Orbital Stability of dn Periodic Wave Solutions of the Boussinesq Equation with Quadratic-Cubic Nonlinear Terms

This paper investigates the problem of the orbital stability of dn periodic wave solutions of the Boussinesq equation with quadratic-cubic nonlinear terms. First, the dn periodic wave solution of the studied equation is solved by using the integral method and the knowledge of elliptic functions, and the existence of smooth curves of dn periodic wave solutions with fixed period L is proved. Then the Floquet theory and Wely’s essential spectrum theorem are applied to the spectral analysis of the operator, and obtain its spectral properties. Finally, according to the ideas for proving the stability of solitary wave solutions from Benjamin and Bona et al., by overcoming the complexity caused by the quadratic-cubic nonlinear terms in the studied equation, we prove the dn periodic wave solution of the studied equation is orbitally stable under small perturbations of the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{2}$$\end{document} norm.


Introduction
In physics , the Boussinesq equation [1][2][3][4] is generally applied to the simulation of atmospheric and oceanic flows. It not only can be used to depict shallow-water waves and many other physical phenomena [5,6], but also can describe the fluid dynamics influenced by gravity, and plays a vital role in studying the Raleigh-Bernard convection [7][8][9][10]. For the Boussinesq equation (1.1), there are many studies, like its Painlevé property, Bäcklund transforms, (1.1) u tt + u xx + (u 2 ) xx + u xxxx = 0 Lax pair, etc. [11][12][13]; and the solvability of the initial value problem of the Boussinesq equation (1.1) was studied in [14] using the inverse scattering method. V. Vatchev and Z. Qiao [15] studied the Wronskian solutions for the good Boussinesq (gB) equation, and provided explicit multi-soliton solutions with fission and fusion type of interactions by Hirota substitution method. Linares [16] used Kato's aptitude theorem to study the global solution of the generalized Boussinesq equation with small initial values.
In the literature [17], the orbital stability of the solitary wave solutions of Eq. with quadratic-cubic nonlinear terms was studied in the literature [18]. Zhang [19]  With the further development of solitary wave theory, some scholars have discussed the orbital stability of the periodic wave solutions of some models [20,[22][23][24] in recent years. For example: in the literature [20], Pava studied the existence of smooth curves for the dn periodic wave solutions of the nonlinear Schrödinger equation and the mKdV equation; and the Lyapunov method and the ideas of Benjamin and Bona for proving the stability of solitary waves were used to verify the dn periodic wave solutions of the nonlinear Schrödinger equation and the mKdV equation are orbitally stable under small perturbations of the period L. Zheng [25] et al. and Shang [26] et al. applied the ideas and methods from literature [20] to show the dn periodic wave solutions of the generalized long-short wave equation with a quadratic nonlinear term (in Ref. [25]) and the generalized Zakharov equation with a quadratic nonlinear term (in Ref. [26]) are orbitally stable under small perturbations of period L, respectively. Lynnyngs Kelly, Arruda [27] studied the existence of smooth curves of dn periodic wave solutions and their orbital stability of the modified Boussinesq equation with fixed period L by using the classical stability theory proposed by M. Grillakis et al.. It is essential to note, previous literatures mainly focus on the case of the equation contains only a single nonlinear term in the study of orbital stability problems for periodic wave solutions, in contrast the orbital stability problem for the dn periodic wave solutions of the Boussinesq equation with both quadratic and cubic nonlinear terms has not been seen in the previous literature.
In this paper, we will consider the orbital stability of dn periodic wave solutions of the Boussinesq equation (i.e. Eq. (1.4) when p = 1 ) with simultaneous quadratic-cubic nonlinear terms. Complexity due to multiple nonlinear terms, the research method used in the literature [27] does not apply to the Eq. (1.6) studied in this paper. To achieve the objectives of this paper, we will mainly apply the ideas of Benjamin in the literature [28] and Bona in the literatures [29,30] for proving the stability of solitary waves. Although this method is also applied in the literatures [25,26], the difference is that our equation has not only quadratic nonlinear terms but also cubic nonlinear terms, which makes our research process much more difficult. In this regard, we will use suitable transformation and appropriate arithmetic technique to overcome the complexity of the problem with quadratic-cubic nonlinear terms and the difficulties in estimating the deteriorate of the continuous conservation functional in Eq. (1.6).
The content of this article is organized as follows: In Sect. 2, the dn periodic wave solution of Eq. (1.6) is found by using the integral method and the knowledge of elliptic functions; in Sect. 3, the spectral properties of the operator D are given according to Floquet theory [31], the Lamé equation [32] and Wely's essential spectrum theorem, that is has three simple eigenvalues, the zero eigenvalue is the second eigenvalue, and the remaining spectrum is made up of discrete multiple eigenvalues, which are used later to prove the orbital stability of dn periodic wave solutions; finally in Sect. 4, we apply the ideas of Benjamin and Bona in the literatures [28][29][30] to prove the stability of solitary waves, and obtain that the dn periodic wave solutions of the Boussinesq equation (1.6) with quadraticcubic nonlinear terms under small perturbations of period L is orbitally stable.

