑢₂∈𝑋, 𝑡∈[0,𝑇], we have 𝑓𝑡,^→𝑢₁−𝑓𝑡,^→𝑢₂=(Δ−1)⁻¹𝑏₂𝑢²₁+𝑏₃𝑢³_1𝑥0−(Δ−1)⁻¹𝑏₂𝑢²₂+𝑏₃𝑢³_2𝑥0=(Δ−1)⁻¹𝑏₂𝑢²₁−𝑢²₂+𝑏₃𝑢³₁−𝑢³_2𝑥0≤𝐾(𝑡)^→𝑢₁−^→𝑢₂. (13) Therefore, 𝑓(𝑡,^→𝑢(𝑡)) satisfies the local Lipschitz manner.
Now we want to verify that 𝐴 is the infinitesimal generator of a 𝐶₀ semigroup in 𝑋 and 𝐷(𝐴)=𝐻¹×𝐿².
According to Lemma 2, we only need to prove that there exists 𝜔, such that (𝜆𝐼−𝐴)⁻¹≤1𝜆−𝜔 (14) if 𝜆>𝜔 and 𝜆∈𝜌(𝐴).
Indeed, since 𝜆∈𝜌(𝐴), for any ^→𝑣=^→𝑣1_→𝑣2∈𝑋, we have ^→𝑢=^→𝑢1_→𝑢2∈𝐷(𝐴) and (𝜆𝐼−𝐴)^→𝑢=^→𝑣. Thus ^→𝑢=(𝜆𝐼−𝐴)^→−1𝑣. Taking the Fourier transform yields 𝑢₁=𝜆𝜆²+𝜉²/𝜉²𝑣+1¹−𝑖𝜉/1+𝜉²𝜆²+𝜉²/1+𝜉²𝑣₂,𝑢₂=𝜆𝜆²+𝜉²/𝜉²𝑣+1²−𝑖𝜉𝜆²+𝜉²/1+𝜉²𝑣₁. (15)(16) By (15), we have (𝜆−𝜔)𝑢₁≤𝜆(𝜆−𝜔)𝜆²+𝜉²/𝜉²𝑣+1¹+(𝜆−𝜔)𝜉𝜆²𝜉²+1+𝜉²𝑣₂≤𝑣₁+(𝜆−𝜔)𝜉𝜆²𝜉²+1+𝜉²𝑣₂. (17)
Since (𝜆−𝜔)𝜉/(𝜆²(𝜉²+1)+𝜉²)→0 as |𝜉|→∞, there exists positive real number 𝑁₁, when |𝜉|≥𝑁₁, such that |(𝜆−𝜔)𝜉/(𝜆²(𝜉²+1)+𝜉²)|≤1; that is, |(𝜆−𝜔)𝑢₁|≤|𝑣₁|+|𝑣₂|.
Solving the inequality |(𝜆−𝜔)𝜉/(𝜆²(𝜉²+1)+𝜉²)|≤|(𝜆−𝜔)𝜉/𝜆²|≤1 when |𝜉|≤𝑁₁, we can obtain that when 𝜔≥𝑁₁/4, (𝜆−𝜔)𝑢₁≤𝑣₁+𝑣₂. (18) By (16), we know (𝜆−𝜔)𝑢₂≤𝜆(𝜆−𝜔)𝜆²+𝜉²/𝜉²𝑣+1²+(𝜆−𝜔)𝜉𝜆²+𝜉²/𝜉²𝑣+1¹≤𝑣₂+(𝜆−𝜔)𝜉𝜆²𝑣₁. (19)
Since (𝜆−𝜔)|𝜉|/𝜆²→0 as |𝜉|→0, there exists a positive real number 𝑁₂, when |𝜉|≤𝑁₂, such that |(𝜆−𝜔)𝜉/𝜆²|≤1; that is, |(𝜆−𝜔)𝑢₂|≤|𝑣₂|+|𝑣₁|.
Solving the inequality (𝜆−𝜔)|𝜉|/𝜆²≤1 when |𝜉|≥𝑁₂, we can obtain that when 𝜔≥𝑁₂/4, (𝜆−𝜔)𝑢₂≤𝑣₂+𝑣₁. (20) Combining (18) and (20) and choosing 𝜔=Max{𝑁₁/4,𝑁₂/4}, then we get (14) due to the definition of the operator norm.
In conclusion, we can obtain Lemma 4 from Lemmas 2 and 3.

3. General Results for the Orbital Stability of Solitary Wave Solutions

Equation (4) can be written in a Hamiltonian form 𝑑^→𝑢𝑑𝑡=𝐽𝐸^′→𝑢, (21) where 𝜕𝐽=()𝜕𝑥1−Δ⁻¹0𝜕01,Δ=²𝜕𝑥²,𝐸^→𝑢=−_𝑅𝑏𝑣𝑢+²3𝑢³+𝑏₃4𝑢⁴𝑑𝑥,𝐸^′→𝑢=𝐸^′(𝑢)𝐸^′(𝑣)=−𝑣−𝑏₂𝑢²−𝑏₃𝑢³.−𝑢 (22)(23)(24)

Let 𝑋=𝐻¹(𝑅)×𝐿²(𝑅), whose dual space is 𝑋^∗=𝐻⁻¹(𝑅)×𝐿²(𝑅), and the inner product of 𝑋 is ^→𝑢₁,^→𝑢₂=_𝑅𝑢₁𝑢₂+𝑣₁𝑣₂+𝑣_1𝑥𝑣_2𝑥∀𝑑𝑥,_→𝑢₁,^→𝑢₂∈𝑋. (25) There exists a natural isomorphism 𝐼∶𝑋→𝑋^∗ defined by ⟨𝐼^→𝑢₁,^→𝑢₂⟩=(^→𝑢₁,^→𝑢₂), where ⟨⋅,⋅⟩ denotes the pairing between 𝑋 and 𝑋^∗, and ^→𝑢₁,^→𝑢₂=_𝑅𝑢₁𝑢₂+𝑣₁𝑣₂𝑑𝑥. (26) From (25) and (26), we know that 𝐼=¹⁰_01−Δ. And we can verify that 𝐽 is an skew symmetric operator; that is, ⟨𝐽^→𝑢,^→𝑣⟩=−⟨^→𝑢,𝐽^→𝑣⟩.

Let 𝑇 be a one-parameter group of unitary operator on 𝑋 defined by 𝑇(𝑠)^→𝑢(⋅)=^→𝑢(⋅−𝑠), where ^→𝑢(𝑠)∈𝑋, for all 𝑠∈𝑅. Obviously, 𝑇^′(0)=^{−𝜕/𝜕𝑥0}_{0−𝜕/𝜕𝑥}. Since 𝑇^′(0)=𝐽𝐵, we can get 𝐵=^Δ−10₀₋₁.

Therefore, we define 𝑉^→𝑢1=−2𝐵^→𝑢,^→𝑢1=−2𝑢𝑣,𝑢𝑣=1Δ−100−12𝑢−𝑢_𝑥𝑥𝑣,𝑢𝑣=12_𝑅𝑢²−𝑢_𝑥𝑥𝑢+𝑣²𝑑𝑥. (27)

Then 𝑉^′(^→𝑢)=^𝑉′_𝑉^(𝑢)_′(𝑣)=(^{𝑢−𝑢𝑥𝑥}_𝑣), 𝑉^′′(^→𝑢)=^1−Δ0₀₁.

The solitary waves (6a) and (7a) of (4) can be written as 𝜙^→_𝑖𝑐=𝜑_𝑖𝑐()𝜓𝑥−𝑐𝑡_𝑖𝑐()()𝑥−𝑐𝑡=𝑇𝑐𝑡𝜙^→_𝑖(𝑥),𝑖=1,2, (28) where 𝜑₁(𝑥) and 𝜑₂(𝑥) are defined by (6b) and (7b), respectively. Now, we consider the orbital stability of solitary waves 𝑇(𝑐𝑡)𝜙^→_𝑖(𝑥). Avoiding repetition, we let 𝜙^→_𝑐(𝑥) be one of 𝜙^→₁(𝑥) and 𝜙^→₂(𝑥). We will verify that (4) and the solitary wave 𝑇(𝑐𝑡)𝜙^→_𝑐(𝑥) satisfy the three assumption conditions of the orbital stability theory presented by Grillakis et al. in [10].

Verification of Assumption  1. From Lemma 4 in Section 2, we obtain that the initial value problem of (4) has a unique solution. And it is easy to prove that 𝐸(^→𝑢) and 𝑉(^→𝑢) defined by (23) and (27) satisfy 𝐸^→𝑢(𝑡)=𝐸^→𝑢(0)=𝐸^→𝑢₀,𝑉^→𝑢(𝑡)=𝑉^→𝑢(0)=𝑉^→𝑢₀, (29) respectively.

This shows that (4) satisfies the Assumption 1 in [10].

Verification of Assumption  2. Firstly, we can prove the following lemma.

Lemma 5. 𝜙^→_𝑐 is a bounded state solution of (4), satisfying 𝐸^′(𝜙^→_𝑐)+𝑐𝑉^′(𝜙^→_𝑐)=0.

Proof. Substituting the solution 𝜙^→_𝑐=^𝜑𝑐_𝜓𝑐 into (4), we obtain 𝜑_𝑐𝜉=𝑐𝜓_𝑐𝜉,−𝑐𝜑_{𝑐𝜉𝜉𝜉}+𝑐𝜑_𝑐𝜉=𝜓_𝑐+𝑏₂𝜑²_𝑐+𝑏₃𝜑³_𝑐𝜉. (30) Integrating (30) once, we get 𝜑_𝑐=𝑐𝜓_𝑐+𝑎₁,−𝑐𝜑_𝑐𝜉𝜉+𝑐𝜑_𝑐=𝜓_𝑐+𝑏₂𝜑²_𝑐+𝑏₃𝜑³_𝑐+𝑎₂, (31) where 𝑎₁, 𝑎₂ are integral constants.
𝜑_𝑐,𝜓_𝑐,𝜑_𝑐𝜉𝜉→0 as |𝜉|→∞, so 𝑎₁=0 and 𝑎₂=0. Thus, 𝜑_𝑐=𝑐𝜓_𝑐,−𝑐𝜑_𝑐𝜉𝜉+𝑐𝜑_𝑐=𝜓_𝑐+𝑏₂𝜑²_𝑐+𝑏₃𝜑³_𝑐. (32) Furthermore, 𝐸𝜙^′→_𝑐+𝑐𝑉𝜙^′→_𝑐=−𝜓_𝑐−𝑏₂𝜑²_𝑐−𝑏₃𝜑³_𝑐−𝜑_𝑐𝜑+𝑐^𝑐−𝜑_𝑐𝜉𝜉𝜓_𝑐=−𝜓_𝑐−𝑏₂𝜑²_𝑐−𝑏₃𝜑³_𝑐−𝑐𝜑_𝑐𝜉𝜉+𝑐𝜑_𝑐−𝜑_𝑐+𝑐𝜓_𝑐. (33) Due to (32), we have 𝐸^′(𝜙^→_𝑐)+𝑐𝑉^′(𝜙^→_𝑐)=0.

