Journal of Applied Mathematics

Volume 2014 (2014), Article ID 963987, 16 pages

http://dx.doi.org/10.1155/2014/963987

Research Article

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

Weiguo Zhang,¹ Xu Chen,¹ Zhengming Li,² and Haiyan Zhang¹

¹College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

²Business School, University of Shanghai for Science and Technology, Shanghai 200093, China

Received 28 February 2014; Accepted 8 May 2014; Published 26 May 2014

Academic Editor: Wan-Tong Li

Copyright © 2014 Weiguo Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 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 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)

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)

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 function, satisfying if and ; as for . In particular, 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.

In this paper, we will consider the orbital stability and instability of solitary wave solutions for the following generalized SRLWE with two nonlinear terms:

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 , where has two nonlinear terms and the symbols of are unfixed. Indeed, is not always positive when , 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 , where and . But the orbital stability of the solitary wave solution 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 and 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.

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

where . Their exact expressions are given by the following theorem.

Theorem 1. Suppose that .

(1)If , or and , then (4) has a bell-profile solitary wave solution

 where

 (2)If , or and , then (4) has another bell-profile solitary wave solution

 where

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 is the infinitesimal generator of a semigroup of if and only if is a closed operator with dense domain and there exist real numbers and , such that when , one has

(1), (2), ,

where is the resolvent set and is the resolvent of .

Lemma 3. Consider the Cauchy problem of nonlinear equation

If the following two conditions hold:

(1) is the infinitesimal generator of a semigroup in ;(2) satisfies the Lipschitz manner, which means for any ,

there exists , such that , for all , , then the initial value problem (8) has a unique solution

in .

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 , there exists , which only depends on , such that (4) has a unique solution with .

Proof. Firstly, (4) can be written as

where . is the pseudodifferential operator. The initial value problem of (10) is equal to

where

Since for any , there exists , such that for any , , we have

Therefore, satisfies the local Lipschitz manner.

Now we want to verify that is the infinitesimal generator of a semigroup in and .

According to Lemma 2, we only need to prove that there exists , such that

if and .

Indeed, since , for any , we have and . Thus . Taking the Fourier transform yields

By (15), we have

Since as , there exists positive real number , when , such that ; that is, .

Solving the inequality when , we can obtain that when ,

By (16), we know

Since as , there exists a positive real number , when , such that ; that is, .

Solving the inequality when , we can obtain that when ,

Combining (18) and (20) and choosing , 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

where

Let , whose dual space is , and the inner product of is

There exists a natural isomorphism defined by , where denotes the pairing between and , and

From (25) and (26), we know that . And we can verify that is an skew symmetric operator; that is, .

Let be a one-parameter group of unitary operator on defined by , where , for all . Obviously, . Since , we can get .

Therefore, we define

Then , .

The solitary waves (6a) and (7a) of (4) can be written as

where and are defined by (6b) and (7b), respectively. Now, we consider the orbital stability of solitary waves . Avoiding repetition, we let be one of and . We will verify that (4) and the solitary wave 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 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.

Lemma 5. is a bounded state solution of (4), satisfying .

Proof. Substituting the solution into (4), we obtain

Integrating (30) once, we get

where , are integral constants.

as , so and . Thus,

Furthermore,

Due to (32), we have .

The above Lemma 5 shows that (4) has the bounded state solutions, and the two solitary waves and 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 as , where

Therefore,

For any , we have . This means that is a self-conjugate operator, that is, , and that is a bounded self-conjugate operator on . The eigenvalues of consist of the real numbers which ensure that are irreversible.

From (30), we have . Namely,

Let . Since the existence of solitary wave solution of (4) is based on the condition that , as , and as , it is easy to know that by Weyl's essential spectrum theorem. Moreover, from (36), we have , where 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 in the case of , whose corresponding eigenfunction is denoted by ; that is, .

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 with , there exists such that .

Let . We have

Let , and then

Let

We have

Thus for any when .

According to the above analysis, when , we can make spectrum decomposition for . Let

For any , , due to .

For any , due to .

For any .

Thus the space can be decomposed as a direct sum , where is the kernel space of , is a finite-dimensional 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).

Theorem 7. Suppose that , and is the solitary wave solution of (4). Then,

(1) is orbitally stable as ;(2) is orbitally unstable as .

Remark 8. The proof of the conclusion 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 . 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 and in this section.

4.1. Discrimination

In view of (42), we have

Next, we simplify (43). According to (6a) and (7a) in Theorem 1, we have . Substituting it into (43), and letting , we obtain

where , , are given by (6c) and (7c).

