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Complexity
Volume 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

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; moc.621@mwzgwz

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 [25]. Equation (1) has important applications in many fields, such as nonlinear lattice, plasma physics, hydrodynamics, and solid state physics [69]. References [28] 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, [1419] 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 solutionamong which(2) Let make ( is negative) meaningful in (such as is odd), if , or and , (2) also has bell-shaped solitary wave solutionamong 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 , satisfyingobviously, .

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 andOtherwise, 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 obtainIntegrating on both sides of (19), we can getAnd when , , then ; that is,So

Next we define operator , , where After calculating, we can getIt 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 57, 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 haveDue toand 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 havewhere , , . 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 , thenSubstituting (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 toSo 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 thatTherefore, 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 satisfiesthen . 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 towe can get Theorem 13 from Propositions 1012 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 getSincethen the coefficient in the right hand of (60) satisfieswhere = .

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 haveThen 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 thatwhere

Since , then we have