Research paper
Orbital stability of solitary waves for generalized Boussinesq equation with two nonlinear terms

https://doi.org/10.1016/j.cnsns.2017.11.018Get rights and content

Highlights

  • The equation studied in this paper contains two nonlinear terms, which is new.

  • The explicit expression has been deduced to judge the orbital stability for the solitary wave.

  • More detail results has been obtained on the orbital stability and instability for the solitary wave.

  • The stable wave speed interval has been presented, which the interaction between the two nonlinear terms affects also has been discussed. This has not appeared in the previous reference.

  • The method used in this paper can be referred to study other nonlinear evolution equations.

Abstract

This paper investigates the orbital stability and instability of solitary waves for the generalized Boussinesq equation with two nonlinear terms. Firstly, according to the theory of Grillakis–Shatah–Strauss orbital stability, we present the general results to judge orbital stability of the solitary waves. Further, we deduce the explicit expression of discrimination d′′(c) to judge the stability of the two solitary waves, and give the stable wave speed interval. Moreover, we analyze the influence of the interaction between two nonlinear terms on the stable wave speed interval, and give the maximal stable range for the wave speed. Finally, some conclusions are given in this paper.

Introduction

Boussinesq equation utt+uxxxxuxxb1(u2)xx=0,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 Painleve´ property, Ba¨cklund 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 utt+δuxxxxauxx(σ(u))xx=0,xR,t>0.

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 uttuxx+(f(u)+uxx)xx=0,respectively. The results showed that: when f(s)=|s|p1s with p > 1, then the solitary wave solutions of Eq. (1.3) are unstable if 1 < p < 5 and c2p14<1, or if p ≥ 5 and c2 < 1; while they are stable, if 1 < p < 5 and p14<c2<1, 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 uttuxxβuxxxxuxxxxxx+f(u)xx=0.The results showed that: when f(s)=|s|p1s with p > 1, if β=0 in Eq. (1.4), then d(c)=(1c2)3p+54(p1)d(0). The solitary wave solutions of Eq. (1.4) are unstable if p ≥ 9 and c2 < 1, or if p < 9 and c2<2(p1)p+7; while they are stable if p<9,c2>2(p1)p+7.

Recently, the following generalized Boussinesq equation [15], [16], [17], [18], [19], [20] with two nonlinear terms uttuxx+uxxxx(b1up+1+b2u2p+1)xx=0,p>0.has been investigated. Obviously, if p=1,b2=0, Eq. (1.5) becomes the classic Boussinesq Eq. (1.1). If p=2, 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 f(u)=b1up+1b2u2p+1, 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 (φi,ψi)T,(i=1,2), 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 p=1, or p=2, 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 c21<0.

(1) If one of these two conditions b2 < 0, and b2=0,b1<0 holds, then Eq. (1.5) has a bell-profile solitary wave solution u1(x,t)=φ1(ξ)=[f1(xct)]1p,where f1(ξ)=A1sech2α12ξ4+B1sech2α12ξ,ξ=xct,α1=p1c2,A1=2(p+2)p+1(1c2)(p+1)b12+b2(c21)(p+2)2,B1=22b1p+1(p+1)b12+b2(c21)(p+2)2.

(2) When (z)1p(z<0) holds, if one of these two conditions b2 < 0, and b2=0,b1>0 holds, then Eq. (1.5) has another bell-profile solitary wave solution u2(x,t)=φ2(ξ)=[f2(xct)]1p,where f2(ξ)=A2sech2α22ξ4+B2sech2α22ξ,α1=α2,A2=A1,B2=2+2b1p+1(p+1)b12+b2(c21)(p+2)2=4B1.

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 u0X(H1(R)×L2(R)), there exists t0 > 0, only depending on u0X, such that Eq. (1.5) has a unique solution uC([0,t0);H1(R)×L2(R))with u(0)=u0.

Section snippets

General conclusion of orbital stability of solitary waves of Eq. (1.5)

Eq. (1.5) has the following equivalent form {ut=vx,vt=(uuxx+b1up+1+b2u2p+1)x.Let u=(uv), and then Eq. (2.1) can be written in a Hamiltonian form dudt=JE(u),uX,where J=(0xx0),E(u)=E(u,v)=R(12u2+12v2+12ux2+b1p+2up+2+b22p+2u2p+2)dx,E(u)=(E(u)E(v))=(uuxx+b1up+1+b2u2p+1v).

Define X=H1(R)×L2(R), whose dual space is X*=H1(R)×L2(R), and the inner product of X is (u1,u2)=R(u1u2+v1v2+v1xv2x)dx,u1,u2X.There exists a natural isomorphism I: X → X* defined by Iu1,u2=(u1,u2),

The case of b1 < 0, b2 < 0, p > 2

Consider the orbital stability discriminant (2.22) of solitary waves (1.6). Since 4cp(d1d1d2d2)=4c2b1b2(p+2)p(1c2)[b12(p+1)b2(1c2)(p+2)2],when b1 < 0, b2 < 0, (3.1) is positive, and 0<d1+d2d1+d2cosh2(d3x)<1. Then 0+(1d1+d2cosh2(d3x))2p+1dx=1d1+d20+(1d1+d2cosh2(d3x))2pd1+d2d1+d2cosh2(d3x)dx<1d1+d20+(1d1+d2cosh2(d3x))2pdx.

Substituting (3.1), (3.2) into (2.22), we have d(c)2(1+2c(d1+d2)p(d1+d2)+cd3d3)·0+(1d1+d2cosh2(d3x))2pdx.In (3.3) d1+d2d1+d2=c(b12(p+1)b2(1c2)(p+2)2b1p+1

Further simplifying the discriminant d′′(c) when 0 < p ≤ 2

Since 0<d1+d2cosh2(d3x) as x → ∞, when 0 < p ≤ 2, (1d1+d2cosh2(d3x))2p1 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. d(c)=(1d1+d2cosh2(d3x*))2p1·I(c),where I(c)=2[1+2cp(d2d2+d3p2d3)]0+1d1+d2cosh2(d3x)dx+4cp(dd1d2d2)0+1(d1+d2cosh2(d3x))2dx.

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 c2(c02,1),(c02(p2,1)), 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, 0+ddc(1d1+d2cosh2(d3x))2pdxconverges. 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).

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