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

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.


Introduction
Symmetric regularized-long-wave equations (SRLWE) 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][4][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)   −   +   + ()  =  (, ,   ) , 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: where () is a  1 function, satisfying () > 0 if  > 0 and |()| = (||  ); |  ()| = (|| −1 ) as  → 0 for  > 1.In particular,   > 0 in the solitary wave solution (  ,   )  ( represents transposition) in Assumption 1 of [7].Moreover,   only has a simple negative eigenvalue, whose kernel is spanned by    .In addition, the rest of its eigenvalues are positive and bounded away from zero.(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 () =  2  2 +  3  3 , where () has two nonlinear terms and the symbols of  2 ,  3 are unfixed.Indeed, () is not always positive when  > 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 (  ,   )  ,  = 1, 2, where  1 () > 0 and  2 () < 0. But the orbital stability of the solitary wave solution ( 2 (),  2 ())  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  2  2 and  3  3 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   () 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  is not onto, we will define a new conservational functional ( ⃗ ) = ∫  ⃗  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.
According to Lemma 2, we only need to prove that there exists , such that if  >  and  ∈ ().Indeed, since  ∈ (), for any ⃗ Taking the Fourier transform yields
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 ( ⃗ ) and ( ⃗ ) defined by ( 23) and (27) satisfy respectively.This shows that (4) satisfies the Assumption 1 in [10].
Verification of Assumption 2. Firstly, we can prove the following lemma.
,   ,   → 0 as || → ∞, so  1 = 0 and  2 = 0. Thus, Furthermore, Due to (32), we have The above Lemma 5 shows that (4) has the bounded state solutions, and the two solitary waves ⃗  1 and ⃗  2 given in Theorem 1 both are the bounded state solutions of the equation.
Verification of Assumption 3. We consider spectrum analysis of the operator   .Now we define the operator where Therefore, For any , and that  −1   is a bounded self-conjugate operator on .The eigenvalues of   consist of the real numbers  which ensure that   −  are irreversible.
From (30), we have Since the existence of solitary wave solution ⃗   = ( 4) is based on the condition that  2 − 1 > 0,  − 1/ > 0 as  > 1, and Moreover, from (36), we have   = 0, where  = 0 is a unique zero point of   .By Sturm-Liouville theorem we know that zero is the second eigenvalue of .Thus  only has one strictly negative eigenvalue − 2 in the case of  > 1, whose corresponding eigenfunction is denoted by ; that is,  = − 2 .
Therefore,   has a unique simple negative eigenvalue, and zero is its eigenvalue and the rest of its spectrums are bounded away from zero.So,   satisfies the Assumption 3 in [10].
According to [10,11], we can get the following lemma.

Lemma 6. For any real function
We have (1/)  ), and then We have According to the above analysis, when  > 1, we can make spectrum decomposition for   .Let (41) . Thus the space  can be decomposed as a direct sum  =  +  + , where  is the kernel space of   ,  is a finitedimensional subspace and  is a closed subspace.
We now define () :  →  as () = ( ⃗   ) + ( ⃗   ), and then 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).
Remark 8.The proof of the conclusion (2) in Theorem 7 will be given by Theorem 26 in Section 5.

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    ().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  3 > 0 and  > 1 in this section.

Discrimination 𝑑 󸀠󸀠
().In view of (42), we have Next, we simplify (43).According to (6a) and (7a) in Theorem 1, we have   = (1/)  .Substituting it into (43), and letting  =     (  > 0), we obtain where   ,   ,   are given by (6c) and (7c).Since −2 <   < 0, we can solve above two integrations.Then, If then ( 45) can be simplified into the following form: 3 If then (45) can be simplified into the following form: 3 By calculating, we have where ) . (52) Furthermore, suppose that Then (50) can be written as And (51) can be written as Therefore, we only need to consider the conditions such that    () > 0 hold in (54) and (55) to study the orbital stability of the solitary waves ⃗   , while needing to consider    () < 0 to study instability.

