Research paperOrbital stability of solitary waves for generalized Boussinesq equation with two nonlinear terms
Introduction
Boussinesq equation an important model in the field of physics and mechanics [1], [2], [3], [4], describes the wave propagation in the weakly nonlinear and dispersive medium. Besides [1], [2], [3], [4], [5], [6], [7] also studied its Painlev property, Bcklund transformation and Lax pairs. Zakharov et al. [8] investigated the solvability of its initial value problem by the inverse scattering method. Moreover, Bona and Sachs [9], Linares [10] studied the existence of the global smooth solutions and stability of solitary waves with the initial value problem in the following extended form
Much has been done on the orbital stability of solitary wave solutions for the Boussinesq-type equation. For example, Bona and Sachs [9], Liu [11] studied the orbital stability and instability of solitary wave solutions for the following generalized Boussinesq equation respectively. The results showed that: when with p > 1, then the solitary wave solutions of Eq. (1.3) are unstable if 1 < p < 5 and or if p ≥ 5 and c2 < 1; while they are stable, if 1 < p < 5 and where c is the wave speed. Recently, Esfahani and Levandosky [12] studied the orbital stability and instability of solitary wave solutions for the generalized sixth-order Boussinesq (GSBQ) equation The results showed that: when with p > 1, if in Eq. (1.4), then . The solitary wave solutions of Eq. (1.4) are unstable if p ≥ 9 and c2 < 1, or if p < 9 and ; while they are stable if
Recently, the following generalized Boussinesq equation [15], [16], [17], [18], [19], [20] with two nonlinear terms has been investigated. Obviously, if Eq. (1.5) becomes the classic Boussinesq Eq. (1.1). If nonlinear terms in Eq. (1.5) are cubic-quintic, deriving from the models of atomic chains and shape-memory alloys [15], [16]. Zhang et al. [17] obtained the exact solitary wave solutions by the method of undetermined coefficients. Chertock et al. [18], Dimova and Vasileva [19] presented the local stationary solutions, and investigated their time evolution relation employing different numerical techniques. Kutev et al. [20] proved the existence and uniqueness of the global solution by Nehari functional, and found the conditions such that the weak solution blow up in the finite time.
In this paper, we will consider the orbital stability of solitary wave solutions for the above generalized Boussinesq Eq. (1.5). It should be pointed out that, previous literature investigating the orbital stability of the solitary wave solutions for Boussinesq-type equation, are limited to the model only containing a single nonlinear term [9], [10], [11], and require the consideration φc > 0 in the solitary wave (φc, ψc)T. However, Eq. (1.5) studied in this paper contains two nonlinear terms. Eq. (1.5) can be converted into Eq. (1.3), by letting where f(u) has two nonlinear terms, in which the symbols of b1, b2 are unfixed. Actually, f(u) is not always positive when u > 0. In addition, Eq. (1.5) has two bell-shaped solitary wave solutions where φ1(ξ) > 0 and φ2(ξ) < 0. The orbital stability (φ2, ψ2)T has not been considered in [9], [10], [11], which will be studied in this paper. So the problem studied in this paper is new. When or Eq. (1.5) corresponds to the practical problems. Therefore, the discussion of the solitary wave solutions may have practical value. Especially, we will discuss the influence of the interaction between nonlinear terms on the orbital stability, which has few been seen before. Obviously, this discussion is valuable to the control in the nonlinear system.
In this paper, we firstly give the general conclusion which can be used to judge the orbital stability for the two solitary waves by applying the theory of Grillakis–Shatah–Strauss orbital stability. Further, we deduce the explicit expression of discrimination d′′(c) according to two exact solitary waves of Eq. (1.5), and give the wave speed intervals which make the two solitary waves stable, called stable wave speed interval for short. We analyze the influence of the interaction between the nonlinear terms of Eq. (1.5) on the stable wave speed interval at last.
