Complexity
Volume 2019 (2019), Article ID 4209275, 17 pages
https://doi.org/10.1155/2019/4209275

Research Article

The Orbital Stability of Solitary Wave Solutions for the Generalized Gardner Equation and the Influence Caused by the Interactions between Nonlinear Terms

Wei-Guo Zhang,¹ Xing-Qian Ling,² Xiang Li,¹ and Shao-Wei Li³

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
³School of Mathematics and Information Engineering, Taizhou University, Taizhou 317000, China

Correspondence should be addressed to Wei-Guo Zhang; zwgzwm@126.com

Received 16 April 2019; Accepted 21 May 2019; Published 20 June 2019

Academic Editor: Dimitri Volchenkov

Copyright © 2019 Wei-Guo 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.

In this paper, the orbital stability of solitary wave solutions for the generalized Gardner equation is investigated. Firstly, according to the theory of orbital stability of Grillakis-Shatah-Strauss, a general conclusion is given to determine the orbital stability of solitary wave solutions. Furthermore, on the basis of the two bell-shaped solitary wave solutions of the equation, the explicit expressions of the orbital stability discriminants are deduced to give the orbitally stable and instable intervals for the two solitary waves as the wave velocity changing. Moreover, the influence caused by the interaction between two nonlinear terms is also discussed. From the conclusion, it can be seen that the influences caused by this interaction are apparently when , which shows the complexity of this system with two nonlinear terms. Finally, by deriving the orbital stability discriminant in the form of Gaussian hypergeometric function, the numerical simulations of several main conclusions are given in this paper.

1. Introduction

The Gardner equation originates from the study of the conservation law of KdV equation by Miura, Gardner, and Kruskal [1]. Because the Gardner equation possesses both quadratic and cubic term, many researchers recall the mixed KdV-MKdV equation,

as Gardner equation [2–5]. Equation (1) has important applications in many fields, such as nonlinear lattice, plasma physics, hydrodynamics, and solid state physics [6–9]. References [2–8] used different methods to solve the solitary wave solutions and some explicit solutions, such as the inverse scattering method and Hirota bilinear method [6, 7], the method of undetermined coefficients, and the extended tanh-function method [2, 3]. Recently, [10, 11] studied the generalized Gardner equation,

and used sub-ODE method and expansion method to solve the solitary wave solutions and other solutions of (2).

Equation (2) is the extension of the Gardner equation in [1]. When , (2) turns into (1). When , (2) turns into the generalized KdV equation,

This paper mainly studies the orbital stability of solitary wave solutions of generalized Gardner equation (2). The theory of the orbital stability of solitary wave solutions can be referred to in [12, 13]. We found that the stability of solitary wave solutions studied by the mostly above literatures has only one nonlinear term in the equation. For example, for the stability of the solitary wave solutions for the generalized KdV equation (3) (it can be seen as the special case of the generalized Gardner equation (2)), many literatures have studied on it. By using variational method, [14–19] studied the stability of solitary wave solutions of the generalized KdV equation (3) and got the following conclusion: if , the solitary wave solution of (3) is stable, and if , it is instable. Meanwhile, [20] studied the orbital stability and instability of the solitary wave solutions of the nonlinear evolution equation which is much more complicated than (3) and obtained the following conclusion: if , the solitary wave solution of (3) is orbitally stable, and if , it is orbitally instable. Besides, [21, 22] obtained some sufficient conditions when the solutions of the equation,

are stable, by using variational method and Lyapunov method, respectively. The reason why previous literature focused on the stability of solitary wave solutions for the evolution equations with only one nonlinear term may be that the analysis to the existence and other relative problems for the solitary wave solutions will become more complex and difficult when there are two or more than two nonlinear terms in the equation.

In this paper, based on our acknowledge on the solving the exact solitary wave solutions for nonlinear evolution equations, we give the exact solitary wave solutions of (2) with two nonlinear terms to directly state the existence. This reduces the high assumptions for the nonlinear parts of the equation when proving the existence of solitary wave solutions by analysis. Thus, it makes us apply the theory of orbital stability of Grillakis-Shatah-Strauss to the study of the orbital stability of solitary wave solutions for (2) possible.