The above Lemma 5 shows that (4) has the bounded state solutions, and the two solitary waves 𝜙^→_1𝑐 and 𝜙^→_2𝑐 given in Theorem 1 both are the bounded state solutions of the equation.

Verification of Assumption  3. We consider spectrum analysis of the operator 𝐻_𝑐.

Now we define the operator 𝐻_𝑐∶𝑋→𝑋^∗ as 𝐻_𝑐=𝐸^′′(𝜙^→_𝑐)+𝑐𝑉^′′(𝜙^→_𝑐), where 𝐸^′′→𝑢=−2𝑏₂𝑢−3𝑏₃𝑢²−1−10,𝑉^′′→𝑢=.1−Δ001 (34) Therefore, 𝐻_𝑐=𝐸𝜙^′′→_𝑐+𝑐𝑉𝜙^′′→_𝑐=−2𝑏₂𝜑_𝑐−3𝑏₃𝜑²_𝑐=−1−10+𝑐1−Δ001−2𝑏₂𝜑_𝑐−3𝑏₃𝜑²_𝑐𝜕−𝑐²𝜕𝑥².+𝑐−1−1𝑐 (35) For any ^→𝑢₁,^→𝑢₂∈𝐻¹(𝑅)×𝐿²(𝑅), we have ⟨𝐻_𝑐^→𝑢₁,^→𝑢₂⟩=⟨𝐻_𝑐^→𝑢₂,^→𝑢₁⟩. This means that 𝐻_𝑐 is a self-conjugate operator, that is, 𝐻_𝑐=𝐻^∗_𝑐, and that 𝐼⁻¹𝐻_𝑐 is a bounded self-conjugate operator on 𝑋. The eigenvalues of 𝐻_𝑐 consist of the real numbers 𝜆 which ensure that 𝐻_𝑐−𝜆𝐼 are irreversible.

From (30), we have −𝑐𝜑_{𝑐𝜉𝜉𝜉}+𝑐𝜑_𝑐𝜉=((1/𝑐)𝜑_𝑐+𝑏₂𝜑²_𝑐+𝑏₃𝜑³_𝑐)_𝜉. Namely, −2𝑏₂𝜑_𝑐−3𝑏₃𝜑²_𝑐𝜕−𝑐²𝜕𝑥²+1𝑐−𝑐𝜑_𝑐𝑥=0. (36) Let 𝐿=−2𝑏₂𝜑_𝑐−3𝑏₃𝜑²_𝑐−𝑐(𝜕²/𝜕𝑥²)+(𝑐−(1/𝑐)). Since the existence of solitary wave solution 𝜙^→_𝑐=^𝜑𝑐_𝜓𝑐 of (4) is based on the condition that 𝑐²−1>0, 𝑐−1/𝑐>0 as 𝑐>1, and −2𝑏₂𝜑_𝑐−3𝑏₃𝜑²_𝑐→0 as |𝜉|→∞, it is easy to know that 𝜎_𝑒𝑠𝑠(𝐿)=[𝑐−1/𝑐,+∞) by Weyl's essential spectrum theorem. Moreover, from (36), we have 𝐿𝜑_𝑐𝑥=0, where 𝑥=0 is a unique zero point of 𝜑_𝑐𝑥. By Sturm-Liouville theorem we know that zero is the second eigenvalue of 𝐿. Thus 𝐿 only has one strictly negative eigenvalue −𝜎² in the case of 𝑐>1, whose corresponding eigenfunction is denoted by 𝜒; that is, 𝐿𝜒=−𝜎²𝜒.

Therefore, 𝐻_𝑐 has a unique simple negative eigenvalue, and zero is its eigenvalue and the rest of its spectrums are bounded away from zero. So, 𝐻_𝑐 satisfies the Assumption 3 in [10].

According to [10, 11], we can get the following lemma.

Lemma 6. For any real function 𝑦∈𝐻¹(𝑅) with ⟨𝑦,𝜒⟩=⟨𝑦,𝜑_𝑐𝑥⟩=0, there exists 𝛿>0 such that ⟨𝐿𝑦,𝑦⟩≥𝛿‖𝑦‖²_𝐻2(𝑅).
Let 𝜒^→_𝑐=^𝜒_(1/𝑐)𝜒. We have 𝐻𝜒_𝑐^→_𝑐,𝜒^→_𝑐=_𝑅−2𝑏₂𝜑_𝑐𝜒²−3𝑏₃𝜑²_𝑐𝜒²−𝑐𝜒_𝑥𝑥1𝜒+𝑐−𝑐𝜒²=𝑑𝑥𝐿𝜒,𝜒=−𝜎²𝜒,𝜒<0. (37)
Let 𝜑^→_𝑐𝑥=^𝜑𝑐𝑥_{(1/𝑐)𝜑𝑐𝑥}, and then 𝐻𝜑_𝑐^→_𝑐𝑥,𝜑^→_𝑐𝑥=_𝑅−2𝑏₂𝜑_𝑐𝜑²_𝑐𝑥−3𝑏₃𝜑²_𝑐𝜑²_𝑐𝑥−𝑐𝜑_{𝑐𝑥𝑥𝑥}𝜑_𝑐𝑥+1𝑐−𝑐𝜑²_𝑐𝑥=𝑑𝑥𝐿𝜑_𝑐𝑥,𝜑_𝑐𝑥=0. (38)
Let 𝑃=𝑝^→_𝑐∈𝑋∣𝑝^→_𝑐=𝑝₁𝑝₂,𝑝₁,𝜒=𝑝₁,𝜑_𝑐𝑥𝑝=0,₂,1𝑐𝜒=𝑝₂,1𝑐𝜑_𝑐𝑥=0. (39) We have 𝐻𝑝_𝑐^→_𝑐,𝑝^→_𝑐=_𝑅−2𝑏₂𝜑_𝑐𝑝²₁−3𝑏₃𝜑²_𝑐𝑝²₁−𝑐𝑝_1𝑥𝑥𝑝₁+1𝑐−𝑐𝑝²₁+1𝑐𝑝₁−𝑐𝑝₂²𝑑𝑥. (40)
Thus ⟨𝐻𝑝_𝑐^→_𝑐,𝑝^→_𝑐⟩≥⟨𝐿𝑝₁,𝑝₁⟩≥𝛿‖𝑝^→_𝑐‖²>0 for any 𝑝^→_𝑐∈𝑃 when 𝑐>1.
According to the above analysis, when 𝑐>1, we can make spectrum decomposition for 𝐻_𝑐. Let 𝑁=𝑘𝜒₁^→_𝑐∣𝑘₁,𝑘∈𝑅𝑍=𝜑₂^→_𝑐𝑥∣𝑘₂∈𝑅,𝑃=𝑝^→_𝑐∈𝑋∣𝑝^→_𝑐=𝑝₁𝑝₂,𝑝₁,𝜒=𝑝₁,𝜑_𝑐𝑥𝑝=0,₂,1𝑐𝜒=𝑝₂,1𝑐𝜑_𝑐𝑥=0. (41)
For any ^→𝑢∈𝑁, ^→𝑢≠0, ⟨𝐻_𝑐^→𝑢,^→𝑢⟩<0 due to ⟨𝐻_𝑐𝑘𝜒₁^→_𝑐,𝑘𝜒₁^→_𝑐⟩=𝑘²₁⟨𝐻𝜒_𝑐^→_𝑐,𝜒^→_𝑐⟩=−𝑘²₁𝜎²⟨𝜒,𝜒⟩<0.
For any ^→𝑧∈𝑍,^→𝑧≠0, ⟨𝐻_𝑐^→𝑧,^→𝑧⟩=0 due to ⟨𝐻_𝑐𝑘𝜑₂^→_𝑐𝑥,𝑘𝜑₂^→_𝑐𝑥⟩=0.
For any 𝑝^→_𝑐∈𝑃,𝑝^→_𝑐≠0, ⟨𝐻𝑝_𝑐^→_𝑐,𝑝^→_𝑐⟩>𝛿‖𝑝^→_𝑐‖²_𝐻2(𝑅).
Thus the space 𝑋 can be decomposed as a direct sum 𝑋=𝑁+𝑍+𝑃, where 𝑍 is the kernel space of 𝐻_𝑐, 𝑁 is a finite-dimensional subspace and 𝑃 is a closed subspace.
We now define 𝑑(𝑐)∶𝑅→𝑅 as 𝑑(𝑐)=𝐸(𝜙^→_𝑐)+𝑐𝑉(𝜙^→_𝑐), and then 𝑑^′(𝑐)=𝐸𝜙^′→_𝑐,𝜙^→′_𝑐+𝑐𝑉𝜙^′→_𝑐,𝜙^→′_𝑐+𝑉𝜙^→_𝑐=𝑉𝜙^→_𝑐. (42) According to the above verification of Assumptions 1–3, (4) and its solitary wave solutions satisfy the three assumptions of Theorem 2 in [10], so we can obtain the following general conclusion on orbital stability of solitary waves for (4).

Theorem 7. Suppose that 𝑏₂𝑏₃≠0,𝑏₃≥0,𝑐>1, and 𝜙^→_𝑐=𝑇(𝑐𝑡)𝜙^→(𝑥) is the solitary wave solution of (4). Then,(1)𝜙^→_𝑐 is orbitally stable as 𝑑^′′(𝑐)>0;(2)𝜙^→_𝑐 is orbitally unstable as 𝑑^′′(𝑐)<0.

