Solitary waves of a generalized Ostrovsky equation

https://doi.org/10.1016/j.nonrwa.2021.103395Get rights and content

Abstract

We consider the existence and stability of traveling waves of a generalized Ostrovsky equation (utβuxxxf(u)x)x=γu, where the nonlinearity f(u) satisfies a power-like scaling condition. We prove that there exist ground state solutions which minimize the action among all nontrivial solutions and use this variational characterization to study their stability. We also introduce a numerical method for computing ground states based on their variational properties. The class of nonlinearities considered includes sums and differences of distinct powers.

Introduction

The Ostrovsky equation utβuxxxf(u)xx=γu,also known as the rotation-modified Korteweg deVries (KdV) equation, was originally proposed by Ostrovsky [1], with f(u)=u2, as a model for the unidirectional propagation of weakly nonlinear long surface and internal waves of small amplitude in a rotating fluid, where the parameter γ is a measure of rotational effects (see also [2], [3], [4], [5], [6], [7]). In [8], Holloway, Pelinovsky and Talipova proposed it as a model for long waves in coastal zones, with a quadratic-cubic nonlinear term f(u)=u2+u3. It has also been derived as a model for the propagation of short pulses in nonlinear media [9], and as a model for magneto-acoustic solitary waves in a plasma [10] and solitary waves in a relaxing medium [11]. We refer the reader to the recent review article by Stepanyants [12] for a more thorough account of its history and applications.

The aim of this paper is to study the existence and stability of traveling waves of the Ostrovsky equation for a general class of nonlinearities [13], [14], [15]. By a traveling wave, we mean a solution of (1.1) of the form u(x,t)=φ(xct) for some cR. The profile φ must then satisfy the stationary equation βφxxcφγx2φ=f(φ).Existence of traveling waves was considered in [16] and [17] for the quadratic nonlinearity f(u)=u2, and in [18] for more general homogeneous nonlinearities. It was shown that solitary waves exist in the space X1 (see the definition below) provided β>0, γ>0 and c<2βγ. The behavior of the solitary waves of the Ostrovsky equation as γ0 was also studied in [17] and [18], where it was shown that solitary waves of the Ostrovsky equation converge in H1 to solitary waves of the KdV equation. This is somewhat surprising, as the Ostrovsky solitary waves have zero mass, while the KdV solitary waves have nonzero mass.

Stability of solitary waves was considered in [17], [19] and [18]. It was shown for homogeneous nonlinearities that the stability of solitary waves is determined by the convexity or concavity of the function d defined in Eq. (4.5). Although there are no known explicit expressions for d, by using the scaling identity satisfied by d, together with numerically computed solitary wave solutions, it is possible to obtain numerical approximations of d that determine the regions of stability and instability in terms of the parameters β, c and γ. This was done in [17], [18] for the class of pure power nonlinearities f(u)=|u|p and f(u)=|u|p1u. Note that it seems difficult to apply the abstract theory by Grillakis, Shatah and Strauss [20]. More precisely, due to the fourth order differential operator appearing in (1.2) it is not easy to obtain the spectral properties of the associated linearized operator of (1.2).

The purpose of this paper is to extend the aforementioned results to a class of nonlinearities that satisfy a power-like scaling condition (see Assumption 2.1 below). This class of nonlinear terms includes sums and differences of powers. The paper is organized as follows. Section 2 contains the proof of the main existence theorem. In Section 3 we show that as γ goes to zero, traveling waves of the Ostrovky equation converge to traveling waves of the KdV equation in H1(R). The main stability and instability results are proved in Sections 4 Stability, 5 Instability. In Section 6 we apply these results to nonlinearities that are sums or differences of powers. Finally, in Section 7 we present a numerical method for computing ground state traveling waves and use it to determine precise regions of stability and instability for several different nonlinear terms.

Notation.

We shall use (u) or uˆ to denote the Fourier transform of u with respect to the spatial variable x, and 1(u) or uˇ to denote the inverse Fourier transform. We denote by x1 the operator defined by x1u=1(iξ)1u.The space Xs is defined by Xs={uHs(R)x1uHs(R)}with norm uXs=uHs+x1uHs.