The generalized Gardner equation (2) has another nonlinear term , comparing with the generalized KdV equation. So people want to know how about the orbital stability of the solitary wave solution of (2) naturally and the differences between the orbital stability of (2) and (3). In addition, there is only one bell-shaped solitary wave solution

in (3), but there are two solutions and in (2) (as shown in Lemma 1 below). Naturally, people want to know whether the orbital stability of the solutions and is the same. In this paper, firstly by the application of the orbital stability theory of Grillakis-Shatah-Strauss, we will get a general conclusion to judge the orbital stability of the generalized Gardner equation. Further, on the basis of the two exact bell-shaped solitary wave solutions of (2), we can deduce the explicit expressions of the orbital stability discriminants of two solutions and then give the intervals which make two solutions stable as the wave velocity changing. And we will also discuss the influences on the stable intervals caused by the two nonlinear terms in (2). The results in this paper show that, when the whole order of the nonlinear terms in (2) is high, just as when , the solitary wave solutions of (2) with two nonlinear terms and (3) with a single nonlinear term possess the same stability, both of which are instable. But when , the orbital stability of (2) and (3) has apparent differences, which reflects the influences caused by two nonlinear terms of (2) on the orbital stability. Especially, in the case of , , we prove the following result: there exists ; if , for any , the solitary wave solution is orbitally stable; if , although the higher nonlinear term of (2) is less than 4, there is still a part of wave velocity interval in making orbitally instable. The same conclusions of solitary wave solution can also be deduced in the situation of , . All the above conclusions indicate the complexity caused by the interaction between two nonlinear terms of (2).

To give explicit explanations of our conclusions, we use the transformation method to establish the orbital stability discriminant of the solitary wave solution in the form of Gauss hypergeometric function. Then the numerical simulating illustrations of several main conclusions can be got.

It is worth pointing out the discussion of the influence of the orbital stability of solitary wave solutions of the equation caused by the interactions between two nonlinear terms; the expression of orbital stability discriminant in the form of Gaussian hypergeometric function and giving the numerical simulations of the wave velocity intervals of the stability are original in this paper.

The method in this paper is an extension of Grillakis-Shatah-Strauss orbital stability theory. If an equation satisfies the conditions of Grillakis-Shatah-Strauss orbital stability theory and the solitary wave solutions of this equation can be solved, then the method in this paper can be referred to study the stability and the influence caused by the nonlinear terms.

Here we give some related results for the generalized Gardner equation as follows.

Lemma 1. Assume .
(1) If or and , then (2) has a bell-shaped solitary wave solution

among which

(2) Let make ( is negative) meaningful in (such as is odd), if , or and , (2) also has bell-shaped solitary wave solution

among which

Remark 2. The above results can be deduced by the conclusion in [10, 23].

Remark 3. It is easy to examine that the function in (6) is always positive under the condition of Lemma 1(1), and the function in (8) is always negative under the condition of Lemma 1(2).

2. General Conclusions of the Orbital Stability of the Solitary Wave Solution for the Generalized Gardner Equation (2)

The generalized Gardner equation (2) can be rewritten as the following Hamilton system:

where with dual space and

The inner product of is

And there is a natural isomorphism between and , , satisfying , where , .

Let be a unitary operator group with a single parameter on , satisfying

obviously, .

We can deduce by , so it can be defined that

From (15), we know solitary wave solutions (6), (8) of (2) in Lemma 1 can be rewritten as . Next, we consider the orbital stability of . To avoid duplication, we denote as one of and .

Definition 4 (see [21]). The -orbit is stable if for all there exists with the following property. If and is a solution of (2) in some interval with , then can be continued to a solution in and

Otherwise, the -orbit is called unstable.

To prove the orbital stability of the solitary wave solution , we need to verify whether (2) and the solution satisfy the three assumptions in [12].

Firstly, from Theorems 1 and 2 in [24], we can deduce the existence of solutions for the initial value problem of system (10).

Lemma 5 (see [19]). Let , for any fixed , there exists a unique solution in (2), which satisfies .

