Blow-up phenomena and stability of solitary waves for a generalized Dullin-Gottwald-Holm equation
© Dündar and Polat; licensee Springer. 2013
Received: 21 May 2013
Accepted: 3 September 2013
Published: 7 November 2013
In this work, we consider the Cauchy problem of the generalized Dullin-Gottwald-Holm equation. We establish a blow-up result for the generalized Dullin-Gottwald-Holm equation. In addition to this, we investigate the stability of solitary wave solutions of the equation.
is, in the dimensionless space-time variable , a model for unidirectional shallow water waves over a flat bottom. Here, is a momentum variable, the constants and are squares of length scales, and (where ) is the linear wave speed for undisturbed water at rest at spatial infinity, where h is the mean fluid depth and g is the gravitational constant. Dullin, Gottwald and Holm derived Eq. (1) by using asymptotic expansions directly in the Hamiltonian for Euler’s equations in the shallow water regime in . Eq. (1) was shown to be bi-Hamiltonian and to have a Lax pair formulation. The Dullin-Gottwald-Holm equation (we call it DGH equation for short) is an integrable system via the inverse scattering transform (IST) method and contains both the Korteweg-de Vries (KdV) and Camassa-Holm (CH) equations  as limiting cases.
Recently, the DGH equation has been studied by many authors. Tian et al.  studied the well-posedness of the Cauchy problem and the scattering problem for the DGH equation. In , Hakkaev proved the orbital stability of the peaked solitary waves for the DGH equation by using the method in . It was shown that the DGH equation has global solutions and blow-up solutions in [6–9].
One of the aims of this paper is to give the precise blow-up scenario and to show that Eq. (4) has blow-up solutions for . In addition to this, we investigate the stability of solitary wave solutions of Eq. (4) with and .
The remainder of the paper is organized as follows. In Section 2, we give our basic notation and recall some required results. In Section 3, we investigate blow-up of solutions for Eq. (4). In Section 4, we prove the stability of solitary wave solutions with the help of the orbital stability theory .
and for its inner product. For the sake of simplicity, we employ the same symbol c for different positive constants.
Some useful lemmas are as follows.
Lemma 2.1 
Here c is a constant depending only on s.
Lemma 2.2 
where is a monotone increasing function depending only on F and s.
Lemma 2.3 
where c is a constant depending only on s.
Lemma 2.4 
Theorem 2.1 
Lemma 2.5 
are constant with respect to t, where .
3 Blow-up phenomena
In this section, we discuss the blow-up phenomena of Eq. (4) with . For Eq. (4), which describes shallow water waves, the blow-up occurs only in the form of wave-breaking, i.e. the solution remains bounded but its slope becomes unbounded in finite time.
We now prove the following result.
by Sobolev’s embedding theorem, we obtain that the solution will blow up in finite time.
This completes the proof of Theorem 3.1. □
We have the following blow-up scenario for Eq. (4).
the invariance of ensures that the solution u is uniformly bounded as long as they exist.
If the slope of the solution becomes unbounded from below in finite time, then by Theorem 2.1 and Sobolev’s embedding theorem, we can see that the solution blows up in finite time.
By Theorem 3.1 and the above inequality, we have that if the slope of the solution is bounded from below in finite time, then the solution will not blow up in finite time. □
Next, we present the following blow-up result.
The conclusion is reached by letting . □
4 Stability of solitary waves
The appropriate notion of stability for the solitary waves here is orbital stability: a wave starting close to a solitary wave should stay close, as long as it exists, to some translate of the solitary wave. The orbit of a solitary wave is the set of all its translates .
We define the orbit , where τ is a one-parameter group of unitary operators on defined by , , . They may be interpreted physically as ‘solitary waves’ or ‘bound states.’
Otherwise, the φ-orbit is called unstable.
Definition can be seen in reference .
Lemma 4.1 For each , has a unique simple negative eigenvalue, zero is a simple eigenvalue, and the rest of its spectrum is bounded away from zero.
Differentiating (13) with respect to σ, we find that the right-hand side of Eq. (15) equals zero, that is, . The behavior of the function φ tells us that has exactly one zero. So, the zero eigenvalue of the operator is simple, and by using the Sturm-Liouville theorem, we have that only has a negative eigenvalue. Using Weyl’s theorem, the essential spectrum of belongs to . That completes the proof. □
As for the above results, it is known  that stability would be ensured by the convexity of the scalar function , . So, we obtain the following theorem under the condition .
Theorem 4.1 Suppose that is a solitary wave solution for , if , then is stable.
Multiplying solitary wave equation (13) by φ and integrating the resulting equation gives .
We say that is a minimizing sequence if for some , and .
Theorem 4.2 Let be a minimizing sequence for some . If , then there exists a subsequence , scalars and such that in . The function ψ achieves the minimum subject to the constraint .
In this section, we show that the stability of solitary wave solutions is determined by the convexity of the function .
We state the basic properties of the function d.
Therefore, d is well defined, and we may deduce its properties by examining the function . □
Lemma 4.2 For fixed , is monotonically increasing in c.
This shows that m is monotonically increasing in c, so that by (18), d must be monotonically increasing as well.
The following lemma is helpful in order to prove the stability of solitary waves.
This implies that , as , since is uniformly bounded for k.
From (22) and (23), is a minimizing sequence for the pair I, K and thus by Theorem 4.2 has a subsequence, named , that converges in to some φ. This contradicts (20). The proof is completed. □
This work was supported by Grant No. 13-FF-46 of the Dicle University of Scientific Research Projects Coordination (DUBAP), Diyarbakir, Turkey.
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