Remark 8. The proof of the conclusion (2) in Theorem 7 will be given by Theorem 26 in Section 5.

4. Orbital Stability and Influence of the Interaction between Nonlinear Terms on It

In this section, by using two exact solitary waves (6a), (6b), and (6c) and (7a), (7b), and (7c) given in Theorem 1, we will give the explicit expressions for the discrimination 𝑑^′′_𝑖(𝑐). Then with the analysis method, we will give several sufficient conditions to judge the orbital stability and instability of the solitary waves. Furthermore, we will also analyze the influence of the interaction between two nonlinear terms on the orbital stability. We assume that 𝑏₃>0 and 𝑐>1 in this section.

4.1. Discrimination 𝑑^′′_𝑖(𝑐)

In view of (42), we have 𝑑^′_𝑖(𝑐)=𝑉𝜙^→_𝑖𝑐=12_𝑅𝜑²_𝑖𝑐−𝜑_{𝑖𝑐𝑥𝑥}𝜑_𝑖𝑐+𝜓²_𝑖𝑐𝑑𝑥,𝑖=1,2. (43) Next, we simplify (43). According to (6a) and (7a) in Theorem 1, we have 𝜓_𝑖=(1/𝑐)𝜑_𝑖. Substituting it into (43), and letting 𝑧=𝑒^𝛼𝑖𝜉(𝛼_𝑖>0), we obtain 𝑑^′_𝑖(𝑐)=12_𝑅𝜑²_𝑖𝑐−𝜑_{𝑖𝑐𝑥𝑥}𝜑_𝑖𝑐+𝜓²_𝑖𝑐=1𝑑𝑥2_𝑅11+𝑐²𝜑²_𝑖𝑐+𝜑²_𝑖𝑐𝑥=1𝑑𝑥2_𝑅11+𝑐²𝜑²_𝑖𝑐+𝜑²_𝑖𝑐𝜉=1𝑑𝜉2𝛼_𝑖0^+∞1𝑧2𝐴_𝑖𝑧(1+𝑧)²+2𝐵_𝑖𝑧²×11+𝑐²+𝛼_𝑖𝑧²−1(1+𝑧)²+2𝐵_𝑖𝑧²=1𝑑𝑧1+𝑐²2𝐴²_𝑖𝛼_𝑖0^+∞𝑧(1+𝑧)²+2𝐵_𝑖𝑧²𝑑𝑧+2𝐴²_𝑖𝛼_𝑖0^+∞𝑧𝑧²−1²(1+𝑧)²+2𝐵_𝑖𝑧⁴𝑑𝑧, (44) where 𝛼_𝑖, 𝐴_𝑖, 𝐵_𝑖 are given by (6c) and (7c).

Since −2<𝐵_𝑖<0, we can solve above two integrations. Then, 𝑑^′_𝑖(𝑐)=11+𝑐²×𝐴²_𝑖𝐵_𝑖+1−𝐵_𝑖𝐵_𝑖×𝐵+2𝜋−2arctan^𝑖+1−𝐵_𝑖𝐵_𝑖×+2−22𝛼_𝑖𝐵_𝑖𝐵_𝑖+2⁻¹+𝐴²_𝑖𝛼_𝑖23+2𝐵_𝑖+𝐵²_𝑖𝐵−3^𝑖+1−𝐵_𝑖𝐵_𝑖×𝐵+2𝜋−2arctan^𝑖+1−𝐵_𝑖𝐵_𝑖×+212𝐵²_𝑖𝐵_𝑖+2²⁻¹. (45) If 𝐵_𝑖=𝐵₁√=−1+2𝑏₂𝑐𝑐2𝑏²₂𝑐+9𝑏₃𝑐²,−1 (46) then (45) can be simplified into the following form: 𝑑^′₁(𝑐)=13𝑏₃𝑐6𝛼𝑏₃𝑐−√2𝑏₂×√𝜋−2arctan2𝑏₂3𝑐𝑏₃𝑐²⋅1−12𝑏₃𝑐+3𝑐2𝑏₃+𝑏²₂9𝑏²₃−𝑐𝛼³3𝑏₃. (47) If 𝐵_𝑖=𝐵₂√=−1−2𝑏₂𝑐𝑐2𝑏²₂𝑐+9𝑏₃𝑐²,−1 (48) then (45) can be simplified into the following form: 𝑑^′₂(𝑐)=13𝑏₃𝑐6𝛼𝑏₃𝑐+√2𝑏₂×√𝜋+2arctan2𝑏₂3𝑐𝑏₃𝑐²⋅1−12𝑏₃𝑐+3𝑐2𝑏₃+𝑏²₂9𝑏²₃−𝑐𝛼³3𝑏₃. (49) By calculating, we have 𝑑^′′₁(𝑐)𝑘=−3𝑐−3/𝑐−𝑘²𝐷₁4𝑏₃𝑐^3/2+√𝑐²−116𝑐⁴+7𝑘²𝑐³−8+3𝑘⁴𝑐²−7𝑘²𝑐+166𝑏₃𝑐³𝑐²−1+𝑘²𝑐,𝑑^′′₂(𝑐)=𝑘3𝑐−3/𝑐−𝑘²𝐷₂4𝑏₃𝑐^3/2+√𝑐²−116𝑐⁴+7𝑘²𝑐³−8+3𝑘⁴𝑐²−7𝑘²𝑐+166𝑏₃𝑐³𝑐²−1+𝑘²𝑐, (50)(51) where √𝑘=2𝑏₂3𝑏₃,𝐷₁=𝜋−2arctan𝑘𝑐𝑐²,𝐷−1₂=𝜋+2arctan𝑘𝑐𝑐².−1 (52) Furthermore, suppose that 𝑀₁=𝑘𝑐𝑐²𝐷−1¹=𝑘𝑐𝑐²𝑘−1𝜋−2arctan𝑐𝑐²,𝑀−1₂=𝑘𝑐𝑐²𝐷−1²=𝑘𝑐𝑐²𝑘−1𝜋+2arctan𝑐𝑐².−1 (53) Then (50) can be written as 𝑑^′′₁(𝑐)=√𝑐²−112𝑏₃𝑐²×−3𝑀₁33𝑐−𝑐−𝑘²+216𝑐⁴+7𝑘²𝑐³−8+3𝑘⁴𝑐²−7𝑘²𝑐+16𝑐𝑐²−1+𝑘²𝑐. (54) And (51) can be written as 𝑑^′′₂(𝑐)=√𝑐²−112𝑏₃𝑐²×3𝑀₂33𝑐−𝑐−𝑘²+216𝑐⁴+7𝑘²𝑐³−8+3𝑘⁴𝑐²−7𝑘²𝑐+16𝑐𝑐²−1+𝑘²𝑐. (55) Therefore, we only need to consider the conditions such that 𝑑^′′_𝑖(𝑐)>0 hold in (54) and (55) to study the orbital stability of the solitary waves 𝜙^→_𝑖𝑐, while needing to consider 𝑑^′′_𝑖(𝑐)<0 to study instability.

4.2. Discussion on 𝑀₁ and 𝑀₂

In this section, we consider 𝑀₁ and 𝑀₂ in the case of 𝑐>1 and 𝑏₃>0.(1)For 𝑀₁. If 𝑏₂>0, then 𝑘>0. Suppose that 𝑥=𝑘𝑐/(𝑐²−1)∈(0,+∞), and then lim_𝑐→+∞𝑥=lim_𝑐→+∞𝑘𝑐𝑐²−1=0,lim_𝑐→1𝑥=lim_𝑐→1𝑘𝑐𝑐²−1=+∞. (56) Let 𝑔(𝑥)=𝑥(𝜋−2arctan𝑥). We have 𝑔^′(𝑥)=𝜋−2arctan𝑥−2𝑥1+𝑥²,𝑔^′′(𝑥)4=−1+𝑥²²<0. (57)Moreover, 𝑔(𝑥) can obtain the local maximum at 𝑥₀, where 𝑥₀ satisfies 𝑔^′(𝑥₀)=0 and 𝑔𝑥₀=𝑥₀𝜋−2arctan𝑥₀=2𝑥²₀1+𝑥²₀<2. (58) Since lim_𝑥→+∞𝑀₁=lim_𝑥→+∞𝑔(𝑥)=lim_𝑥→+∞𝑥()𝑀𝜋−2arctan𝑥=2,₁∈(0,2). (59)

When 𝑏₂<0, it is easy to know that 𝑥∈(−∞,0), and then 𝑀₁∈(−∞,0).(2)For 𝑀₂. When 𝑏₂>0, it is clear that 𝑥∈(0,+∞), and then 𝑀₂∈(0,+∞). But if 𝑏₂<0, then 𝑘<0 and 𝑥=𝑘𝑐/(𝑐²−1)∈(−∞,0). Therefore lim_{𝑥→−∞}𝑀₂=lim_{𝑥→−∞}𝑥(𝜋+2arctan𝑥)=−2. (60)Similarly, we can get 𝑀₂∈(−2,0).

4.3. Orbital Stability of Solitary Waves for (4) in the Case of 3𝑐−3/𝑐−𝑘²>0

Based on (54), (55), and above discussion on 𝑀₁,𝑀₂, we want to obtain much more simple conditions on the orbital stability of solitary waves 𝜙^→_1𝑐 and 𝜙^→_2𝑐.

4.3.1. Orbital Stability of 𝜙^→_1𝑐

(1) If 𝑏₂>0, then 𝑀₁∈(0,2). At this time, −3𝑀₁(3𝑐−3/𝑐−𝑘²)<0. In order to find 𝑐 such that 𝑑^′′₁(𝑐)>0, we only need to consider 𝑀₁=2 in (54). It is easy to see that 𝑑^′′₁(𝑐)>0 when 𝑐 satisfies 7𝑐⁴+𝑘²𝑐³+10𝑐²−𝑘²𝑐+7>0. (61) Thus, 𝜙^→_1𝑐 is orbitally stable.