It is easy to show that which are defined in (12) and (16) satisfy

Lemma 6. is the bounded solution of (2) and satisfies

Proof. By substituting into (2), we can obtain

Integrating on both sides of (19), we can get

And when , , then ; that is,

So

Next we define operator , , where

After calculating, we can get

It is easy to know that is a self-conjugate operator, , which means is a bounded self-conjugate operator in and the eigenvalues of consist of the real number such that irreversible.

From (19), we can know is one eigenvalue of ,

Let , which contains the core of .

Now we are ready to prove the following lemma.

Lemma 7. For each , has only one negative single eigenvalue. Its kernel space is spanned by , and its other eigenvalues are positive, bounded, and far away from 0.

Proof. Since is the only zero of , we know 0 is the second eigenvalue of from Sturm-Liouville Theorem [25]. So only has one negative eigenvalue , and its corresponding eigenfunction is denoted as ; then , .
For , when , , so . From Wely essential spectrum Theorem [26], the essential spectrum of satisfies , , so the desired result holds.

From the above discussion, we can make a spectral decomposition to . Let

For any , , so .

For any , , so .

For any , according to Lemma 3.2 in [12], for any real function which satisfies , there exists a positive making , where is irrelevant to ; then .

Therefore, space can be decomposed into direct sum , where is the kernel space of , is a finite-dimension space, and is a closed subspace.

We define as . As a result of Lemmas 5–7, according to [12], we can get the following general conclusion of the solitary wave solution of (2).

Theorem 8. Let , be the solitary wave solutions of the generalized Gardner equation (2) given in Lemma 1, among which coefficients and wave velocity satisfy the hypothesis in Lemma 1, respectively. Denote as one of and ; then if , is orbitally stable; else if , is orbitally instable.

Remark 9. Since the skew-symmetric operator is not onto, by directly using the conclusion in [20] or making similarly deduction, we can obtain the conclusion that if , is orbitally instable in Theorem 8.

It can be inferred from Theorem 8 that we can examine the symbol of to study the orbital stability of the solitary wave solution of the generalized Gardner equation (2). Here

For convenience, we give the explicit expressions of the orbital stability discriminants of the solitary wave solution (6) and the solitary wave solution (8) of the generalized Gardner equation (2), respectively.

Firstly, we consider the orbital stability discriminant of the solitary wave solution (6) for (2). The solitary wave solution (6) can be rewritten as

among which

In (27), we take . Noting that is an even function, we have

Due to

and when , for any fixed , each term on the right hand of formula (33) converges in ,

So when , for any fixed , converges.

For any real number , if wave velocity satisfies , , and in , then there exists a constant , independent of , satisfying

If , not only converges, but also uniformly converges in any closed subintervals in . Hence, according to (27), (33), and the derivation rule of the integral with parameter, we can deduce the orbital stability discriminant of the solitary wave solution (6) of (2),

To avoid confusion, we denote the stability discriminant of the solitary wave solution (8) for the generalized Gardner equation (2) as . By taking in (27), we have

where , , . According to (8) and (9), we have

Similar to the discussion of the solitary wave solution (6), since not only converges, but also uniformly converges in any closed subintervals in when , then according to (37) and the derivation rule of the integral with parameter, we have the orbital stability discriminant of the solitary wave solution (8),

According to (38),

By substituting (40) into (39), we can get

3. The Orbital Stability of the Bell-Shaped Solitary Wave Solution of the Generalized Gardner Equation (2) When

3.1. The Orbital Stability of the Solitary Wave Solution (6) When

3.1.1. In the Case of , ,

Note that the orbital stability discriminant of the solitary wave solution (6) of (2) is (36); we can get

From (43), we have that when , , , and , then

Substituting (43) and (44) into (36), we can get when , ,

In (45),

so the coefficients in the right hand of (45) satisfy

where = .

We denote the numerator in the right hand of (47) as , which means

In (48), it can be shown that if , , , then , and follows from (45) and (47). When , is equivalent to

So in the case of , , , if and , then , and follows from (45) and (47).

We can get following Proposition 10 from the above discussions.