Section snippets

Existence of traveling waves

In this section we prove existence of traveling wave solutions of Eq. (1.1). We will restrict attention to the case β>0, γ>0 and c<2βγ. Let I(u)=I(u;β,c,γ)=12Rβux2cu2+γ(x1u)2dx,and K(u)=RF(u)dx,where F=f and F(0)=0. Then uX1 is a weak solution of (1.2) if and only if I(u)=K(u), or equivalently if u is a critical point of S, where S(u)=I(u)K(u)In [17] and [18] the existence of critical points of S was obtained for homogeneous nonlinearities by showing that there exist minimizers of I(u)

Gamma limit

We next consider the behavior of the ground states as the parameter γ approaches zero (see Fig. 1). Setting γ=0 in the Ostrovsky equation and integrating, one obtains the generalized KdV equation utβuxxxf(u)x=0.As in [17] and [18], we show that traveling waves of the Ostrovsky equation converge strongly in H1 to traveling waves of the KdV equation, which are solutions of βφxxcφ=f(φ).We first show that (3.2) has exactly one or two nontrivial solutions.

Theorem 3.1

Suppose f satisfies Assumption 2.1. Then

Stability

In this section we consider the stability of traveling waves of the Ostrovsky equation. We first make the following local well-posedness assumption.

Assumption 4.1

For each u0X1, there exists T>0 and a unique solution u of (1.1) in C([0,T],X1) satisfying E(u)Rβ2ux2+γ2|x1u|2F(u)dx=E(u0)and V(u)12Ru2dx=V(u0)for all t[0,T].

By the results of Tsugawa [29] and Wang and Yan [30] this assumption holds for f(u)=u2 and f(u)=u3. Given a subset S of X1 and ε>0, we define Uε(S)={uX1:uvX1<ε for some vS}.

Definition 4.1

We say

Instability

In this section we consider stability of the orbit of a particular ground state. We once again will suppose Assumption 4.1 holds throughout this section. Given φG(β,c,γ), we define the orbit of φ by O(φ)={τrφ:rR}where τrφ(x)=φ(xr). The instability theorem in [18] (Theorem 4.4) for the case of homogeneous nonlinearities relies only on the well-posedness of (1.1) and the fact that ground states are minimizers of S subject to the constraint P=0 (Lemma 4.5). Thus, by exactly the same method of

Applications

In this section we apply the existence and stability theorems of the previous sections to nonlinearities that are sums and differences of powers. We consider three cases, the sum of two odd powers f(u)=|u|q1u+|u|p1u, the sum of an odd power and an even power f(u)=|u|q1u+|u|p, and the difference of two odd powers f(u)=|u|q1u|u|p1u. We assume that Assumption 4.2 holds. By replacing u with u, it follows that the results for the difference of an odd and an even power are the same as those

Numerical computations

As in [17], [18] and [34], we use numerical approximations of ground states to approximate the function d(c) and its derivatives, and thereby determine approximately the ranges of parameters β, c and γ for which the ground states are stable and unstable. The numerical method used here is based on the variational characterization of ground states as minimizers of S subject to the constraint P=0. It is essentially a gradient descent method where at each step a rescaling is done to preserve the

Acknowledgment

The authors wish to thank the unknown referee for the valuable suggestions which helped to improve the paper. A. E. was supported by the Social Policy Grant (SPG) funded by Nazarbayev University, Kazakhstan .

References (35)

  • OstrovskyL.A.

    Nonlinear internal waves in a rotating ocean

    Okeanologia

    (1978)
  • CazenaveT. et al.

    Orbital stability of standing waves for some nonlinear Schrodinger equations

    Comm. Math. Phys.

    (1982)
  • OstrovskyL.A. et al.
  • PetviashviliV.I.

    Equation of an extraordinary soliton

    Sov. J. Plasma Phys.

    (1976)
  • VarlamovV. et al.

    Cauchy Problem for the Ostrovsky equation

    Discrete Dynam. Syst.

    (2004)
  • VarlamovV.

    Oscillatory integrals generated by the Ostrovsky equation

    Z. Angew. Math. Phys.

    (2005)
  • ZhangP. et al.

    Symmetry and uniqueness of the solitary-wave solution for the Ostrovsky equation

    Arch. Ration. Mech. Anal.

    (2010)
  • Cited by (4)

    View full text