Stability or instability of solitary waves to double dispersion equation with quadratic-cubic nonlinearity

Abstract

The solitary waves to the double dispersion equation with quadratic-cubic nonlinearity are explicitly constructed.

Grillakis, Shatah and Strauss’ stability theory is applied for the investigation of the orbital stability or instability of solitary waves to the double dispersion equation. An analytical formula, related to some conservation laws of the problem, is derived. As a consequence, the dependence of orbital stability or instability on the parameters of the problem is demonstrated. A complete characterization of the values of the wave velocity, for which the solitary waves to the generalized Boussinesq equation are orbitally stable or unstable, is given.

In the special case of a quadratic nonlinearity our results are reduced to those known in the literature.

Introduction

The aim of this paper is to investigate the orbital stability or instability of the solitary waves to the double dispersion equation

$$u_{tt}-u_{xx}-h_2{u}_{ttxx}+h_1{u}_{xxxx}+f{\left(u\right)}_{xx}=0\text{,}x\in \mathbb{R},t\in {\mathbb{R}}^{+}\text{,}$$$$f\left(u\right)=a{u}^2+b{u}^3\text{,}$$

with constant coefficients $a$, $b$, $h_1$, $h_2$, $h_1>0$, $h_2>0$ and initial data

$$u(0,x)=u_0\left(x\right)\text{,}{u}_t(0,x)=u_1\left(x\right)\text{;}{u}_0\left(x\right)\in H^1\left(\mathbb{R}\right)\text{,}{u}_1\left(x\right)\in L^2\left(\mathbb{R}\right)\text{,}{(-\Delta )}^{-\frac{1}{2}}{u}_1\left(x\right)\in L^2\left(\mathbb{R}\right)\text{.}$$

Here ${(-\Delta )}^{-s}u=F^{-1}\left({\left|\xi \right|}^{-2s}F\left(u\right)\right)$, for $s>0$ and $F\left(u\right)$, $F^{-1}\left(u\right)$ are the Fourier and the inverse Fourier transform. By a solitary wave we mean a travelling with velocity $c$ wave of the form ${\phi }_c(x-ct)$, vanishing at infinity. Let us recall, that a solitary wave ${\phi }_c(x-ct)$ is said to be orbitally stable if any solution $u(t,x)$ of (1), (2), (3) with initial data close to the solitary wave for $t=0$ stays close to ${\phi }_c(x-ct)$ for every $t>0$.

The orbital stability of the solitary waves to the generalized Boussinesq equation

$$u_{tt}-u_{xx}+h_1{u}_{xxxx}+f{\left(u\right)}_{xx}=0\text{,}x\in \mathbb{R},t\in {\mathbb{R}}^{+}$$

with a single nonlinearity

$$f\left(u\right)={\left|u\right|}^{p-1}u\text{,}p>1$$

and $h_1=1$ is obtained by Bona and Sachs  [2] for $1<p<5$ and $\frac{p-1}{4}<c^2<1$. Later on, Liu  [12] proves instability result for the same problem for $c^2\le \frac{p-1}{4}$ and $1<p<5$ or $c^2<1$ and $p\ge 5$. Strong instability to (4), (5), i.e. instability by means of blow up of the solutions, is obtained in  [13], [14] when $0<c^2<\frac{p-1}{2(p+1)}$.

The stability of solitary waves to the generalized Boussinesq equation (4) with $h_1=1$ and quadratic-cubic nonlinearity (2) is analytically studied in  [5]. The authors prove that for some values of the constants $a$ and $b$ all solitary waves are stable (see Table 2).

The double dispersion equation (1) with a single nonlinearity (5) is studied in  [4]. The authors find conditions on $c$ and the parameters $a$, $b$ and $p$, for which the solitary waves are orbitally stable. Moreover, strong instability is proved for $c^2<c_0^2$, where the constant $c_0$ is explicitly given.

In the present paper we study the orbital stability or instability of the solitary waves to the double dispersion equation (1) with quadratic-cubic nonlinearity (2) and parameters, satisfying the conditions

$$a>0\text{,}b<0\text{,}{h}_1>h_2\text{.}$$

This choice is motivated by the investigations in  [7], [8], where Eq. (4) with nonlinearity (2) has been proposed to be relevant to pulse propagation in biomembranes. Recently, the model (4), (2) is revised in  [3] from the viewpoint of the solid mechanics. More precisely, a higher order term $h_2{u}_{ttxx}$ with $h_2$—a small positive constant, $h_2<h_1$, is added to (4). With this term (4) transforms into the double dispersion equation (1). In both models  [3], [7], [8], the constants in the equation are supposed to satisfy (6). For more details about the discussed models see  [3], [7], [8] and the references therein.

In this study we first derive explicitly all positive solitary waves to (1), (2). These waves are obtained in four regimes, depending on the relations between parameters $a$, $b$, $h_1$ and $h_2$.

The orbital stability or instability of solitary waves is studied by constructing of a function $d\left(c\right)$, related to some conservation laws of (1), (2), (3) and applying the convexity or concavity criterion in  [2], [6], [12]. By a stability or instability region we mean the set of wave velocities $c$, for which the solitary waves are orbitally stable or unstable.

In Theorem 2 we obtain an analytical formula for $d^{″}\left(c\right)$. This formula allows one to check the orbital stability or instability of the solitary waves with velocity $c$ for every choice of the constants $a$, $b$, $h_1$ and $h_2$, satisfying (6). As a consequence, in Theorem 3 we give a complete description of the regions of orbital stability or instability of the solitary waves to generalized Boussinesq equation (4), (2). In this way we improve the results in  [5] for all admissible values of the velocity $c$ (see Remark 4). For double dispersion equation (1), (2) the orbital stability or instability of the solitary waves is studied in neighbourhoods of the end points of the admissible values of the velocity $c$, see Theorem 5. In the special case of the generalized Boussinesq equation and double dispersion equation with a single quadratic nonlinearity our results reduce to the well known ones in the literature.

The paper is organized in the following way. In Section  2 explicit formulas for the solitary waves to (1), (2) and (4), (2) are obtained. An analytical formula for $d^{″}\left(c\right)$, which determines the orbital stability or instability of the solitary waves to (1), (2), is derived in Section  3. As an application of this formula, in Section  4 a complete investigation of the stability or instability of the solitary waves to (4), (2) is done, while Section  5 deals with an application to the double dispersion equation (1), (2).

For simplicity, throughout the paper, for functions depending on $x$ and $t$ we use the short notation $\Vert u\Vert ={\Vert u(\cdot ,t)\Vert }_{L^2\left(\mathbb{R}\right)}$. By $x_{+}$ we denote the first truncated power function: $x_{+}=0$, if $x\le 0$, and $x_{+}=x$, if $x>0$.