Existence of dn Periodic Traveling Wave Solutions
In this section, we focus on the existence of dn periodic traveling wave solutions of the Boussinesq equation (1.6) with quadratic-cubic nonlinear terms of the form as follow, where c ∈ R , = x − ct , , ∶ R → R are smooth periodic functions with any fixed period L > 0.
Substitute (2.1) into (1.6), the following system of equations can be obtained By substituting the first equation in (2.2) into the second equation, we can collate Integrating on the both sides of Eq.
. Next, multiply both sides of Eq.
(2.5) by U ′ , and integrate once about , it can be obtained that where M U is a necessary non-zero integration constant, and satisfy For the sake of convenience, let F(U) = U 4 + 2mU 2 + 4nU + M U , then we have . As in the (x, y) plane, the number of finite singularities of a planar dynamical system depends on the number of real roots of f (x) = x 3 + mx + n = 0 , denote the discriminant of the real root of f (x) = 0 as Δ = ( n 2 ) 2 + ( m 3 ) 3 , and if Δ < 0 , Eq. (1.6) has a periodic traveling wave solution, so at this point, we may assume that n = 0 , Δ < 0 , consider if b 2 < 0 , for −F(U) , due to where A 2 and B 2 satisfy

8)
It is easy to know 0 < B < √ −m < A < √ −2m . By substituting −F(U) into Eq. (2.7), we get According to the properties of elliptic functions [33], we make k � = |B| |A| , where A 2 = − 2m 2−k 2 and B 2 = − 2m(1−k 2 ) 2−k 2 . From (2.9), we have thus we can get it is easy to get For the sake of discussion, let its period is where K is the first type of elliptic integral. We transform (2.13) into a function with respect to B only, namely