Orbital Stability of Solitary
Waves for (4) in the Case of 3 − 3/ −  2 > 0. Based on (54), (55), and above discussion on  1 ,  2 , we want to obtain much more simple conditions on the orbital stability of solitary waves ⃗  1 and ⃗  2 .

Orbital Stability of
In order to find  such that   1 () > 0, we only need to consider  1 = 2 in (54).It is easy to see that   1 () > 0 when  satisfies Thus, ⃗  1 is orbitally stable.
In addition, we know that 3 − 3/ − ], but we always assume  > 1 through this section.So if we assume Summarizing above results, we have the following theorem.
Summarizing above results, we have the following theorem.
(2) ⃗  2 is orbitally stable if  2 < 0 and the wave speed  satisfies (62).⃗  2 is orbitally unstable if  2 > 0 and the wave speed  satisfies (63), or  2 < 0 and the wave speed  satisfies (64).(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).
According to the above Corollaries 11 and 12, we know that if (4) has only one nonlinear term  2 ( 2 )  or  3 ( 3 )  , that is,  3 = 0 or  2 = 0, the solitary waves of ( 4) are both orbitally stable if  > 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  2 ( 2 )  and  3 ( 3 )  , the stability of solitary waves will be affected by the interaction between them.For convenience, we call the solitary wave whose wave speed  satisfies  >  0 3 )/27 3 ) the big wave speed solitary wave, while we call the solitary wave whose wave speed  satisfies  <  0 the small wave speed solitary wave.Generally, we have the results from Theorems 9 and 10 as follows.
(1) For given  3 > 0, when | 2 | 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  2 .For the big wave speed solitary wave, the wave speed interval which makes it stable will become larger if  3 is bigger and the wave speed interval will become smaller if  3 is smaller.For the small wave speed solitary wave, the wave speed interval which makes it stable will become smaller if  3 is bigger and the wave speed interval will become larger if  3 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  2 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  3 is smaller, and ( 2  2 +√ 4 2 + 729 2 3 )/27 3 → +∞ as  3 → 0, it has little influence on the stability to ignore  3  3 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  3 is smaller, so it is not suitable to ignore  3  3 in the application here.

Instability of the Solitary Waves
In this section, we will prove the conclusion (2) given in Theorem 7; that is, the solitary wave solution ⃗   is orbitally unstable if   () < 0.
Since  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 We will prove that   () < 0 is the sufficient condition to judge orbital instability of solitary wave solutions by estimating to the solution of initial value problem.
The next theorem is the key step in the proof of instability, and it is the main result of this section.
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.
The following lemma concerns the decay of the linear evolution operator.
Lemma 19.Suppose that () the evolution operator of the linear equation where  0 is a constant.
Proof.The solution of the linear equation is where ̂⃗  0 is the Fourier transform of ⃗  0 .According to Fubini's theorem and Lemmas 17 and 18, we have Choosing  =  1/5 , we have where  > 0. This completes the proof of Lemma 19.
By the implicit function theorem, the function () can be determined.In fact, where   = (  1,  2, ) with  1, = ( − ) 2, , and  is the unique negative eigenvalue of   and   =   ,   > 0,  1, > 0, Λ 2   > 0. Therefore It is easy to see that So it suffices to show that ⟨  ⃗ , ⃗ ⟩ < 0. Since then Lemma 21 (see [14]).There exists  > 0 and a unique  1 map  :   → , such that for any ⃗  ∈   and  ∈ , where Next we define an auxiliary operator  which will play a critical role in the proof of instability.
The next lemma summarizes the properties of .

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   .In Section 4, we gave the explicit expressions for the discrimination    (),  = 1, 2, of orbital stability in terms of the two exact solitary waves (  ,   )  ,  = 1, 2, of (4).Furthermore, we deduced Theorems 9 and 10 which could easily judge the orbital stability of the two solitary waves (  ,   )  ,  = 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  is not onto.We constructed a formal Lyapunov function and proved Theorem 26.
is the resolvent of .
Proof.Let   be the unique negative eigenfunction of   , which has been proved in Section 2. Next we define ⃗ Φ  = ⃗   +  () ⃗   , for  → ,