Influence of Nonlinear Terms on Orbital Stability of Solitary Wave Solutions to the Generalized Symmetric Regularized-Long-Wave Equation

The influence of two nonlinear terms on the orbital stability of solitary wave solutions to the generalized symmetric regularized-long-wave(gsrlw) equation is investigated in this paper. Based on the general conclusion to judge the orbit stability of solitary wave solution to the equation, the stable and unstable wave velocity intervals of solitary wave solutions to the gsrlw equation with two low order nonlinear terms are given. By appropriate transformation and scaling, the complexity caused by two high-order nonlinear terms is overcome, and the stable and unstable wave velocity intervals of solitary wave solutions to the gsrlw equation with high-order nonlinear terms are also obtained. Last, the influences of the coefficients and the order of the nonlinear terms on the stability of solitary wave solutions are studied.


Introduction
The symmetric regularized-long-wave equation (called srlw equation for short) is discovered and named by Seyler and Fenstermacher in 1984 [1], it describes a kind of weak nonlinear ion acoustic wave and space charge wave, where and u describe the fluid velocity and electron charge density, respectively. It also appears (1) in many other physical phenomena and is an important model equation in physics and fluid mechanics [2][3][4]. Recent years, many researchers have studied the solitary wave solutions of Eq. (1) by various methods [5][6][7][8], such as exp(− ( ) ) extend method [5], tanh function extend method [6], analytical method [7] and exponential wave integrator pseudospectral method [8]. For the stability [9], studied the local and global well posedness problems of solutions to Eq. (1) in H s (R) × H s−1 (R) , and proved the existence and the orbit stability of elliptic periodic wave solutions with period L. And [10] proved the instability of periodic wave solutions to (1).
It is generally believed that the solitary wave is the result of the balance between dispersion effect and nonlinear effect. For the symmetric regularized long wave Eq.
(1) with weak nonlinearity, the nonlinear term can not be ignored. When the nonlinear term is changed, the amplitude u related to space and time will also change. Therefore, if the nonlinear term of symmetric regularized long wave equation is changed, whether traveling wave solutions of Eq. (1) and the stability of solutions will be affected, it is worth considering.
Gradually strengthening the nonlinear effect from Eq. (1), Shang considered the solutions of the modified regularized long wave equation in [11], [12] used the reductive perturbation method to study the amplitude modulation of symmetric regularized long-wave equation (3) with quartic nonlinearity, [13] used lion compact concentration theorem to consider the existence of the solutions to the generalized symmetric regularized-long-wave equation(called gsrlw equation for short) with only one high nonlinear term, and gave the convergence and error estimation of its approximate solutions. [14] designed a compact finite difference scheme to solve the gsrlw Eq. (4).
Besides, some scholars attempted to consider the gsrlw equation with two nonlinear terms, such as [15] obtained the exact solitary wave solutions and kink wave solutions to a class of gsrlw equation, Recently, [16] studied the orbital stability of solitary wave solutions to Eq. (5) in the case of p = 1 . However, the exact stable and unstable wave velocity intervals have (3) u xxt − u t = ( + 1 4 u 4 ) x , t + u x = 0.
(4) u xxt − u t = ( + 1 p u p ) x , t + u x = 0, not been given in [16]. For the more general form [17], studied the orbital stability and instability of the gsrlw equation, here f ∈ C 1 , and f (s) > 0 when s > 0 . However, [17] only gave the orbital stable or instable velocity intervals when equation (6) has only one nonlinear term , the solution is orbital instable. So here we study the orbital stability of solitary wave solutions to the gsrlw Eq. (5) with two high order nonlinear terms, and our purpose is to obtain exact orbital stable and instable wave velocity intervals of the solitary wave solutions to the gsrlw Eq. (5) with two nonlinear terms, and find the influences of the coefficients and the order of two nonlinear terms on the stability of solitary wave solutions to Eq. (5), which should make an impact on future studies in stability of the similar equations.
In this paper, we first verify that solitary wave solution to Eq. (5) satisfy the three assumptions of Grillakis-Shatah-Strauss theory [18,19], and give the general conclusion of orbital stability of solitary wave solutions to Eq. (5). Based on the exact expressions of the discriminant d �� (v) of the orbital stability of solitary wave solutions in the case of p = 2 , we obtain the specific stable and instable wave velocity intervals of the solitary wave solutions to Eq. (5). Besides, we point out the problems in [16] and correct its conclusions for the gsrlw equation. We also give the specific stable and instable wave velocity intervals for case of p = 1 . These results are new. Then by appropriate transformation and scaling, we overcome the complexity caused by the two high-order nonlinear terms, and obtain the stable and instable wave velocity ranges of solitary wave solutions to Eq. (5) in the case of p > 2 . Last, we analysis the influence of the coefficients and the order of nonlinear terms on the orbital stability and instability of solutions to the gsrlw Eq. (5), and get the conclusion that the ratio of the coefficients b 2 , b 3 affects the orbital stable and instable wave velocity intervals, and the influence of the order of nonlinear terms on the orbital stability of solutions may be stronger than the influence of the coefficient of nonlinear terms. For the convenient in the following discussions, here quote two exact solitary wave solutions of Eq. (5) in [15] as follows.  Here introduce some background knowledge before giving the three assumptions of Grillakis-Shatah-Strauss theory [18,19]. The gsrlw Eq. (5) can be written into the following Hamilton system, ,