B ≡ B(m) such that the fundamental period of the dn periodic wave solution = (⋅, A(m), B(m)) and = (⋅, A(m), B(m)) is T (B) = T (B) = L.
(1) There exist intervals W(m 0 ) and V(B 0 ) around m 0 and B 0 , respectively, and the unique smooth function is a strictly decreasing function. Due to k(B, m) is a strictly decreasing function about B, and the complementary modulus , derive Λ with respect to m, we can obtain Next, we will use the perturbation theorem, Floquet theory [31] and the Lamé equation [32] eigenvalue problem to study the properties of the operator D, for proving the stability of dn periodic wave solutions.
First of all, consider the problem of periodic eigenvalues of the operator D on [0, L] according to the theory of tight self-adjoint operators, the spectrum of D determined by (3.4) is a countable infinite set { n |n = 0, 1, …} , that satisfies and if n → +∞ , then n → +∞ . We use n to denote the eigenfunction corresponding to the eigenvalue n , it is easy to see that n as a continuously differentiable function of period L can be extended to the whole interval R.
Next, by Floquet's theory [31], consider the periodic eigenvalue problem (3.4) for the semi periodic problem on [0, L], it is known that the problem (3.6) is also a self-adjoint problem, thus eigenvalues { n |n = 0, 1, …} (n → +∞, n → +∞) satisfy Here, we use n to denote the eigenfunction corresponding to the eigenvalue n . Since for all x, function g with property g(x + L) = −g(x) is called a semi periodic function with semi periodic L, in other words, the period of the function g is 2L, therefore, when and only when = n ,n = 0, 1, … , equation has a resolution with a period L; when and only when = n , n = 0, 1, … , Eq. Finally, the perturbation theorem associated with Eq. (3.8) follows, the sequences (3.5), (3.7) are interleaved, that is Clearly, solution of Eq. (3.8) is stable in the intervals ( 0 , 0 ), ( 1 , 1 ), … , these intervals are called stable intervals; while the solution in the intervals (−∞, 0 ), ( 0 , 1 ), ( 1 , 2 ), … is unstable, these intervals are called unstable intervals, and the unstable interval (−∞, 0 ) is always present. Based on the previous analysis, we can obtain the following theorem. Proof First, prove that 0 = 1 < 2 , we derive the Eq. (2.5) with respect to x, it can be obtained D � m = 0 , thus zero is the eigenvalue of D corresponding to the eigenfunction ′ m . Since ′ m has two zeros on [0, L], the above case of the number of zeros of n and n indicates the zero eigenvalues is 1 or 2 . Next, prove that zero is the second eigenvalue of D. Using transformation
The first three eigenvalues 0 , 1 , 2 and their corresponding eigenfunctions Ψ 0 , Ψ 1 , Ψ 2 will be given in the following form.
. From the literature [33], it is known that the function of period 2K Because Ψ 0 has no zeros on [0, 2K), Ψ 2 has two zeros on [0, 2K), and for each k ∈ (0, 1) there is 0 < 1 < 2 . And then according to the relationship (3.11) between and , the following relationship can be obtained From this, we have that is an increasing function with respect to , where To complete the proof, we need to give the first two eigenvalues 0 , 1 of the semi periodic problem (3.6). According to the equation we can see the eigenvalue i associated with the semi periodic problem is related to i , and the relationship between i and i is (3.12) Ψ �� + ( − 6k 2 sn 2 (x;k))Ψ = 0, Ψ(0) = −Ψ(2k), Ψ � (0) = −Ψ � (2k), Notice that 0 = 1 + k 2 and 1 = 1 + 4k 2 are the first two eigenvalues of the Lamé equation [32] in (3.10) under the semi periodic problem. Their corresponding eigenfunctions are respectively. Therefore, from the relation between i and i , we obtain the eigenvalues 0 = − 3m 2−k 2 , 1 = − 3m(k 2 −1) 2−k 2 . From (3.9), the eigenvalues 0 , 1 , 2 are simple, the remaining eigenvalues are all multiple eigenvalues. ◻ For the sake of discussion, we introduce the following lemma.

Lemma 3.1 [34]. Let A is a self-companion operator with only one negative feature
Based on the above discussion, we can obtain the following theorem.
Consider the smooth curve of the dn wave m given in Theorem 2.1, then Proof (1) Since the dn periodic wave solution m is bounded, it can be deduced that 0 is finite. From D � m = 0 and ⟨ m , � m ⟩ = 0 , we can deduce that 0 ≤ 0 . Therefore, the following is the first step to proving that the infimum 0 can be obtained. Here we first let { j } be a column of functions of In addition, since the weak convergence is lower semi continuous, we can obtain Therefore, the infimum 0 can be reached on a desirable function Φ ≠ 0.
According to Lemma 3.1 and Theorem 3.1, it is known that D has the spectral properties in Lemma 3.1. Therefore, we need to find such that D = m , ⟨ , m ⟩ ≤ 0 . By Theorem 2.1, we know that the mapping m ↦ m is continuously differentiable. So, derivate (2.5) with respect to m, there is = − observe that Eq. (1.6) is invariant under the effect of T(⋅) , the following definition of orbital stability is given:    then, the deviation of the solution u(t) from orbit Θ Ψ is measure by Therefore, by using the method in the literature [28,33,34] again, it is known that, there exists an interval [0, T], such that for each t ∈ [0, T] , inf Ω t (y).

Conclusion
In this paper we are interested in studying the orbital stability of dn periodic wave solutions of the Boussinesq equation (1.6) with quadratic-cubic nonlinear terms. According to the ideas for proving the stability of solitary wave solutions from Benjamin [28] and Bona et al. [29,30], by overcoming the complexity caused by multiple nonlinear terms, we proved the orbital stability of the periodic wave solution of Eq. (1.6) under small perturbations of the L 2 norm. The results on the orbital stability of periodic wave solutions for the Boussinesq equation with quadratic-cubic nonlinear terms studied in this paper, can improve and extend the previous stability results of Boussinesq equation.
Acknowledgements The authors will be grateful for comments from the referees and from the editor.
Author Contributions WZ and SH formal analysis and writing (original draft) the manuscript.
Funding This work was supported by National Natural Science Foundation of China [grant number 11471215 ].

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