Proposition 10. In the case of , , , if , or and , the stability discriminant of the solitary wave solution (6) satisfies .

3.1.2. In the Case of , ,

Since , from (43), we have , which implies that

Therefore, according to (36), we can get when ,

After calculating, we can get that the coefficients in the right hand of (51) satisfy

We use to denote the numerator of (52), which means

From (53), we can obtain the following conclusion: if , , , then ; if and , then . So we can get the following Proposition 11 from (51).

Proposition 11. In the case of , , , if , or and , the stability discriminant of the solitary wave solution (6) satisfies .

3.1.3. The Orbital Stability of the Solitary Wave Solution (6) in the Case of , ,

From (43), we can get that if . So we only need to consider which value takes; the first coefficient of (36) will be not less than 0; then we will have . From (52) and (53), the first coefficient of (36) can be rewritten as

From (53) and (54), we can obtain that, if satisfies

then . So we can deduce from (36). And under this condition, we have Proposition 12.

Proposition 12. In the case of , , , if satisfies

then the stability discriminant of the solitary wave solution (6) satisfies .

Due to

we can get Theorem 13 from Propositions 10–12 and Theorem 8.

Theorem 13. Assume , , .
(1) If , or and , the solitary wave solution (6) is orbitally instable no matter or .
(2) In the case of , if and , the solitary wave solution (6) is orbitally instable.
(3) In the case of , if , the solitary wave solution (6) is orbitally stable.

3.2. The Orbital Stability of the Solitary Wave Solution (8) When

3.2.1. In the Case of , ,

Note that , and in ; combining this with (38), we can deduce the following conclusion: no matter or , , which means ,

And from (43), we can deduce that when , . So from (58), we have

Then we substitute (59) into (41) and get

Since

then the coefficient in the right hand of (60) satisfies

where = .

Here we set the numerator in the right hand of (62)

Since , similar to (48) and (49), from (63), we can deduce that if , or and , then . And we can get Proposition 14 from (60) and (62).

Proposition 14. In the case of , , , if , or and , the stability discriminant of the solitary wave solution (8) satisfies .

3.2.2. In the Case of , ,

From (43), we can deduce that when , So from (41), we have

Then we can deduce the following proposition from (52) and (53).

Proposition 15. In the case of , , , if , or and , the stability discriminant of the solitary wave solution (8) satisfies .

3.2.3. The Orbital Stability of the Solitary Wave Solution (8) When , ,

In the case of , , we have , , so the second term of (41) is greater than 0. And only if the coefficient of the first term in (41) is greater than 0. That is, . And from the discussion in Section 3.1.3, we can deduce that if , , so , which means the solitary wave solution (8) is orbitally stable.

Combining with the above discussion and Propositions 14 and 15, we can get the following theorem.

Theorem 16. Assume , and set make ( is negative) meaningful in .
(1) If , or and , the solitary wave solution (8) is orbitally instable no matter or .
(2) In the case of , if and , the solitary wave solution (8) is orbitally instable.
(3) In the case of , if , the solitary wave solution (8) is orbitally stable.

4. The Orbital Stability of the Generalized Gardner Equation (2) When

4.1. The Orbital Stability of the Solitary Wave Solution (6) of the Generalized Gardner Equation (2) When

4.1.1. The Simplification of the Orbital Stability Discriminant of the Solitary Wave Solution (6) When

Since (), in the case of , converges to the bounded function which converges to 0 when or constant 1. So we can apply Mean Value Theorem of Integrals to the orbital stability discriminant (36) of the solitary wave solution (6). Then there exists , such that

where

Since , then we have

By making the substitution to the above formula, we can get

By the same substitution, we can get

By substituting (68) and (69) into (66), and noticing that , then we have

By simple calculating, we have

Then by substituting (71)-(75) into (70), we have

Notice that

and then

Substituting (77)-(79) into (76), then we have

so (65) can be rewritten as

From Lemma 1, we know that is always positive in (81), and , , can be given by (29), (30), and (31), respectively.