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here E is one natural invariant of Eq. (5), and E � (⃗ u) is the Frechet derivative of E, J is a skew symmetric operator, and X = H 1 (R) × L 2 (R) is a Hilbert space with the inner product There exists a natural isomorphism I ∶ X → X * with its dual space X * , satisfying Let T be a unitary operator group with single parameter, which is defined as, are given by (7), (10), respectively.
To prevent repetition, we denote v (x) as one of u 1 (x) and u 2 (x).
Next, we will show that T(vt) ⃗ v (x) satisfies three assumptions of Grillakis-Shatah-Strauss theory. Here we introduce the assumption in [18,19] as the following lemma.
Lemma 2 (Existence of Solutions) Set X = H 1 (R) × L 2 (R) , for each x 0 ∈ X, there exits t 0 > 0 depending only on , where ||u 0 || ≤ , and there exists a solution u of (13) By semigroup theory [20], the existence of the initial value problem of equation can be proved by the similar method in [17], so we omit the proof process here. Next, we give the Lemma 3, that is the assumption 2 in [18,19]. We make integration on the both sides of Eq. (18), and set the constant as 0, then we have

Lemma 3 There exist real numbers
Then we give the assumption 3 in [18,19] as Lemma 4 here.

Lemma 4 For each
, H v has exactly one negative simple eigenvalue and has its kernel spanned by T � (0) , and the rest of its spectrum is positive and bounded away from zero.
, and making spectrum analysis, where ,we can know x = 0 is the only zero point of vx , then according to Sturm-Liouville theory, we can get that 0 is the second eigenvalue of L. So under the condition v > 1 , L only has one negative eigenvalue − 2 , and its corresponding eigenfunction is , that is

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According to the orbital stability theory proposed by Grillakis-Shatah-Strauss and Lemmas 2-4, we can get the following general conclusion of orbital stability of solitary wave solution to the gsrlw Eq. (5).

Remark 1
Since the skew-symmetric operator J is not onto, by making similarly deduction in [16], we can obtain the conclusion From Theorem 1, in order to consider the orbital stability of solitary wave solutions to Eq. (5), we only need to consider the sigh of the second derivative d �� Then we consider the orbital stability of solitary wave solutions to the gsrlw equation in the case of p = 1 , p = 2 , p > 2 . For the convenient of discussion, here always assume b 3 > 0.

Orbital Stability of Solitary Wave Solution to Eq. (5) When p = 1,2
Here, we consider the case of p = 2 first. Solitary wave solution (7) can be rewritten into , into (27), we can get Similarly, for the solitary wave solution (10), we have . In order to consider the orbital stability of the solitary wave solution (25), we need to consider the sign of d �� (v) . Here we discuss the orbital stability of (25) first.
, in order to consider the orbital stability of u 1 ( ) , we take k m D 1 = 2 , and consider when d �� (v) > 0 . After simplification, we have Similar to the discussion in ①, we take k m D 1 = 2 and consider when d �� (v) > 0 . From (30), we can get Then we consider the instability of the solitary wave solution u 1 ( ).
In order to make d �� (v) < 0 , we only need to consider when Similarly, we can get the following conclusions:

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Journal of Nonlinear Mathematical Physics (2021) 28:390-413 In conclusion, we can get the following theorem.
For the solitary wave solution u 2 ( ) , we can obtain the following conclusions similarly.
, the solitary wave solution u 2 ( ) is orbital stable.