For convenience, according to Zhang et al. [17], the exact expressions of two solitary wave solutions of Eq. (1.5) are given by the following Lemma.
Lemma 1.1 Suppose . (1) If one of these two conditions b2 < 0, and holds, then Eq. (1.5) has a bell-profile solitary wave solution
where
(2) When holds, if one of these two conditions b2 < 0, and holds, then Eq. (1.5) has another bell-profile solitary wave solution
where
Obviously, under the conditions of Lemma 1.1, f1(ξ) is always positive, while f2(ξ) is always negative.
Moreover, according to Lemma 1.1 in [11], we can get the following lemma, which describes the local existence for the solution of Cauchy problem for Eq. (1.5).
Lemma 1.2 For any there exists t0 > 0, only depending on such that Eq. (1.5) has a unique solution
with .
Section snippets
General conclusion of orbital stability of solitary waves of Eq. (1.5)
Eq. (1.5) has the following equivalent form Let and then Eq. (2.1) can be written in a Hamiltonian form where
Define whose dual space is and the inner product of X is There exists a natural isomorphism I: X → X* defined by
The case of b1 < 0, b2 < 0, p > 2
Consider the orbital stability discriminant (2.22) of solitary waves (1.6). Since when b1 < 0, b2 < 0, (3.1) is positive, and . Then
Substituting (3.1), (3.2) into (2.22), we have In (3.3)
Further simplifying the discriminant d′′(c) when 0 < p ≤ 2
Since as x → ∞, when 0 < p ≤ 2, is a bounded function which tends to zero as x → ∞, or is equivalent to 1. Thus, we can apply the Integral Mean Value Theorem to the discriminant (2.22) of orbital stability for solitary wave (1.6). Therefore, ∃x* ∈ R, s.t. where
Now, we assume that d2 can be positive or negative, to compute the
Influences of nonlinear terms of Eq. (1.5) on the stable wave speed interval
We will discuss the effect by the nonlinear terms on orbital stability of the solitary waves (1.6) and (1.11) of Eq. (1.5) with Theorems 4.1 and 4.3 under the condition of b2 < 0, 0 < p ≤ 2, 0 < c2 < 1. According to Theorem 4.1, when b1 < 0, and the wave speed c satisfies c2 ∈ (t1, 1), (i.e. c satisfies (4.17)), or when b1 > 0 and the wave speed c satisfies the solitary wave (1.6) of Eq. (1.5) is orbitally stable; but according to Theorem 4.3, we know when b1 < 0,
Conclusions
In this paper, we apply the theory of Grillakis–Shatah–Strauss orbital stability to deduce the general conclusion (Theorem 2.1 in our paper) of the orbital stability for the solitary waves of Eq. (1.5). We found and prove that if p > 0, for any fix c satisfying 0 < c2 < 1, converges. Especially, it is internally closed uniform convergence if 0 < c2 < 1. Based on this, we established the explicit expression of the orbital stability discriminant (2.22) and (2.27) for
Acknowledgment
This research is supported by National Natural Science Foundation of China (No. 11471215).
References (22)
Global existence of small solutions for a generalized Boussinesq equation
J Differ Equ
(1993)- et al.
Stability theory of solitary waves in the presence of symmetry I
J Funct Anal
(1987) - et al.
Stability theory of solitary waves in the presence of symmetry II
J Funct Anal
(1990) Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal
J Math Pures Appl
(1872)Linear and nonlinear wave
(1974)On stochastization of one-dimensional chains of nonlinear oscillation
Sov Phys JETP
(1974)Boussinesq’s equation on the circle
Pure Appl Math
(1981)- et al.
The Painlevé property for partial differential equations
J Math Phys
(1983) The Painlevé property for the partial differential equations: Bäcklund transformation, Lax pairs, and the Schwarzian derivative
J Math Phys
(1983)The Painlevé property and Bäcklund transformation for the sequence of Boussinesq equations
J Math Phys
(1985)
Theory of solitons: the inverse scattering method
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