If 𝑏₂<0, −3𝑀₁(3𝑐−3/𝑐−𝑘²)>0. In order to make 𝑑^′′₁(𝑐)>0, that is, 𝜙^→_1𝑐 is orbitally stable, only when 𝑐 satisfies 16𝑐⁴+7𝑘²𝑐³−8+3𝑘⁴𝑐²−7𝑘²𝑐+16>0. (62)

(2) If 𝑏₂>0, then 𝑀₁∈(0,2). Here, −3𝑀₁(3𝑐−3/𝑐−𝑘²)<0 in (54). In order to make 𝑑^′′₁(𝑐)<0, we only need to consider 𝑀₁=0 in (54). Then, it is easy to see that 𝑑^′′₁(𝑐)<0 when 𝑐 satisfies 16𝑐⁴+7𝑘²𝑐³−8+3𝑘⁴𝑐²−7𝑘²𝑐+16<0. (63) Thus, 𝜙^→_1𝑐 is orbitally unstable.

4.3.2. Orbital Stability of 𝜙^→_2𝑐

(1) If 𝑏₂>0, then 𝑀₂∈(0,+∞). In order to make 𝑑^′′₂(𝑐)>0, it is easy to know that 𝜙^→_2𝑐 is orbitally stable if 𝑐 satisfies (62).

If 𝑏₂<0, then 𝑀₂∈(−2,0). In order to make 𝑑^′′₂(𝑐)>0, we only need to consider 𝑀₂=−2 in (55). It is easy to know that 𝜙^→_2𝑐 is orbitally stable if 𝑐 satisfies (61).

(2) If 𝑏₂<0, in order to make 𝑑^′′₂(𝑐)<0, we only need to consider 𝑀₂=0 in (55). Then, it is easy to know that 𝜙^→_2𝑐 is orbitally unstable if 𝑐 satisfies (63).

In addition, we know that 3𝑐−3/𝑐−𝑘²>0 is equal to 𝑐>(1/27𝑏₃)[𝑏²₂+𝑏⁴₂+729𝑏²₃] or 𝑐<(1/27𝑏₃)[𝑏²₂−𝑏⁴₂+729𝑏²₃], but we always assume 𝑐>1 through this section. So if we assume 𝑐₀=(1/27𝑏₃)[𝑏²₂+𝑏⁴₂+729𝑏²₃], then 3𝑐−3/𝑐−𝑘²>0 is equal to 𝑐>𝑐₀.

Summarizing above results, we have the following theorem.

Theorem 9. Suppose that 𝑏₃>0 and 𝑐>𝑐₀, where 𝑐₀=(𝑏²₂+𝑏⁴₂+729𝑏²₃)/27𝑏₃>1.(1)𝜙^→_1𝑐 is orbitally stable if 𝑏₂>0 and the wave speed 𝑐 satisfies (61), or 𝑏₂<0 and the wave speed 𝑐 satisfies (62); 𝜙^→_1𝑐 is orbitally unstable if 𝑏₂>0 and the wave speed 𝑐 satisfies (63).(2)𝜙^→_2𝑐 is orbitally stable if 𝑏₂>0 and the wave speed 𝑐 satisfies (62), or 𝑏₂<0 and the wave speed 𝑐 satisfies (61); 𝜙^→_2𝑐 is orbitally unstable if 𝑏₂<0 and the wave speed 𝑐 satisfies (63).

4.4. Orbital Stability of Solitary Waves for (4) in the Case of 3𝑐−3/𝑐−𝑘²<0

4.4.1. Orbital Stability of 𝜙^→_1𝑐

(1) If 𝑏₂>0, −3𝑀₁(3𝑐−3/𝑐−𝑘²)>0. In order to make 𝑑^′′₁(𝑐)>0, we only need to consider 𝑀₁=0 in (54). It is easy to see that 𝜙^→_1𝑐 is orbitally stable if c satisfies (62).

(2) If 𝑏₂>0. In order to find 𝑐 such that 𝑑^′′₁(𝑐)<0, we only need to consider 𝑀₁=2 in (54). It is easy to see that 𝑑^′′₁(𝑐)<0 when 𝑐 satisfies 7𝑐⁴+𝑘²𝑐³+10𝑐²−𝑘²𝑐+7<0. (64) Thus, 𝜙^→_1𝑐 is orbitally unstable.

If 𝑏₂<0, −3𝑀₁(3𝑐−3/𝑐−𝑘²)<0. In order to make 𝑑^′′₁(𝑐)<0, it is easy to know that 𝜙^→_1𝑐 is orbitally unstable if 𝑐 satisfies (63).

4.4.2. Orbital Stability of 𝜙^→_2𝑐

(1) If 𝑏₂<0, then 𝑀₂∈(−2,0). At this time, 3𝑀₂(3𝑐−3/𝑐−𝑘²)>0. In order to find 𝑐 such that 𝑑^′′₂(𝑐)>0, we only need to consider 𝑀₂=0 in (55). It is easy to see that 𝜙^→_2𝑐 is orbitally stable if 𝑐 satisfies (62).

(2) If 𝑏₂>0, then 𝑀₂∈(0,+∞). In order to make 𝑑^′′₂(𝑐)<0, it is easy to know that 𝜙^→_2𝑐 is orbitally unstable if c satisfies (63).

If 𝑏₂<0, then 𝑀₂∈(−2,0). Here, 3𝑀₂(3𝑐−3/𝑐−𝑘²)>0 in (55). In order to find 𝑐 such that 𝑑^′′₂(𝑐)<0, we only need to consider 𝑀₂=−2 in (55). It is easy to see that 𝜙^→_2𝑐 is orbitally unstable if 𝑐 satisfies (64).

Summarizing above results, we have the following theorem.

Theorem 10. Suppose that 𝑏₃>0 and 1<𝑐<𝑐₀, where 𝑐₀=(𝑏²₂+𝑏⁴₂+729𝑏²₃)/27𝑏₃>1.(1)𝜙^→_1𝑐 is orbitally stable if 𝑏₂>0 and the wave speed 𝑐 satisfies (62). 𝜙^→_1𝑐 is orbitally unstable if 𝑏₂>0 and the wave speed 𝑐 satisfies (64), or 𝑏₂<0 and the wave speed 𝑐 satisfies (63).(2)𝜙^→_2𝑐 is orbitally stable if 𝑏₂<0 and the wave speed 𝑐 satisfies (62). 𝜙^→_2𝑐 is orbitally unstable if 𝑏₂>0 and the wave speed 𝑐 satisfies (63), or 𝑏₂<0 and the wave speed 𝑐 satisfies (64).

4.5. Corollaries and Influences of Nonlinear Terms on Orbital Stability of the Solitary Waves for (4)

In this part, we will firstly consider the orbital stability of the solitary waves for (4) with only one nonlinear term. Secondly, we will discuss the effect of nonlinear terms on orbital stability of the solitary waves for (4).

Corollary 11. Suppose that 𝑏₃>0 and 𝑐>1. If 𝑏₂=0, (4) has the solitary wave solutions 𝜙^→_𝑖𝑐=^𝜑𝑖_𝜓^(𝜉)_𝑖(𝜉),  𝑖=1,2, where 𝜑₁(𝜉)=𝐴₁secℎ²𝛼₁/2𝜉2+𝐵₁secℎ²𝛼₁/2𝜉,𝜓₁(𝜉)=1𝑐𝜑₁(𝜉),𝛼₁=√𝑐²−1𝑐,𝐴₁=2𝑐²−1𝑏₃𝑐,𝐵₁𝜑=−1,₂(𝜉)=𝐴₂secℎ²𝛼₂/2𝜉2+𝐵₂secℎ²𝛼₂/2𝜉,𝜓₂(𝜉)=1𝑐𝜑₂(𝜉),𝛼₂=√𝑐²−1𝑐,𝐴₂=2𝑐²−1𝑏₃𝑐,𝐵₂=−1. (65) Under the given conditions, we can easily conclude that the solitary waves 𝜙^→_𝑖𝑐,𝑖=1,2, are both orbitally stable.

Proof. When 𝑏₂=0, the above solitary waves (65) of (4) can be deduced from Theorem 1 directly.
Actually, it is clear that 𝑘=0 as 𝑏₂=0. Substituting 𝑘=0 into (50) and (51), we have 𝑑^′′₁(𝑐)=𝑑^′′₂(𝑐)=43𝑏₃𝑐³2𝑐⁴−𝑐²+2√𝑐².−1 (66) We know that 2𝑐⁴−𝑐²+2>0 in (66), so 𝑑^′′_𝑖(𝑐)>0,𝑖=1,2, if 𝑐>1 and 𝑏₃>0. Thus, we know that the solitary waves 𝜙^→_𝑖𝑐,𝑖=1,2, of (4) are both orbitally stable according to Theorem 7.

Corollary 12. When 𝑐>1 and 𝑏₃=0, then (4) has the solitary wave solution 𝜙^→_𝑐=^𝜑(𝜉)_𝜓(𝜉), where 𝜑(𝜉)=𝐴secℎ²(𝛼/2)𝜉2+𝐵secℎ²()𝜉(𝜉)=1𝛼/2,𝜓𝑐𝜑(𝜉),√𝛼=𝑐²−1𝑐3𝑐,𝐴=²−1𝑏₂𝑐,𝐵=0. (67) Under the given conditions, we know that the solitary wave 𝜙^→_𝑐 of (4) is orbitally stable.

Proof. When 𝑏₃=0, the above solitary wave (67) of (4) can be deduced from Theorem 1 directly.
Moreover, similar to deducing (50) and (51), by calculating, we can obtain 𝑑^′(𝑐)=12_𝑅𝜑²_𝑐−𝜑_𝑐𝑥𝑥𝜑_𝑐+𝜓²_𝑐=1𝑑𝑥2_𝑅11+𝑐²𝜑²_𝑐+𝜑²_𝑐𝑥=1𝑑𝑥2_𝑅11+𝑐²𝜑²_𝑐+𝜑²_𝑐𝜉=1𝑑𝜉2𝛼₀^+∞1𝑧2𝐴𝑧(1+𝑧)²+2𝐵𝑧²×11+𝑐²+𝛼𝑧²−1(1+𝑧)²+2𝐵𝑧²=1𝑑𝑧1+𝑐²2𝐴²𝛼₀^+∞𝑧(1+𝑧)²+2𝐵𝑧²𝑑𝑧+2𝐴²𝛼₀^+∞𝑧𝑧²−1²(1+𝑧)²+2𝐵𝑧⁴𝑑𝑧. (68)
Substituting (67) into the above formula yields 𝑑^′(𝑐)=65𝑐²−1^3/23𝑐²+2𝑐³𝑏²₂. (69) Therefore 𝑑^′′(𝑐)=185√𝑐²−12𝑐⁴+𝑐²+2𝑐⁴𝑏²₂. (70) We know that 2𝑐⁴+𝑐²+2>0 in (70). According to Theorem 7, 𝜙^→_𝑐 is orbitally stable if 𝑐>1 and 𝑏₃=0. Thus, Corollary 12 holds.