Since , we can solve above two integrations. Then,

If

then (45) can be simplified into the following form:

If

then (45) can be simplified into the following form:

By calculating, we have

where

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 hold in (54) and (55) to study the orbital stability of the solitary waves , while needing to consider to study instability.

4.2. Discussion on and

In this section, we consider and in the case of and .

(1)For . If , then . Suppose that , and then

 Let . We have

Moreover, can obtain the local maximum at , where satisfies and

Since

When , it is easy to know that , and then .

(2)For . When , it is clear that , and then . But if , then and . Therefore

Similarly, we can get .

4.3. Orbital Stability of Solitary Waves for (4) in the Case of

Based on (54), (55), and above discussion on , we want to obtain much more simple conditions on the orbital stability of solitary waves and .

4.3.1. Orbital Stability of

If , then . At this time, . In order to find such that , we only need to consider in (54). It is easy to see that when satisfies

Thus, is orbitally stable.

If , . In order to make , that is, is orbitally stable, only when satisfies

If , then . Here, in (54). In order to make , we only need to consider in (54). Then, it is easy to see that when satisfies

Thus, is orbitally unstable.

4.3.2. Orbital Stability of

If , then . In order to make , it is easy to know that is orbitally stable if satisfies (62).

If , then . In order to make , we only need to consider in (55). It is easy to know that is orbitally stable if satisfies (61).

If , in order to make , we only need to consider in (55). Then, it is easy to know that is orbitally unstable if satisfies (63).

In addition, we know that is equal to or , but we always assume through this section. So if we assume , then is equal to .

Summarizing above results, we have the following theorem.

Theorem 9. Suppose that and , where .

(1) is orbitally stable if and the wave speed satisfies (61), or and the wave speed satisfies (62); is orbitally unstable if and the wave speed satisfies (63).(2) is orbitally stable if and the wave speed satisfies (62), or and the wave speed satisfies (61); is orbitally unstable if and the wave speed satisfies (63).

4.4. Orbital Stability of Solitary Waves for (4) in the Case of

4.4.1. Orbital Stability of

If , . In order to make , we only need to consider in (54). It is easy to see that is orbitally stable if c satisfies (62).

If . In order to find such that , we only need to consider in (54). It is easy to see that when satisfies

Thus, is orbitally unstable.

If , . In order to make , it is easy to know that is orbitally unstable if satisfies (63).

4.4.2. Orbital Stability of

If , then . At this time, . In order to find such that , we only need to consider in (55). It is easy to see that is orbitally stable if satisfies (62).

If , then . In order to make , it is easy to know that is orbitally unstable if c satisfies (63).

If , then . Here, in (55). In order to find such that , we only need to consider in (55). It is easy to see that is orbitally unstable if satisfies (64).

Summarizing above results, we have the following theorem.

Theorem 10. Suppose that and , where .

(1) is orbitally stable if and the wave speed satisfies (62). is orbitally unstable if and the wave speed satisfies (64), or and the wave speed satisfies (63).(2) is orbitally stable if and the wave speed satisfies (62). is orbitally unstable if and the wave speed satisfies (63), or and the wave speed 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 and . If , (4) has the solitary wave solutions ,  , where

Under the given conditions, we can easily conclude that the solitary waves , are both orbitally stable.

Proof. When , the above solitary waves (65) of (4) can be deduced from Theorem 1 directly.

Actually, it is clear that as . Substituting into (50) and (51), we have

We know that in (66), so , if and . Thus, we know that the solitary waves , of (4) are both orbitally stable according to Theorem 7.

Corollary 12. When and , then (4) has the solitary wave solution , where

Under the given conditions, we know that the solitary wave of (4) is orbitally stable.

Proof. When , 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

Substituting (67) into the above formula yields

Therefore

We know that in (70). According to Theorem 7, is orbitally stable if and . Thus, Corollary 12 holds.

According to the above Corollaries 11 and 12, we know that if (4) has only one nonlinear term or , that is, or , the solitary waves of (4) are both orbitally stable if . That is to say the wave speed intervals which make the two solitary waves stable are both . But according to Theorems 9 and 10, when (4) has two nonlinear items and , the stability of solitary waves will be affected by the interaction between them. For convenience, we call the solitary wave whose wave speed satisfies () the big wave speed solitary wave, while we call the solitary wave whose wave speed satisfies the small wave speed solitary wave. Generally, we have the results from Theorems 9 and 10 as follows.