the solitary wave solution u 2 ( ) is orbital stable.
Remark 2 Since the case of p = 1 has been discussed in [16], we omit the computing process here. And for the convenience to comprehend, we unify the velocity c in [16] as v here. And here introduce the discriminant d �� (v) in the case of p = 1, However, it can be proved that equation (61) in [16] (31) is always established, then equation (64) in [16] is always incorrect. Similarly, in the case of 3v − 3 v − k 2 > 0 , it can be proved that equation (62) is always established, and equation (63) is always incorrect. And in the case of can be obtained by solving (63) in [16]. According to the above analysis, we make improvements and corrections here for the theorem 9, 10 in [16], and obtain the following theorems.
The conclusion for d �� (v) can be obtained after similar discussion.

The Orbital Stability of the Solitary Wave Solution to Eq. (5) When p > 2
When Journal of Nonlinear Mathematical Physics (2021) 28:390-413

Substituting (32) into (24), we can get
Because of the complexity of the integral formula, we can not obtain the stable or instable velocity changing interval directly. So we discuss the sign of discriminant by using reasonable expansion and contraction here. At the same time, without losing generality, we only discuss the orbit stability of solitary wave solution (7). The orbit stability of solitary wave solution (10) can be discussed similarly.

The Discriminant of Orbital Stability of the Solitary Wave Solution (7)
First, solitary wave solution u 1 ( ) can be rewritten as . Then we only need to consider the sign of d �� (v).

Remark 3 If we take
According to the appropriate transformation and Wallis formula, we can deduce the conclusion which is consistent with Theorem 6.1 in reference [17].

we can get
By the relation m < √ m 2 + 4 < m + 2 , G 3 can be expressed as follows, (52) where Combining the above analysis, we can get the following theorem. (55) .  (7) is orbital instable when .
(3) In the case of p > 10 3 , there exists ̄3 � such that for any , the solitary wave solution (7) is orbital stable.

The Influence of Two Nonlinear Terms on the Stability of the Solitary Wave Solution
In this section, we consider the influence of the two nonlinear terms on the orbital stability of the solitary wave solution to the gsrlw Eq. (5). From the conclusions in the Sects. 3, 4, it can be found that the stable and instable wave velocity intervals are mainly determined by two factors, one is the coefficient of the nonlinear term, the other is the order of the nonlinear term. Here we discuss the influence of the coefficient of the nonlinear term first.
Without loss of generality, we take the solitary wave solution (7) as an example. According to the Theorems 2 and 4, we can get that the stable and instable wave velocity intervals, which are shown in the following tables.
It can be found that when there are two nonlinear terms, the stable and unstable intervals of the solitary wave solutions are affected by the ratio b 2 ∕ √ b 3 of the coefficients b 2 , b 3 of the nonlinear terms, but not by the coefficient of a single nonlinear term. In other words, if the ratio b 2 ∕ √ b 3 of the two nonlinear terms is a constant and whether the coefficients of the two nonlinear terms are increasing or decreasing proportionally, the orbital stability and instability of the solitary wave solutions with two low order nonlinear terms will not be affected.
Besides, from Table 1, we can find that when the ratio �b 2 ∕ √ b 3 � is larger, the orbital stable interval will become smaller. Correspondingly, the orbital instable interval will become larger.
Then we discuss the influence of the order of nonlinear term. According to Theorem 6, when p > 4 , the instable wave velocity interval of solitary wave solution (7) to Eq. (5) � . It can be found that when the order p(p > 4) is increasing, the instable wave velocity interval of solution will become larger. Correspondingly, its stable wave velocity interval will become smaller. Furthermore, it can be seen that the influence of the coefficients of the nonlinear terms on the orbital stability of solitary wave solution (7) is not revealed in the Theorem 6, which may caused by the over scaling of the discriminant d �� (v) . In other words, after the scaling of the discriminant d �� (v) , the order of the nonlinear terms still influence the orbital stable wave velocity intervals of the solitary wave solution (7) to Eq. (5), while the influence of the coefficients of the nonlinear terms cannot be shown. So we can guess that when the gsrlw equation has two high order nonlinear terms, the influence of the order of nonlinear terms on the orbital stability of solitary wave solution (7) is stronger than the influence of the coefficient of nonlinear terms.