According to the above Corollaries 11 and 12, we know that if (4) has only one nonlinear term 𝑏₂(𝑢²)_𝑥 or 𝑏₃(𝑢³)_𝑥, that is, 𝑏₃=0 or 𝑏₂=0, the solitary waves of (4) are both orbitally stable if 𝑐>1. That is to say the wave speed intervals which make the two solitary waves stable are both (1,+∞). But according to Theorems 9 and 10, when (4) has two nonlinear items 𝑏₂(𝑢²)_𝑥 and 𝑏₃(𝑢³)_𝑥, the stability of solitary waves will be affected by the interaction between them. For convenience, we call the solitary wave whose wave speed 𝑐 satisfies 𝑐>𝑐₀ (𝑐₀=(𝑏²₂+𝑏⁴₂+729𝑏²₃)/27𝑏₃) the big wave speed solitary wave, while we call the solitary wave whose wave speed 𝑐 satisfies 𝑐<𝑐₀ the small wave speed solitary wave. Generally, we have the results from Theorems 9 and 10 as follows.(1)For given 𝑏₃>0, when |𝑏₂| is larger, the wave speed interval which makes the solitary waves stable will become smaller for the big wave speed solitary wave, but the wave speed interval which makes the solitary waves stable will become larger for the small wave speed solitary wave.(2)For given 𝑏₂. For the big wave speed solitary wave, the wave speed interval which makes it stable will become larger if 𝑏₃ is bigger and the wave speed interval will become smaller if 𝑏₃ is smaller. For the small wave speed solitary wave, the wave speed interval which makes it stable will become smaller if 𝑏₃ is bigger and the wave speed interval will become larger if 𝑏₃ is smaller.

Summarizing the above results, it is significant to analyze the effect by multiple nonlinear terms on orbital stability of the solitary waves, at least in the application. For example, fix 𝑏₂ in (4). If we need to know the orbital stability of the small wave speed solitary wave in practical problems, since the wave speed interval which makes it stable will become larger as 𝑏₃ is smaller, and (𝑏²₂+𝑏⁴₂+729𝑏²₃)/27𝑏₃→+∞ as 𝑏₃→0, it has little influence on the stability to ignore 𝑏₃𝑢³ in the application. But if we need to consider the orbital stability of the big wave speed solitary wave, the wave speed interval which makes it stable will become smaller as 𝑏₃ is smaller, so it is not suitable to ignore 𝑏₃𝑢³ in the application here.

5. Instability of the Solitary Waves

In this section, we will prove the conclusion (2) given in Theorem 7; that is, the solitary wave solution 𝜙^→_𝑐 is orbitally unstable if 𝑑^′′(𝑐)<0.

Since 𝐽 given in Section 3 is not onto, we cannot apply Grillakis-Shatah-Strauss theory on the system (4) directly. In order to prove instability, we define a new conservational functional 𝐼^→𝑢=^∞→_−∞𝑢(𝑥)𝑑𝑥. (71) We will prove that 𝑑^′′(𝑐)<0 is the sufficient condition to judge orbital instability of solitary wave solutions by estimating to the solution of initial value problem.

5.1. Estimate to the Solution of Initial Value Problem for (4)

Lemma 13. The unique solution ^→𝑢(𝑡) of (4) with initial data ^→𝑢(0)=^→𝑢₀ satisfies 𝐸^→𝑢(𝑡)=𝐸(𝑢,𝑣)=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡,𝑡∈𝑅⁺,𝑉^→𝑢(𝑡)=𝑉(𝑢,𝑣)=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡,𝑡∈𝑅⁺, (72) where 𝐸(^→𝑢)=−∫_𝑅(𝑣𝑢+(𝑏₂/3)𝑢³+(𝑏₃/4)𝑢⁴)𝑑𝑥 and 𝑉(^→𝑢)=(1/2)∫_𝑅(−𝑢_𝑥𝑥𝑢+𝑢²+𝑣²)𝑑𝑥.
From Lemma 4, (23), and (27), we can prove Lemma 13 easily. We now prove that 𝐼(^→𝑢)=∫^∞→_−∞𝑢(𝑥)𝑑𝑥 is an invariance.

Lemma 14. If ∫_𝑅𝑢₀(𝑥)𝑑𝑥 and ∫_𝑅𝑣₀(𝑥)𝑑𝑥 converge, then 𝐼(𝑢)=∫_𝑅𝑢𝑑𝑥 and 𝐼(𝑣)=∫_𝑅𝑣𝑑𝑥 converge and are constants for any 𝑡∈𝑅⁺.

Proof. Integrating (4) separately yields ^𝑏_𝑎𝑢(𝑥,𝑡)𝑑𝑥−^𝑏_𝑎𝑢()=𝑥,0𝑑𝑥_{𝑡0𝑏𝑎}𝜕𝜕𝜕𝑥𝜕𝑥²−1⁻¹𝑣+𝑏₂𝑢²+𝑏₃𝑢³𝑑𝑥𝑑𝜏,_𝑏𝑎𝑣(𝑥,𝑡)𝑑𝑥−^𝑏_𝑎𝑣(𝑥,0)𝑑𝑥=−^𝑡₀^𝑏_𝑎𝜕𝑢()𝜕𝑥𝑥,𝜏𝑑𝑥𝑑𝜏. (73) Now we analyze the second formula and have −^𝑡₀^𝑏_𝑎𝑢_𝑥𝑑𝑥𝑑𝜏=−^𝑡₀[𝑢(𝑏,𝜏)−𝑢(𝑎,𝜏)]𝑑𝜏. (74) For any fixed 𝜏, 𝑢(𝑏,𝜏)→0 and 𝑢(𝑎,𝜏)→0, as 𝑎→−∞ and 𝑏→+∞, −^𝑏_𝑎^𝑡₀𝑢_𝑥𝑑𝜏𝑑𝑥⟶0. (75) Thus ^𝑏_𝑎𝑣(𝑥,𝑡)𝑑𝑥−^𝑏_𝑎𝑣(𝑥,0)𝑑𝑥⟶0. (76) Similarly, ^𝑏_𝑎𝑢(𝑥,𝑡)𝑑𝑥−^𝑏_𝑎𝑢(𝑥,0)𝑑𝑥⟶0. (77) Hence 𝐼(𝑢)=∫_𝑅𝑢𝑑𝑥 and 𝐼(𝑣)=∫_𝑅𝑣𝑑𝑥 exist and are equal to ∫_𝑅𝑢₀(𝑥)𝑑𝑥 and ∫_𝑅𝑣₀(𝑥)𝑑𝑥, respectively. This completes the proof of Lemma 14.

The next theorem is the key step in the proof of instability, and it is the main result of this section.

Theorem 15. Let Λ¹𝑢₀∈𝐿¹ and 𝑣₀∈𝐿¹, where Λ^𝑘=(1−𝜕²/𝜕𝑥²)^𝑘/2(𝑘∈𝑍). Assume that ^→𝑢=(^𝑢_𝑣) satisfies (4) and 𝑢(0,𝑥)=𝑢₀. Then sup_{−∞<𝑥<∞}^𝑥→_−∞𝑢(𝑧,𝑡)𝑑𝑧≤𝑐₀1+𝑡^2/3+𝑡^9/10, (78) where the constant 𝑐₀ only depends on ^→𝑢₀.

In order to prove Theorem 15, we need a series of lemmas. The first one is the well-known Van der Corput lemma [12]. The proofs of the following Lemmas 17 and 18 are similar to those which are given in [13], and we omit the details.

Lemma 16 (Van der Corput lemma). Let ℎ(𝜉) be either convex or concave on [𝑎,𝑏] with −∞≤𝑎<𝑏≤∞. If ℎ^′′(𝜉)≠0 in [𝑎,𝑏]; then ^𝑏_𝑎𝑒^{𝑖ℎ(𝜉)}𝑑𝜉≤4min_[]𝑎,𝑏ℎ^′′(𝜉)^−1/2. (79)

Lemma 17. Suppose 𝑡>0,𝑛>0, one has sup_{−∞<𝛼<+∞}^𝑛_−𝑛𝑒^{𝑖𝑡ℎ(𝜉,𝛼)}𝑑𝜉≤𝑐₀𝑡^−1/3+𝑡^−1/2𝑛², (80) where 𝑐₀ is a constant and ℎ(𝜉,𝛼)=𝜉/1+𝜉²+𝛼𝜉.

Lemma 18. For 1≤𝑝≤∞, if 𝑢∈𝐿^𝑝(𝑅), we have Λ⁻¹𝑢∈𝐿^𝑝 and ‖Λ⁻¹𝑢‖_𝑝≤‖𝑢‖_𝑝.

The following lemma concerns the decay of the linear evolution operator.

Lemma 19. Suppose that 𝑆(𝑡) the evolution operator of the linear equation ^→𝑢_𝑡+0(1−Δ)⁻¹10^→𝑢_𝑥=0,^→𝑢(0)=^→𝑢₀=𝑢₀𝑣₀. (81) That is to say, 𝑆(𝑡)^→𝑢(0)=^→𝑢(𝑡). If Λ¹𝑢₀∈𝐿¹ and 𝑣₀∈𝐿¹, we have 𝑆(𝑡)^→𝑢₀∈𝐿^∞ and 𝑆(𝑡)^→𝑢_0∞≤𝑐₀𝑡^−1/3+𝑡^−1/10Λ¹𝑢₀₁+𝑣₀₁, (82) where 𝑐₀ is a constant.

