Remarks on solitary waves of the generalized two dimensional Benjamin–Ono equation

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Abstract

In this paper we study the generalized 2D-BO equation in two dimensions:(ut+βHuxx+upux)x+ϵuyy=0,(x,y)R2,t0.We classify the existence and non-existence of solitary waves depending on the sign of ϵ, β and on the nonlinearity. We also prove nonlinear stability and some regularity properties of such waves.

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)ut+βHuxx+ϵvy+upux=0,uy=vx,(x,y)R2,t0,where the constant ϵ measures the transverse dispersion effects and is normalized to ±1, the constant β is real and H is the Hilbert transform defined byHu(x,y,t)=p.v.1πRu(z,y,t)x-zdz,where 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,ut-Huxx+upux=0,xR,tR+.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)ut+αuxxx+βHuxx+ϵvy+upux=0,uy=vx,(x,y)R2,t0.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, X and X the closure of x(C0(R2)) for the normsxφX2=φL2(R2)2+φxxL2(R2)2andxφX2=φL2(R2)2+Dx1/2φxL2(R2)2.By a solitary wave solution of 2D-B equation, we mean a solution of (1.1) of the type u(x  ct, y), where uX and cR is the speed of propagation of the wave. So we are looking for localized solutions of the equation-cux+αuxxx+βHuxx+ϵvy+upux=0uy=vx.A solitary wave solution of 2D-BO is defined in the same vein:-cu+βHux+ϵx-2uyy+1p+1up+1=0.

Remark 1.1

Note that the wave speed c can be normalized to ±1, since the scale change w(x,y)=|c|-1/pux|c|,y|c|3/2 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 F(u)=12uL2(R2)2 andE(u)=12R2αux2-ϵv2-βuHux-2(p+1)(p+2)up+2dxdy.Note 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 φˆ the Fourier transform of φ, defined asφˆ(ξ,η)=R2φ(x,y)e-i(xξ+yη)dxdy.For sR, we denote by Hs(R2), the nonhomogeneous Sobolev space defined byHs(R2)=φS(R2):φHs(R2)<,whereφHs(R2)=(1+ξ2+η2)s2φˆ(ξ,η)L2(R2)and S(R2) is the space of tempered distributions. We also define Hxs(R2) byHxs(R2)=φL2(R2);φHxs(R2)<,whereφHxs(R2)=(1+ξ2)s2φˆ(ξ,η)L2(R2).We also denote 〈 , 〉 as L2(R2)-inner product; and ,X as inner product of X 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

Letα + β > 0. Eq. (1.6) admits no nontrivial solitary waves,

  • (I)

    if ϵ = 1 and one of the following cases occurs:

    • (i)

      α, β  0, c < 0 and p  4/3,

    • (ii)

      α  0, β  0, c > 0 and p  4/3,

    • (iii)

      α  0, β  0, c < 0 and p  4,

    • (iv)

      α  0, β  0, c  0 and p is arbitrary,

    • (v)

      α, β  0, c > 0 and p  4,

    or

  • (II)

    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 uX,uLp+2(R2)p+2CuL2(R2)(4-3p)/3x-1uyL2(R2)p/2uHx1/2(R2)(9p+4)/6.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. LetI(u)=S(u),u=uX2-1p+1uLp+2(R2)p+2.Then a solution of (1.7) with least energy is a solution of the following minimization problem:m=inf{S(u);uN},whereN={uX;u0,I(u)=0}.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 SX is X-stable, if for any ε > 0, there exists δ > 0 withinfwSu0-wX<δ,for any u0XXs with s  2, the solution u(t) of (1.1) with initial data u(0) = u0 can be extended to a global solution in C([0,);XXs) and satisfiessup0t<infwSu(t)-wX<ε.Otherwise, S is called X

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

References (42)

  • M.J. Ablowitz et al.

    Long internal waves in fluids of great depth

    Stud. Appl. Math.

    (1980)
  • J.P. Albert

    Positivity properties and stability of solitary wave solutions of model equations for long waves

    Commun. PDE

    (1992)
  • D. Bennet et al.

    The stability of internal waves

    Math. Proc. Cambridge Philos. Soc.

    (1983)
  • O.V. Besov et al.
    (1978)
  • N. Burq et al.

    On well-posedness for the Benjamin–Ono equation

    Math. Ann.

    (2008)
  • A.P. Calderón

    Commutators of singular integral operators

    Proc. Nat. Acad. Sci. USA

    (1965)
  • T. Cazenave et al.

    Orbital stability of standing waves for some nonlinear Schrödinger equations

    Commun. Math. Phys.

    (1982)
  • J. Chen et al.

    Blow up and instability of the solitary wave solutions to a generalized Kadomtsev–Petviashvili equation and two-dimensional Benjamin–Ono equations

    Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.

    (2008)
  • R.R. Coifman, Y. Meyer, Au delà des opérateurs pseudo-différentiels, Astrisque 57, Société Mathématique de France,...
  • J. Colliander et al.

    Low regularity solutions for the Kadomtsev–Petviashvili I equation

    Geom. Funct. Anal.

    (2003)
  • A. de Bouard et al.

    Symmetries and decay of the generalized Kadomtsev–Petviashvili solitary waves

    Siam J. Math. Anal.

    (1997)
  • Cited by (0)

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