(1)For given , when 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 . For the big wave speed solitary wave, the wave speed interval which makes it stable will become larger if is bigger and the wave speed interval will become smaller if is smaller. For the small wave speed solitary wave, the wave speed interval which makes it stable will become smaller if is bigger and the wave speed interval will become larger if 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 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 is smaller, and as , it has little influence on the stability to ignore 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 is smaller, so it is not suitable to ignore in the application here.

5. Instability of the Solitary Waves

In this section, we will prove the conclusion given in Theorem 7; that is, the solitary wave solution is orbitally unstable if .

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 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 of (4) with initial data satisfies

where and .

From Lemma 4, (23), and (27), we can prove Lemma 13 easily. We now prove that is an invariance.

Lemma 14. If and converge, then and converge and are constants for any .

Proof. Integrating (4) separately yields

Now we analyze the second formula and have

For any fixed , and , as and ,

Thus

Similarly,

Hence and exist and are equal to and , 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 and , where . Assume that satisfies (4) and . Then

where the constant only depends on .

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 be either convex or concave on with . If in ; then

Lemma 17. Suppose , one has

where is a constant and .

Lemma 18. For , if , we have and .

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

Lemma 19. Suppose that the evolution operator of the linear equation

That is to say, . If and , we have and

where is a constant.

Proof. The solution of the linear equation is

where is the Fourier transform of .

According to Fubini's theorem and Lemmas 17 and 18, we have

Therefore,

Choosing , we have

where . This completes the proof of Lemma 19.

Proof of Theorem 15. Let . Then satisfies

The solution of the nonlinear (4) can be written as

Let , , and . Then .

We estimate both two terms in the above formula on the right-hand side separately. Firstly, from the equation for , we can obtain

Therefore

Since , we obtain

Using Lemma 19, substituting for , we have

Let

Using Lemma 19 again, and substituting for , we obtain

In view of Lemma 18, we have

Therefore

Summarizing the estimate of and above yields the result of Theorem 15; that is,

5.2. Proof of Instability

Theorem 20. Let be fixed. If , then there is a curve such that , , and on which has a strict local maximum at .

Proof. Let be the unique negative eigenfunction of , which has been proved in Section 2. Next we define

where satisfies and .

By the implicit function theorem, the function can be determined. In fact,

where with and is the unique negative eigenvalue of and . Therefore

It is easy to see that

So it suffices to show that . Since

then

Note that . We derivate it with respect to , and then

Namely,

So

Hence, . The result in Theorem 20 holds.

Lemma 21 (see [14]). There exists and a unique map , such that for any and ,

where

Next we define an auxiliary operator which will play a critical role in the proof of instability.

Definition 22. For is defined by the formula

By Lemma 21, can also be written as

where .

The next lemma summarizes the properties of .

Lemma 23 (see [14]). is a function. Moreover, commutes with translations, and for any .

Lemma 24 (see [14]). There exists a function

which is invariant under translations, such that

for any with and is not a translate of .

Lemma 25 (see [14]). According to Theorem 20, there is a curve which satisfies for , , and changes sign as passes through , with .

Theorem 26. If (4) has a bell-profile solitary wave solution , when , then is orbitally unstable.

Proof. Firstly, we consider . Let , small enough, and be the tubular neighbourhood defined above. By Lemma 25 we can choose which is arbitrarily close to , such that , , and . To prove the instability of , it suffices to show that there are some elements which are arbitrarily close to , but the solution with the initial data exits from in finite time. Let be the maximal interval for which lies continuously in , where . Let be the maximum existence time for the solution with initial data . If is finite, it is easy to see that is orbital instability by definition, so we may assume that and our purpose now is to show that ; 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 enjoys the following properties:

where depends on and , depends on , , and .

Let , where is defined by Lemma 21 and define

where the function serves as a Lyapunov function, and

Due to the assumptions above, it is observed that

Therefore , such that .

Indeed, if is the Heaviside function and , then

Now we can have

Hence, by (113),

It follows from Minkowski's inequality that

Similarly, . Therefore

Since , and , now we estimate , by calculating, and have

Since , where and , it follows that

Since , from Lemma 24, we can deduce that

Since and it is continuous, we can obtain . Therefore for all , . Moreover, since and , we may assume that , , by choosing smaller if necessary. So for all , by Lemma 24, we have

Integrating (126) on , we have

And then

Since , we can conclude that . 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 . In Section 4, we gave the explicit expressions for the discrimination , of orbital stability in terms of the two exact solitary waves , of (4). Furthermore, we deduced Theorems 9 and 10 which could easily judge the orbital stability of the two solitary waves , 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.

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).

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