Proof. The solution of the linear equation is𝑢^→(𝑡)=𝑆(𝑡)𝑢^→₀(𝑥)=1∫2𝜋^∞−∞𝑒^𝑖𝑥𝜉𝜉cos1+𝜉²𝑡1𝑖1+𝜉²𝜉sin1+𝜉²𝑡1+𝜉²𝑖𝜉sin1+𝜉²𝑡𝜉cos1+𝜉²𝑡⋅𝑢^→₀(𝜉)=1𝑑𝜉∫2𝜋^∞−∞𝑒^𝑖𝑥𝜉−𝜉cos1+𝜉²𝑡𝑖1+𝜉²−𝜉sin1+𝜉²𝑡𝑖1+𝜉²−𝜉sin1+𝜉²𝑡−𝜉cos1+𝜉²𝑡⋅𝑢^→₀(𝜉)𝑑𝜉, (83)where ^→𝑢₀ is the Fourier transform of ^→𝑢₀.
According to Fubini's theorem and Lemmas 17 and 18, we have ^→𝑢(𝑡=𝑆(𝑡))_→𝑢₀(𝑥≤1)4𝜋^+∞_−∞𝑢₀±11+𝜉²𝑣₀𝑒^{𝑖𝑡(−𝜉/1+𝜉2±(𝑥/𝑡)𝜉)}+1𝑑𝜉4𝜋^+∞_−∞𝑣₀±1+𝜉²𝑢₀𝑒^{𝑖𝑡(−𝜉/1+𝜉2±(𝑥/𝑡)𝜉)}≤1𝑑𝜉4𝜋^+∞_−∞𝑢₀±Λ⁻¹𝑣₀𝑒^{𝑖𝑡(−𝜉/1+𝜉2±(𝑥/𝑡)𝜉)}+1𝑑𝜉4𝜋^+∞_−∞𝑣₀±Λ¹𝑢₀𝑒^{𝑖𝑡(−𝜉/1+𝜉2±(𝑥/𝑡)𝜉)}≤1𝑑𝜉2𝜋_𝜉>𝑛𝑢₀+Λ⁻¹𝑣₀+𝑣₀+Λ¹𝑢₀+1𝑑𝜉4𝜋_𝑅𝑢₀𝑦±Λ⁻¹𝑣₀𝑦×^𝑛_−𝑛𝑒^{𝑖𝑡(−𝜉/1+𝜉2±(𝑥/𝑡)𝜉−(𝑦/𝑡)𝜉)}+1𝑑𝜉𝑑𝑦4𝜋_𝑅𝑣₀𝑦±Λ¹𝑢₀𝑦×^𝑛_−𝑛𝑒^{𝑖𝑡(−𝜉/1+𝜉2±(𝑥/𝑡)𝜉−(𝑦/𝑡)𝜉)}𝑑𝜉𝑑𝑦. (84)
Therefore, ^→𝑢(𝑡)≤𝑐₀Λ¹𝑢_0𝐿1+𝑣_0𝐿1×_𝜉>𝑛1+𝜉⁻²𝑑𝜉^1/2+𝑐₀^→𝑢_{0𝐿1×𝐿1}+Λ⁻¹𝑣_0𝐿1+Λ¹𝑢_0𝐿1×sup_{−∞<𝛼<∞}^∞_−∞𝑒^{𝑖𝑡ℎ(𝜉,𝛼)}𝑑𝜉≤𝑐₀𝑛^−1/2+𝑡^−1/3+𝑡^−1/2𝑛²𝑣_0𝐿1+Λ¹𝑢_0𝐿1. (85) Choosing 𝑛=𝑡^1/5, we have 𝑆(𝑡)^→𝑢₀≤𝑐₀𝑡^−1/3+𝑡^−1/10𝑣_0𝐿1+Λ¹𝑢_0𝐿1, (86) where 𝑡>0. This completes the proof of Lemma 19.

Proof of Theorem 15. Let 𝑤^→(𝑡)=𝑆(𝑡)^→𝑢₀. Then 𝑤^→(𝑡) satisfies 𝑤^→_𝑡+0𝜕1−²𝜕𝑥²⁻¹10^→𝑤_𝑥=0,with𝑤^→(0)=^→𝑢₀. (87) The solution ^→𝑢(𝑡) of the nonlinear (4) can be written as ^→𝑢(𝑡)=𝑤^→(𝑡)+𝜕𝜕𝑥^𝑡₀𝑆(𝑡−𝜏)−Λ⁻²𝑏₂𝑢²+𝑏₃𝑢³0𝑑𝜏. (88)
Let 𝑈^→(𝑥,𝑡)=∫^𝑥→_−∞𝑢(𝑦,𝑡)𝑑𝑦, 𝑈^→(𝑥,0)=∫^𝑥→_−∞𝑢(𝑦,0)𝑑𝑦, and 𝑊^→(𝑥,𝑡)=∫^𝑥→_−∞𝑤(𝑦,𝑡)𝑑𝑦. Then 𝑈^→(𝑥,𝑡)=𝑊^→(𝑡)−∫^𝑡₀𝑆(𝑡−𝜏)^{Λ−2(𝑏2𝑢2+𝑏3𝑢3)}₀𝑑𝜏.
We estimate both two terms in the above formula on the right-hand side separately. Firstly, from the equation for 𝑤^→(𝑡), we can obtain 𝑤^→(𝑡)=^→𝑢₀−𝜕_𝑥^𝑡₀0(1−Δ)⁻¹10^→𝑤(𝜏)𝑑𝜏. (89) Therefore 𝑊^→(𝑡)=𝑈^→₀−^𝑡₀0(1−Δ)⁻¹10^→𝑤(𝜏)𝑑𝜏. (90) Since 𝑤^→(𝑡)=𝑆(𝑡)^→𝑢₀, we obtain 𝑊^→(≤𝑥,𝑡)_→𝑢_{0𝐿1×𝐿1}+^𝑡₀0(1−Δ)⁻¹𝑆(𝜏)𝑢10𝑣⁰₀=𝑑𝜏_→𝑢_{0𝐿1×𝐿1}+^𝑡₀𝑆(𝜏)(1−Δ)⁻¹𝑣₀𝑢₀≤𝑑𝜏_→𝑢_{0𝐿1×𝐿1}+^𝑡₀𝑆(𝜏)(1−Δ)⁻¹𝑣₀𝑢₀𝑑𝜏. (91) Using Lemma 19, substituting (1−Δ)⁻¹𝑣₀𝑢₀ for 𝑢₀𝑣₀, we have 𝑊^→(𝑥,𝑡)≤^→𝑢_{0𝐿1×𝐿1}+𝑐₀^𝑡₀𝜏^−1/3+𝜏^−1/10𝑑𝜏Λ⁻¹𝑣_0𝐿1+𝑢_0𝐿1≤𝑐₀1+𝑡^2/3+𝑡^9/10𝑣_0𝐿1+𝑢_0𝐿1. (92) Let 𝑌^→(𝑥,𝑡)=^𝑡₀𝑆()Λ𝑡−𝜏⁻²𝑏₂𝑢²+𝑏₃𝑢³0𝑑𝜏. (93)
Using Lemma 19 again, and substituting Λ⁻²(𝑏₂𝑢²+𝑏₃𝑢³)0 for 𝑢₀𝑣₀, we obtain 𝑌^→(𝑥,𝑡)≤^𝑡₀𝑆()Λ𝑡−𝜏⁻²𝑏₂𝑢²+𝑏₃𝑢³0𝑑𝜏≤𝑐₀^𝑡₀(𝑡−𝜏)^−1/3+(𝑡−𝜏)^−1/10Λ⁻¹𝑏₂𝑢²+𝑏₃𝑢³_𝐿1𝑑𝜏. (94) In view of Lemma 18, we have Λ⁻¹𝑏₂𝑢²+𝑏₃𝑢³_𝐿1≤𝑏₂𝑢²+𝑏₃𝑢³_𝐿1. (95) Therefore 𝑌^→(𝑥,𝑡)≤𝑐₀^𝑡₀(𝑡−𝜏)^−1/3+(𝑡−𝜏)^−1/10𝑢²_𝐿1+𝑢³_𝐿1𝑑𝜏≤𝑐₀𝑡^2/3+𝑡^9/10. (96)
Summarizing the estimate of 𝑊^→(𝑥,𝑡) and 𝑌^→(𝑥,𝑡) above yields the result of Theorem 15; that is, sup_{−∞<𝑥<∞}^𝑥→_−∞𝑢(𝑧,𝑡)𝑑𝑧≤𝑐₀1+𝑡^2/3+𝑡^9/10. (97)

5.2. Proof of Instability

Theorem 20. Let 𝑐≠0 be fixed. If 𝑑^′′(𝑐)<0, then there is a curve 𝜔→Φ_𝜔 such that 𝑉(Φ^→_𝜔)=𝑉(𝜑^→_𝑐), Φ^→_𝑐=𝜑^→_𝑐, and on which 𝐸(^→𝑢) has a strict local maximum at ^→𝑢=𝜑^→_𝑐.