4.1.2. The Orbital Stability of the Solitary Wave Solution (6)

(1) Case of . From (81), we can get the following: if , then ; if , then . So the solitary wave solution (6) is orbitally stable when and orbitally instable when in the case of .

(2) Case of , . Firstly, we consider in (80) and (81). When , we easily have from (77) and (79). Then we can get from (81), and the solitary wave solution (6) of (2) is orbitally stable.

(3) Case of , . From (77), it is easy to know when . So when . Now we substitute (77) into (81) and set ; then (81) can be rewritten as

From (82), we can deduce that, in the case of (now ), is equivalent to . Here

and

From , we can get

It is easy to obtain that if , then , and if , then . So is monotone decreasing if , monotone increasing if , and take the minimum value if . And we have

So according to Intermediate Value Theorem of the continuous function, we can get the following proposition.

Proposition 17. Set , .
(1) If the minimum value , then for any .
(2) If the minimum , then there exists (zero points of ); if and , then ; if , then .

Since , according to Proposition 17 and (82) and (83), we have the following proposition.

Proposition 18. Set .
(1) If minimum , then for any ; the solitary wave solution (6) of (2) is orbitally stable in .
(2) If minimum , then there exists (the zero points of ); if , then ; the solitary wave solution (6) is orbitally stable; if , then ; the solitary wave solution (6) is orbitally instable.

Now substituting into , we can get the expression of about ,

Since , , , and is monotone decreasing about in ; from Intermediate Value Theorem of the continuous function, we can deduce that there exists making the minimum (from (82); we can know also makes ). Then for any , the minimum value , while, for any , the minimum value . Combining the above with Proposition 18, we can deduce the following proposition.

Proposition 19. We set , , and denote as the zero point of in (87), and then we have the following.
(1) When (now ), for any , the solitary wave solution (6) of equation (2) is orbitally stable in .
(2) When (now ), there exists (the zero points of ); if , the solitary wave solution (6) is orbitally stable; if , the solitary wave solution (6) is orbitally instable.

Now we set , . Through the analysis, we can get the following theorem.

Theorem 20. Set , for the solitary wave solution (6), we have the following conclusions.
(1) In the case of , the solitary wave solution (6) is orbitally stable when and orbitally instable when .
(2) In the case of , , the solitary wave solution (6) is orbitally stable for any .
(3) In the case of , there exists such that we have the following.
① If , for any , the solitary wave solution (6) is orbitally stable.
② When , there exists ; if , the solitary wave solution (6) is orbitally stable; if , the solitary wave solution (6) is orbitally instable.

4.2. The Orbital Stability of the Solitary Wave Solution (8) of the Generalized Gardner Equation (2) When

4.2.1. The Simplification of the Orbital Stability Discriminant of the Solitary Wave Solution (8) When

Since is always negative in the solitary wave solution (8), we assume that makes ( is negative) meaningful in (The case of is out of our discussion). In the case of , we make simplification for the discriminant of the solitary wave solution (8). Similar to the discussion of the orbital stability of , by applying Mean Value Theorem of Integrals to (41), we can get that there exists such that

where

Since and satisfy (38) and (40), according to (71)-(75), we can obtain

Due to (78) and (79), we set

and then we have

So we have

According to Lemma 1, in (94) is always negative. From the relationships between and in (38), we can know , , .

4.2.2. The Orbital Stability of the Solitary Wave Solution (8) When

(1) The Orbital Stability of the Solitary Wave Solution (8) When , . Firstly, from (77), it is easy to get . Then we can substitute it into (92) and easily obtain . Furthermore, can be deduced from (94), so the solitary wave solution (8) of (2) is orbitally stable.

(2) The Orbital Stability of the Solitary Wave Solution (8) When , . Firstly, from (77) and (92), it is easy to know that, in the case of , if and , .

Similar to discussions in Section 4.1.2, now we substitute (77) into (94) and set ; then (94) can be rewritten as

Since is always negative, from (95), it is easy to get that is equivalent to . Here

Since

we have

from So we can deduce that if , , and if , . So is monotone increasing if and monotone decreasing if , and takes the maximum value when ; meanwhile,

On the basis of Intermediate Value Theorem of continuous function, we have Proposition 21 about and .

