Remarks on solitary waves of the generalized two dimensional Benjamin–Ono equation
Introduction
In this paper we are interested in studying a model which is a natural two-dimensional extension of (1.2), namely, the two-dimensional generalized Benjamin–Ono equation (2D-BO henceforth)where the constant ϵ measures the transverse dispersion effects and is normalized to ±1, the constant β is real and is the Hilbert transform defined bywhere p.v. denotes the Cauchy principal value. When −β = p = 1, Eq. (1.1) was introduced by Ablowitz and Clarkson in [1] and Ablowitz and Segur in [2], which arises in the study of internal waves in deep stratified fluids (see also [39]). Eq. (1.1) is an extension of one-dimensional generalized Benjamin–Ono (BO) equation,See also [16], [17], [18], [19], [20], [21]. The integro-differential equation (1.2), when p = 1, serves as a generic model for the study of weakly nonlinear long waves incorporating the lowest order effects of nonlinearity and nonlocal dispersion; in particular, the propagation of one-dimensional internal waves in stratified fluids of great depth. Eq. (1.2) has been extensively studied by several authors considering both the initial value problem and the nonlinear stability. The initial value problem associated to Eq. (1.2) has been studied, recently, for instance in [7], [27], [28], [34], [38], [41], whereas the issue of existence and stability of solitary waves has been studied in [3], [5].
Eq. (1.1) is a spacial case of the generalized two dimensional Benjamin (2D-B henceforth)Eq. (1.3) contains other classical equations: when β = 0, α = p = 1, (1.3) is known as the KP-II equation (ϵ = 1) or KP-I equation (ϵ = −1). Many rigorous results have recently appeared concerning the Cauchy problem for the KP equations [12], [24], [25], [26], [35], [36], whereas the issue of existence and stability of solitary waves has been studied in [13], [31], [32]. Regarding on Eq. (1.3), Boling and Yongqian investigated the well-posedness of the local solutions for the Cauchy problem in [22]. They used a general local existence theory has been established by Ukai [40] and Saut [39] for KP equations. When α = 1, recently, the existence, blow up and instability of solitary waves of (1.3) has been studied by Chen et al. in [10].
In this paper, we shall investigate the existence and nonexistence of the nontrivial solitary waves of Eqs. (1.3), (1.1), when the exponent p in (1.1) will be a rational number of the form p = k/m, where m is odd and m and k are relatively prime.
In order to give a precise definition of our needed spaces, we use the following spaces. We shall denote, and the closure of for the normsandBy a solitary wave solution of 2D-B equation, we mean a solution of (1.1) of the type u(x − ct, y), where and is the speed of propagation of the wave. So we are looking for localized solutions of the equationA solitary wave solution of 2D-BO is defined in the same vein: Remark 1.1 Note that the wave speed c can be normalized to ±1, since the scale change transforms Eq. (1.7) in u into the same in w with ∣c∣ = 1. Remark 1.2 Note that the constant β ≠ 0 can be normalized to ±1, since the scale change w(·) = u(∣β∣·) transforms Eq. (1.7) in u into the same in w with ∣β∣ = 1. Remark 1.3 It is easy to check that there is no scaling for (1.6) to normalize the wave speed c. We shall prove that the solitary waves of (1.1) are stable in some sense, when p < 4/5. We demonstrate that these solution are ground states, namely, they have minimal energy. It worth remarking that the flow associated to (1.3) satisfies the conservation quantities F and E, where andNote that, when α = 0, F and E are two invariants of (1.3).
This paper is organized as follows. In Section 2, we shall obtain the conditions of the nonexistence of the solitary wave of 2D-B and 2D-BO equations. Section 3 is devoted to the existence and the regularity properties of solitary waves of 2D-BO equation. In Section 4, we shall obtain some variational properties of solitary waves obtained in Section 3; and we show that these solutions are ground states. In Section 5, the nonlinear stability of our solitary waves of (1.1) will be investigated.
Before stating the main results let us introduce some notations that will be used throughout this article.
We shall denote by the Fourier transform of φ, defined asFor , we denote by , the nonhomogeneous Sobolev space defined bywhereand is the space of tempered distributions. We also define bywhereWe also denote 〈 , 〉 as -inner product; and as inner product of space. For any positive numbers a and b, the notation a ≲ b means that there exists a positive (harmless) constant c such that a ⩽ cb.
Section snippets
Nonexistence
In this section, we are going to obtain the conditions of nonexistence of solitary wave solutions of 2D-B and 2D-BO equations. Theorem 2.1 Let ∣α∣ + ∣β∣ > 0. Eq. (1.6) admits no nontrivial solitary waves, if ϵ = 1 and one of the following cases occurs: α, β ⩾ 0, c < 0 and p ⩾ 4/3, α ⩽ 0, β ⩾ 0, c > 0 and p ⩽ 4/3, α ⩽ 0, β ⩾ 0, c < 0 and p ⩾ 4, α ⩾ 0, β ⩽ 0, c ≠ 0 and p is arbitrary, α, β ⩽ 0, c > 0 and p ⩽ 4,
or
if ϵ = −1 and one of the following cases occurs:
- (i)
α, β ⩾ 0, c < 0 and p ⩽ 4,
- (ii)
α ⩽ 0, β ⩾ 0, c ≠ 0 and p is arbitrary,
- (iii)
α ⩾ 0, β ⩽ 0, c > 0 and p ⩾ 4,
- (iv)
α ⩾ 0, β ⩽ 0, c < 0 and p ⩽ 4/3,
Existence and regularity
In this section, we are going to prove the existence and regularity property of solitary wave solutions of 2D-BO. Henceforth, we assume that α = 0, c > 0 and ϵ = −1. For simplicity and without loss of generality, we can also suppose that β = −1. By Remark 1.1, we can also assume that c = 1.
First, we are going to obtain an embedding appropriate to Eq. (1.1). Lemma 3.1 Let p ⩽ 4/3. Then there is a constant C > 0 (depending on p) such that for any ,As
Variational characterizations
In this section, we are going to obtain some variational properties of the solution constructed in last section which plays an important role in our stability/instability analysis (cf. [15]). In particular, we shall show that these solutions are ground states, i.e. solutions with minimal energy. LetThen a solution of (1.7) with least energy is a solution of the following minimization problem:The following theorem
Stability
In this section, the nonlinear stability of the solitary waves is the issue we are going to investigate. Throughout this section, we assume c > 0. First, it worth reminding the definition of the nonlinear stability. Definition 5.1 We say that the set is -stable, if for any ε > 0, there exists δ > 0 withfor any with s ⩾ 2, the solution u(t) of (1.1) with initial data u(0) = u0 can be extended to a global solution in and satisfiesOtherwise, S is called
Summary
We have considered the two dimensional Benjamin–Ono equation, which derived as a model for internal waves in deep stratified fluids. The nonexistence/existence of the solitary-wave solutions of this equation (and also Benjamin equation) have been shown via Mountain-Pass theorem. The key ingredient in proving this result is a suitable embedding closely related to the embedding theorems for anisotropic Sobolev spaces studies in [6]. Moreover, we have also demonstrated these solitary waves are
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