Proof. Let 𝜒_𝑐 be the unique negative eigenfunction of 𝐻_𝑐, which has been proved in Section 2. Next we define Φ^→_𝜔=𝜑^→_𝜔+𝑠(𝜔)𝜒^→_𝑐,for𝜔⟶𝑐, (98) where 𝑠(𝜔) satisfies 𝑠(𝑐)=0 and 𝑉(Φ^→_𝜔)=𝑉(𝜑^→_𝑐).
By the implicit function theorem, the function 𝑠(𝜔) can be determined. In fact, 𝜕𝑉𝜕𝑠^→𝜑_𝜔+𝑠𝜒^→_{{𝑠=0,𝜔=𝑐}}=𝑉𝜑^′→_𝑐,𝜒^→_𝑐=^∞_−∞Λ²𝜑_𝑐⋅𝜒_1,𝑐+𝜓_𝑐⋅𝜒_2,𝑐𝑑𝑥, (99) where 𝜒_𝑐=^𝜒1,𝑐_𝜒2,𝑐 with 𝜒_1,𝑐=(𝑐−𝜆)𝜒_2,𝑐, and 𝜆 is the unique negative eigenvalue of 𝐻_𝑐 and 𝜑_𝑐=𝑐𝜓_𝑐,𝜑_𝑐>0,𝜒_1,𝑐>0,Λ²𝜑_𝑐>0. Therefore 𝜕𝑉𝜕𝑠^→𝜑_𝜔+𝑠𝜒^→_{{𝑠=0,𝜔=𝑐}}=^+∞_−∞Λ²𝜑_𝑐⋅𝜒_1,𝑐+𝜓_𝑐⋅𝜒_2,𝑐=𝑑𝑥_+∞−∞Λ²𝜑_𝑐⋅𝜒_1,𝑐+1𝑐()𝜑𝑐−𝜆^𝑐⋅𝜒_1,𝑐𝑑𝑥≠0. (100) It is easy to see that 𝑑²𝑑𝜔²𝐸Φ^→_{𝜔𝜔=𝑐}=𝐻𝑦_𝑐^→,𝑦^→,where𝑦^→=𝜕Φ^→_𝜔𝜕𝜔_𝜔=𝑐=𝜕𝜑^→_𝑐𝜕𝑐+𝑠^′(𝑐)𝜒^→_𝑐. (101) So it suffices to show that ⟨𝐻𝑦_𝑐^→,𝑦^→⟩<0. Since 0=𝑑𝑉𝜑^→_𝑐=𝑑𝜔𝑑𝑉Φ^→_{𝜔𝜔=𝑐}=𝑉𝑑𝜔_′→𝜑_𝑐,𝑦^→=𝑉𝜑^′→_𝜔,𝑦^→_𝜔=𝑐, (102) then 𝑑^′′(𝑐)=𝑉𝜑^′→_𝑐,𝑑𝜑^→_𝑐=𝑉𝑑𝑐^′→𝜑_𝑐,𝑦^→−𝑠^′(𝑐)𝜒^→_𝑐=𝑉𝜑^′→_𝑐,𝑦^→−𝑉𝜑^′→_𝑐,𝑠^′(𝑐)𝜒^→_𝑐=−𝑠^′(𝑐)𝑉𝜑^′→_𝑐,𝜒^→_𝑐,𝐻𝑦_𝑐^→=𝐻_𝑐𝜕𝜑^→_𝑐𝜕𝑐+𝐻_𝑐𝑠^′(𝑐)𝜒^→_𝑐=𝐸𝜑^′′→_𝑐+𝑐𝑉𝜑^′′→_𝑐𝜕𝜑^→_𝑐𝜕𝑐+𝐻_𝑐𝑠^′(𝑐)𝜒^→_𝑐. (103) Note that 𝐸^′(𝜑^→_𝑐)+𝑐𝑉^′(𝜑^→_𝑐)=0. We derivate it with respect to 𝑐, and then 𝐸𝜑^′′→_𝑐𝜕𝜑^→_𝑐𝜕𝑐+𝑉𝜑^′→_𝑐+𝑐𝑉𝜑^′′→_𝑐𝜕𝜑^→_𝑐𝜕𝑐=0. (104) Namely, 𝐸𝜑^′′→_𝑐+𝑐𝑉𝜑^′′→_𝑐𝜕𝜑^→_𝑐𝜕𝑐+𝑉𝜑^′→_𝑐=0. (105) So 𝐻𝑦_𝑐^→=−𝑉𝜑^′→_𝑐+𝑠^′(𝑐)𝐻𝜒_𝑐^→_𝑐,𝐻𝑦_𝑐^→,𝑦^→=−𝑉𝜑^′→_𝑐+𝑠^′(𝑐)𝐻𝜒_𝑐^→_𝑐,𝜕𝜑^→_𝑐𝜕𝑐+𝑠^′(𝑐)𝜒^→_𝑐=−𝑉𝜑^′→_𝑐,𝜕𝜑^→_𝑐+𝜕𝑐−𝑉𝜑^′→_𝑐,𝑠^′(𝑐)𝜒^→_𝑐+𝑠^′(𝑐)𝐻𝜒_𝑐^→_𝑐,𝜕𝜑^→_𝑐+𝑠𝜕𝑐^′(𝑐)𝐻𝜒_𝑐^→_𝑐,𝑠^′(𝑐)𝜒^→_𝑐=𝑠^′(𝑐)𝐻𝜒_𝑐^→_𝑐,𝜕𝜑^→_𝑐+𝑠𝜕𝑐^′(𝑐)²𝐻𝜒_𝑐^→_𝑐,𝜒^→_𝑐=𝑠^′(𝑐)𝐻_𝑐𝜕𝜑^→_𝑐,𝜕𝑐^→𝜒_𝑐+𝑠^′(𝑐)²𝐻𝜒_𝑐^→_𝑐,𝜒^→_𝑐=𝑠^′(𝑐)−𝑉𝜑^′→_𝑐,𝜒^→_𝑐+𝑠^′(𝑐)²𝐻𝜒_𝑐^→_𝑐,𝜒^→_𝑐=𝑑^′′(𝑐)+𝑠^′(𝑐)²𝐻𝜒_𝑐^→_𝑐,𝜒^→_𝑐<0. (106) Hence, (𝑑²/𝑑𝜔²)𝐸(Φ^→_𝜔)|_𝜔=𝑐=⟨𝐻𝑦_𝑐^→,𝑦^→⟩<0. The result in Theorem 20 holds.

Lemma 21 (see [14]). There exists 𝜀>0 and a unique 𝐶¹ map 𝛼∶𝑈_𝜀→𝑅, such that for any ^→𝑢∈𝑈_𝜀 and 𝑟∈𝑅, (1)^→𝑢⋅+𝛼^→𝑢,𝑇^′(0)𝜑^→_𝑐(2)𝛼=0,_→𝑢(⋅+𝑟)=𝛼^→𝑢(3)𝛼−𝑟,_′→𝑢=(𝜕/𝜕𝑥)𝜑^→_𝑐⋅−𝛼^→𝑢_→𝑢,𝜕²/𝜕𝑥𝜑^2→_𝑐⋅−𝛼^→𝑢, (107) where 𝑈_𝜀=^→𝑢∈𝑋∶inf_𝑠∈𝑅^→𝑢−𝜏𝜑_𝑠^→_𝑐𝑋<𝜀. (108)
Next we define an auxiliary operator 𝐵 which will play a critical role in the proof of instability.

Definition 22. For ^→𝑢∈𝑈_𝜀,𝐵(^→𝑢) is defined by the formula 𝐵^→𝑢=𝑦^→⋅−𝛼^→𝑢−𝐼^→𝑢,𝑦^→⋅−𝛼^→𝑢𝜕𝐼𝜕𝑥⁻¹𝛼^′→𝑢. (109) By Lemma 21, 𝐵(^→𝑢) can also be written as 𝐵^→𝑢=𝑦^→⋅−𝛼^→𝑢−𝐼^→𝑢,𝑦^→⋅−𝛼^→𝑢_→𝑢,𝜕²/𝜕𝑥𝜑^2→_𝑐⋅−𝛼^→𝑢𝐼⁻¹𝜕²𝜕𝑥²𝜑_𝑐⋅−𝛼^→𝑢, (110) where 𝐼=^1−Δ0₀₁.

The next lemma summarizes the properties of 𝐵.

Lemma 23 (see [14]). 𝐵(^→𝑢)∶𝑈_𝜀→𝑋 is a 𝐶¹ function. Moreover, 𝐵 commutes with translations, 𝐵(𝜑^→_𝑐)=𝑦^→ and for any ^→𝑢∈𝑈_𝜀,⟨𝐵(^→𝑢),𝐼^→𝑢⟩=0.

Lemma 24 (see [14]). There exists a 𝐶¹ function Π∶^→𝑣∈𝑈_𝜀∶𝑉^→𝑣=𝑉𝜑^→_𝑐⟶𝑅, (111) which is invariant under translations, such that 𝐸𝜑^→_𝑐<𝐸^→𝑣+Π^→𝑣𝐸^′→𝑣,𝐵^→𝑣, (112) for any ^→𝑣∈𝑈_𝜀 with 𝑉(^→𝑣)=𝑉(𝜑^→_𝑐) and ^→𝑣 is not a translate of 𝜑^→_𝑐.

Lemma 25 (see [14]). According to Theorem 20, there is a curve 𝜔→Φ_𝜔 which satisfies 𝐸(Φ^→_𝜔)<𝐸(𝜑^→_𝑐) for 𝜔≠𝑐, 𝑉(Φ^→_𝜔)=𝑉(𝜑^→_𝑐), and ⟨𝐸^′(Φ^→_𝜔),𝐵(Φ^→_𝜔)⟩ changes sign as 𝜔 passes through 𝑐, with 𝑐≠0.

Theorem 26. If (4) has a bell-profile solitary wave solution 𝜙^→_𝑐, when 𝑑^′′(𝑐)<0, then 𝜙^→_𝑐 is orbitally unstable.