Proposition 21. (1) If the maximum , then for any .
(2) If the maximum , there exists ; if and , then ; if , then .

Since , ; then we can deduce Proposition 22 from Proposition 21 and (95).

Proposition 22. Set .
(1) If the maximum , then, for any , , the solitary wave solution (8) of (2) is orbitally stable.
(2) If the maximum , then there exist ; if , , the solitary wave solution (8) is orbitally stable; if , , the solitary wave solution (8) is orbitally instable.

Now substituting into , we can get the expression of as follows:

Since , , , and is monotone increasing in , from Intermediate Value Theorem of the continuous function, we can deduce, there exists making the maximum (from (95), we can know that also makes ). So for any , maximum , and for any , maximum . Summarizing the above discussion, we can get the following proposition.

Proposition 23. Set , , as the zero point of .
(1) When , for any , the solitary wave solution (8) of (2) is orbitally stable on .
(2) When , there exist , ; if , the solitary wave solution (8) of (2) is orbitally stable; if , the solitary wave solution (8) is orbitally instable.

We set , . From the above analysis, we can get the following theorem.

Theorem 24. Set p make ( is negative) meaningful in and , , ; then for the solitary wave solution (8); we have the following conclusions.
(1) In the case of , for any , the solitary wave solution (8) is orbitally stable.
(2) In the case of , there exists with the following properties:
① When , for any , the solitary wave solution (8) is orbitally stable.
② When , there exists , , if , such that the solitary wave solution (8) is orbitally stable; if , the solitary wave solution (8) is orbitally instable.

4.3. The Numerical Simulation of the Orbital Stability of the Solitary Wave Solution

For making the results in Sections 3, 4.1, and 4.2 more intuitive, we use MAPLE to give the numerical simulations. To avoid duplication and save the space, we only give the illustrations for the stability of the solitary wave solution (6) for (2) (others can be similarly explained). We start from the orbital stability discriminant (36) of the solitary wave solution (6) for (2).

We denote the first integral formula in (36) as ; i.e.,

By using and setting , we can get

Then we set and have

With the help of MAPLE, the integral in (103) can be expressed as

when hypergeom is Guass generalized hypergeometric function.

Similarly, we set the second integral formula in (36) as ; i.e.,

With similar transformation, we have

where can be obtained from (104) by replacing with .

By substituting in (29)-(31) and and into (36) and setting , , we have

where

We can use (107) to make the numerical simulation of the orbital stability of the solitary wave solution (6) on the basis of the expression which is expressed by Guass generalized hypergeometric function.

4.3.1. The Numerical Simulation of the Orbital Stability of the Solitary Wave Solution (6) When , ,

We set , , in (107) and get the 3D image of about shown in Figure 1, where dark blue surface is the image of and green flat surface is the horizontal plane of .

Figure 1: The 3D image of when , , , .

From Figure 1, we can get that the 3D image of has both positive and negative area when , , , , so there will be wave velocity interval that makes both positive and negative. From Figure 2, we can get that when , the 3D image of is under the surface . So is always negative when .

Figure 2: The 3D image of when , , , .

In Figure 3, the first curve is given by , the second curve is , and the third curve is given by . From Figure 3, we can obtain that the area above the curve is the instable area, while the area below this curve is the stable area. But the instable area of the solitary wave solution (6) given by conclusion (1) of Theorem 13 lies above the first curve, and the stable region of the solitary wave solution (6) given by conclusion (3) lies below the third curve in Figure 3. The reason is we scale the inequalities in the analysis process, since it is hard to calculate the integral and . So conditions of the stability and instability of the solitary wave solution (6) given in Theorem 13 when are only sufficient conditions.

Figure 3: The stability region of the solitary wave solution (6) when , .

4.3.2. The Numerical Simulation of the Orbital Stability of the Solitary Wave Solution (6) When , ,

Using (107), by taking , , , , we get the simulation results of which are shown in Figure 4. From Figure 4, we can see clearly that the curve takes the minimum point when . If , the curve will fall into the area below the curve . And if , the curve will fall into the area above the curve . From Figure 4, we can see that under this case.

Figure 4: The curve of when , , , .

In the case of , , , when , , , the curve is given in Figure 5. We can see that the curve is wholly above the straight line when (now ), which means for any . The curve is above the straight line in and below the straight line in when , which means, for any , while for any . Here , . The curve is above the straight line in , and below the straight line in when , which means for any and for any . Here , . In Figures 4 and 5, it can be shown that simulation results prove the correctness of Conclusion 3 in Theorem 20.

Figure 5: The curve image of when .

5. The Discussion of the Influence Caused by Interaction between Two Nonlinear Terms on the Stability of the Solitary Wave Solution

In this paper, we study the orbital stability of the solitary wave solution of the generalized Gardner equation (2). And we obtain Theorems 13 and 20 for the solitary wave solution (6) and obtain Theorems 16 and 24 for the solitary wave solution (8).

Comparing with results that the solitary wave solution of the KdV equation (3) is orbitally instable when and orbitally stable when , it can be deduced that, when the order of the nonlinear terms in (3) and (2) is wholly high (in this paper ), the solitary wave solutions of both equations are orbitally instable. However when , the influence caused by interaction between two nonlinear terms on the orbital stability of the solitary wave solution for the generalized Gardner equation is obvious. Details are as follows.

(1) In the case of , the order of the higher order nonlinear term in (2) is greater than 4, and the effect of the lower order nonlinear term in (2) is obvious. According to Theorem 13, we can get that both orbitally stable and instable intervals exist for the solitary wave solution (6). If is larger, the interval which makes the solitary wave solution (6) stable is larger. If is larger, the interval which makes the solitary wave solution (6) stable is smaller, and the interval that makes the solitary wave solution (6) instable is larger. This seems that the lower order nonlinear term plays a role in promoting stability in (2) when .

(2) In the case of , , the order of the higher order nonlinear term in (2) is smaller than 4. And both the solitary wave solution (6) of (2) and the solitary wave solution of (3) are orbitally stable. But if , , from Theorem 20 in this paper, we can know that (2) can be rewritten as

and its corresponding solitary wave solution (6) is orbitally stable for any when . But according to [20], we can get that if we let in (109), then the solitary wave solution of the KdV equation

is orbitally instable. By comparing, when , the lower order nonlinear term in (109) seems to play a role in promoting the stability of the solitary wave solution (6).

(3) In the case of , we consider the solitary wave solution (6). From Theorem 20, we can get that there exists , if , the solitary wave solution (6) is orbitally stable in , but if , although the order of the higher order nonlinear term in (2) is less than 4, there is still part of wave velocity interval in which the solution is orbitally instable. It seems that, for the solitary wave solution (6), when , the term in (2) plays a role against the stability in equation.

(4) For the solitary wave solution (8) of (2), we can make the similar analysis as conclusions (1)-(3). From Theorems 16 and 24, we can get that, for the solitary wave solution (8), the term in (2) plays a role promoting the stability in the equation if and plays a role against the stability in equation if .

The method in this paper is an extension of the theory of orbital stability of Grillakis-Shatah-Strauss. The most fundamental result is that we find and prove, converges for any fixed when and uniformly converges in any subintervals in . Then we apply our method to calculate the integral with a parameter to establish the explicit expressions (36) and (41) of the orbital stability discriminant of the solitary wave solutions (6) and (8), respectively. All conclusions in this paper are obtained on the basis of the above two formulas and by the analysis method.

From the study in this paper, we can find that if an equation satisfies the conditions of the theory of orbital stability of Grillakis-Shatah-Strauss and the solitary wave solutions of equation can be solved, then the method in this paper can be referred to study the stability and instability of its solitary wave solutions. Since the exact explicit solitary wave solutions of a large number of soliton equations have been obtained in the field of Mathematics and Physics since 1980s, in order to further study the stability of these solitary wave solutions with explicit expressions, we think the method in this paper is worth being referred.

Data Availability

The data used to support the findings of this study are included within this article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

This work was supported by National Nature Science Foundation of China [Grant no. 11471215].

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