Proof. Firstly, we consider 𝑐≠0. Let 𝜀>0, small enough, and 𝑈_𝜀 be the tubular neighbourhood defined above. By Lemma 25 we can choose ^→𝑢₀∈𝑋 which is arbitrarily close to 𝜑^→_𝑐, such that 𝑉(^→𝑢₀)=𝑉(𝜑^→_𝑐), 𝐸(^→𝑢₀)<𝐸(𝜑^→_𝑐), and ⟨𝐸^′(^→𝑢₀),𝐵(^→𝑢₀)⟩>0. To prove the instability of 𝜑^→_𝑐, it suffices to show that there are some elements ^→𝑢₀∈𝑋 which are arbitrarily close to 𝜑^→_𝑐, but the solution ^→𝑢(𝑥,𝑡) with the initial data ^→𝑢₀ exits from 𝑈_𝜀 in finite time. Let [0,𝑡₁) be the maximal interval for which ^→𝑢(𝑥,𝑡) lies continuously in 𝑈_𝜀, where 𝑡₁>0. Let 𝑇 be the maximum existence time for the solution ^→𝑢(𝑥,𝑡) with initial data ^→𝑢₀. If 𝑇 is finite, it is easy to see that 𝜑^→_𝑐 is orbital instability by definition, so we may assume that 𝑇=+∞ and our purpose now is to show that 𝑡₁<+∞; that is to say, it is instability if it blows up at a finite time. The proof is as follows.
Firstly, in view of Lemmas 4, 13, and 14 and Theorem 15, we know that ^→𝑢 enjoys the following properties: ^→𝑢∈𝐶0,𝑡₁;𝑋,^→𝑢(0,𝑥)=^→𝑢₀(𝑥),𝐸^→𝑢(𝑡),𝑉^→𝑢(𝑡)areconstant,for𝑡∈0,𝑡₁,𝐼(𝑢(𝑡)),𝐼(𝑣(𝑡))convergeandareconstant,for𝑡∈0,𝑡₁,sup_{0≤𝑡<𝑡1}^→𝑢(𝑥,𝑡)_𝑋≤𝑐₁,sup_𝑥∈𝑅^𝑥→_−∞𝑢𝑦,𝑡𝑑𝑦≤𝑐₂1+𝑡^2/3+𝑡^9/10, (113) where 𝑐₁ depends on 𝜑^→_𝑐 and 𝜀, 𝑐₂ depends on 𝑐₁, ‖𝑣₀‖_𝐿1, and ‖^→𝑢₀‖_𝐿1.
Let 𝛽(𝑡)=𝛼(^→𝑢(𝑡)), where 𝛼 is defined by Lemma 21 and define 𝑦^→(𝑧)=𝑑Φ^→_𝑐=𝑑𝑑𝑐^→𝜑_𝑐𝑑𝑐+𝑠^′(𝑐)𝜒^→_𝑐,_→𝑌(𝑥)=^𝑥_−∞𝐼𝑦^→(𝑧)𝐴(𝑡)𝑑𝑧,=^∞→_−∞𝑌𝑥−𝛽(𝑡)⋅^→𝑢(𝑥,𝑡)𝑑𝑥,0≤𝑡<𝑡₁, (114) where the function 𝐴(𝑡) serves as a Lyapunov function, and ^→𝛾=^∞→_−∞𝑦(𝑥)𝑑𝑥. (115) Due to the assumptions above, it is observed that ^∞_−∞(1+|𝑥|)^1/2𝜕𝜑^→_𝑐𝜕𝑐𝑑𝑥<∞,_∞−∞(1+|𝑥|)^→1/2𝜒_𝑐𝑑𝑥<∞. (116) Therefore ∫^∞_−∞(1+|𝑥|)^1/2|𝑦^→(𝑥)|𝑑𝑥<∞, such that |^→𝛾|<+∞.
Indeed, if 𝐻 is the Heaviside function and 𝑊^→(𝑥)=∫^𝑥→_−∞𝑦(𝑧)𝑑𝑧−^→𝛾𝐻(𝑥), then 𝑊^→(𝑥)=^𝑥→_−∞𝑦(𝑧)𝑑𝑧for𝑥<0,_→𝑊(𝑥)=−𝑦^∞_𝑥^→(𝑧)𝑑𝑧for𝑥≥0. (117) Now we can have 𝐴(𝑡)=𝑌^∞_−∞^→𝑥−𝛽(𝑡)−^→𝛾𝐻𝑥−𝛽(𝑡)⋅^→𝑢()+𝑥,𝑡𝑑𝑥_→𝛾^∞→_−∞𝐻𝑥−𝛽(𝑡)⋅^→𝑢()=𝑥,𝑡𝑑𝑥_∞−∞→𝑌𝑥−𝛽(𝑡)−^→𝛾𝐻𝑥−𝛽(𝑡)⋅^→𝑢()+𝑥,𝑡𝑑𝑥_→𝛾^∞_𝛽(𝑡)^→𝑢(𝑥,𝑡)𝑑𝑥. (118) Hence, by (113), |𝐴(𝑡)|≤𝑌^→−^→𝛾𝐻₀^→𝑢(𝑡)₀+𝑐₂1+𝑡^2/3+𝑡^9/10. (119) It follows from Minkowski's inequality that ⁰_−∞𝑌^→(𝑥)−^→𝛾𝐻(𝑥)²𝑑𝑥^1/2=𝑌⁰_−∞^→(𝑥)²𝑑𝑥^1/2≤⁰_−∞⁰_𝑧𝐼𝑦^→(𝑧)²𝑑𝑥^1/2=𝑑𝑧_{0−∞0𝑧→}𝑦(𝑧)²𝑑𝑥^1/2𝑑𝑧=𝑦⁰_−∞^→(𝑧)√−𝑧𝑑𝑧<∞. (120) Similarly, (∫^∞₀|𝑌^→−^→𝛾𝐻(𝑥)|²𝑑𝑥)^1/2<∞. Therefore |𝐴(𝑡)|≤𝑐₂1+𝑡^2/3+𝑡^9/10,for0≤𝑡<𝑡₁. (121) Since 𝐴(𝑡)=⟨𝑌^→(𝑥−𝛽(𝑡)),^→𝑢(𝑥,𝑡)⟩, and 𝛽(𝑡)=𝛼(^→𝑢(𝑡)), now we estimate 𝑑𝐴(𝑡)/𝑑𝑡, by calculating, and have 𝑑𝐴(𝑡)=𝑑𝑑𝑡_→𝑌𝑥−𝛽(𝑡),𝑑𝑡^→𝑢(𝑥,𝑡)+𝑌^→(𝑡),𝜕𝑥−𝛽𝜕𝑡^→𝑢(=−𝐼𝑥,𝑡)_→𝑦𝑥−𝛽,^→𝑢𝛼^′→𝑢+𝑌^→𝑥−𝛽,𝜕𝜕𝑡^→𝑢(𝑥,𝑡). (122) Since 𝑑^→𝑢/𝑑𝑡=𝐽𝐸^′(^→𝑢), where 𝐽=(𝜕/𝜕𝑥)𝐼⁻¹ and (𝜕/𝜕𝑥)𝑌^→=𝐼𝑦^→, it follows that 𝑑𝐴=𝐼𝑑𝑡^→𝑦𝑥−𝛽(𝑡),^→𝑢𝜕𝛼𝜕𝑥^′→𝑢−𝐼𝑦^→𝑥−𝛽(𝑡),𝐼⁻¹𝐸^′→𝑢=𝐼𝑦^→𝑥−𝛽(𝑡),^→𝑢𝜕𝐼𝜕𝑥⁻¹𝛼^′→𝑢−𝑦^→𝑥−𝛽(𝑡),𝐸^′→𝑢=−𝐵^→𝑢,𝐸^′→𝑢. (123) Since 0<𝐸(𝜑^→₀)−𝐸(^→𝑢₀)=𝐸(𝜑^→_𝑐)−𝐸(^→𝑢(𝑡)), from Lemma 24, we can deduce that 0<Π^→𝑢(𝑡)𝐵^→𝑢(𝑡),𝐸^′→𝑢(𝑡). (124) Since ⟨𝐸^′(^→𝑢₀),𝐵(^→𝑢₀)⟩>0 and it is continuous, we can obtain ⟨𝐸^′(^→𝑢(𝑡)),𝐵(^→𝑢(𝑡))⟩>0. Therefore for all 0<𝑡<𝑡₁, Π(^→𝑢(𝑡))>0. Moreover, since ^→𝑢(𝑡)∈𝑈_𝜀 and Π(𝜑^→_𝑐)=0, we may assume that 0<Π(^→𝑢(𝑡))<1, 0<𝑡<𝑡₁, by choosing 𝜀 smaller if necessary. So for all 𝑡∈[0,𝑡₁), by Lemma 24, we have 𝐵^→𝑢(𝑡),𝐸^′→𝑢≥Π^→𝑢(𝑡)𝐵^→𝑢(𝑡),𝐸^′→𝑢(𝑡)>𝐸𝜑^→_𝑐−𝐸^→𝑢(𝑡)=𝐸𝜑^→_𝑐−𝐸^→𝑢₀=𝜀₀>0,−𝑑𝐴𝑑𝑡≥𝐸𝜑^→_𝑐−𝐸^→𝑢₀=𝜀₀>0,for𝑡∈0,𝑡₁. (125)(126) Integrating (126) on [0,𝑡₁), we have 𝐴(0)−𝐴𝑡₁≥𝜀₀𝑡₁=𝐸𝜑^→_𝑐−𝐸^→𝑢₀𝑡₁. (127) And then 𝑡₁≤𝐴(0)−𝐴𝑡₁𝐸𝜑^→_𝑐−𝐸^→𝑢₀≤2|𝐴(𝑡)|𝐸𝜑^→_𝑐−𝐸^→𝑢₀. (128) Since |𝐴(𝑡)|≤𝑐₂(1+𝑡^2/3+𝑡^9/10), we can conclude that 𝑡₁<∞. This completes the proof of Theorem 26.

6. Conclusions

In this paper, we studied the orbital stability and instability of solitary waves for (4) with two nonlinear terms. By using the orbital stability theory proposed in [10, 11], we obtained a general theorem judging the orbital stability for solitary waves of (4) in Section 3 based on proof of the local existence of the solutions, existence of the bounded state solution, and the spectral analysis of operator 𝐻_𝑐. In Section 4, we gave the explicit expressions for the discrimination 𝑑^′′_𝑖(𝑐),𝑖=1,2, of orbital stability in terms of the two exact solitary waves (𝜑_𝑖,𝜓_𝑖)^𝑇,𝑖=1,2, of (4). Furthermore, we deduced Theorems 9 and 10 which could easily judge the orbital stability of the two solitary waves (𝜑_𝑖,𝜓_𝑖)^𝑇,𝑖=1,2, and analyzed the influence of the two nonlinear terms on the orbital stability. Finally, we studied instability in Section 5. We defined a new conservational functional and estimated to the solution of initial value problem to overcome the difficulty that we could not apply Grillakis-Shatah-Strauss theory on the system directly since 𝐽 is not onto. We constructed a formal Lyapunov function and proved Theorem 26.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (Grant no. 11071164), Innovation Program of Shanghai Municipal Education Commission (Grant no. 13ZZ118), and Shanghai Leading Academic Discipline Project (Grant no. XTKX2012).