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Blow-up phenomena and stability of solitary waves for a generalized Dullin-Gottwald-Holm equation
Boundary Value Problems volume 2013, Article number: 226 (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.
The nonlinear evolution 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.
Using the notation , we can rewrite the initial value problem of Eq. (1) as
Eq. (2) relates two separately integrable soliton equations for water waves. Formally, when and , this equation becomes the KdV equation
For and , Eq. (2) becomes the Camassa-Holm equation
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].
If the term is replaced with in the DGH equation, then it is known as the generalized DGH equation and has the following form:
Recently, the local well-posedness problem for the following generalization of the DGH equation:
Eq. (4) can be written as the following Hamiltonian form:
where is a skew-symmetric operator and
where . We note that the functional is formally conserved. Moreover, the other conserved quantity is
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 .
Firstly, we start by summarizing some notations. , ; with the norm
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 
Assume that . Then we have
Here c is a constant depending only on s.
Lemma 2.2 
Assume that with . Then, for every , we have that
where is a monotone increasing function depending only on F and s.
Lemma 2.3 
Assume that . Then is an algebra. Moreover,
where c is a constant depending only on s.
Lemma 2.4 
Let and . Then, for every , there exists at least one pair of points such that
and , are absolutely continuous in . Moreover,
Theorem 2.1 
Assume that , and . Given , , there exists a maximal and a unique solution u to Eq. (4) such that
Moreover, the solution depends continuously on the initial data, i.e., the mapping
Lemma 2.5 
Let be a solution of Eq. (4). Then the functionals
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.
Set , , then for all . Using this identity, we can rewrite Eq. (4) as follows:
or in the equivalent form:
We now prove the following result.
Theorem 3.1 Let , , and , . If T is the existence time of the corresponding solution of initial data , then the -norm of to Eq. (4) (or (6)) blows up on if and only if
Proof Let be the solution of Eq. (4) with the initial data , , which is guaranteed by Theorem 2.1. If
by Sobolev’s embedding theorem, we obtain that the solution will blow up in finite time.
Next, applying the operator to Eq. (6), multiplying by , and integrating over R, we obtain
Here, . Assume that there exists such that
Similar to , using Lemma 2.1 with , we get
On the other hand, we estimate the second term of the right-hand side of Eq. (7)
Here, we applied Lemma 2.2 with and , Lemma 2.3 with . From (7)-(9), we obtain
Thus, by Gronwall’s inequality, we get
This completes the proof of Theorem 3.1. □
We have the following blow-up scenario for Eq. (4).
Theorem 3.2 Assume that , . Given , , the solution of Eq. (4) is uniformly bounded. Blow-up in finite time occurs if and only if
Proof is an invariant for Eq. (4). According to the inequalities
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.
Next, if the slope of the solution is bounded from below in finite time, then we deduce that the solution will not blow up in finite time. Differentiating Eq. (5) with respect to x, in view of the identity , we have
Note that and
By Young’s inequality, we get
Define . Since for all , it follows that a.e. on
If , then and is decreasing. Otherwise, . Thus we get
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.
Theorem 3.3 Assume that , . Given , , , assume that we can find with
Then the corresponding solution to Eq. (4) for blows up in finite time. Moreover, the maximal time of existence T satisfies the inequality
Proof Now define by Lemma 2.4, and let be a point where this infimum is attained. From Eq. (10), we have
For , since , we arrive at
By Young’s inequality, we get
So, it follows that
where (note that )
The absolute continuity of the locally Lipschitz function allows us to perform an integration over and to have
We claim now that for all , where is fixed arbitrarily provided that . In fact, assuming the contrary, in view of being continuous, ensure the existence of such that in and . Then we deduce that
and a contradiction appears as we take . Using (11), we get
where . Since , and is locally Lipschitz, it follows that is locally Lipschitz as well. This gives
Integration of this inequality yields
Since , we obtain
In fact, as a consequence of these considerations, we obtain that the maximal existence time
An estimation from the above for T is obtained immediately, namely
The conclusion is reached by letting . □
4 Stability of solitary waves
where . When and , the conservation law takes the form
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.’
Definition The φ-orbit is stable if for all , there exists with the following property. If and u is a solution of Eq. (12) in some interval with , then u can be continued to a solution in and
Otherwise, the φ-orbit is called unstable.
Definition can be seen in reference .
Substituting into Eq. (12) and integrating once with respect to σ, we obtain
In terms of the functionals E and F,
and according to Eq. (13), we can obtain that , where and are the Fréchet derivatives of E and F, respectively. To study the orbital stability of the solitary waves of Eq. (12), we need the operator and the function . The linearized operator around φ is defined by
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.
Proof For any , we have and
By (14), we know that is a self-adjoint operator and
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.
Define the functionals
For , we consider the following constrained minimization problem on :
Then if achieves the minimum of problem (16), for some , then there exists a Lagrange multiplier ϑ such that
Hence is a solution of solitary wave equation (13). By homogeneity of and , φ satisfies
and it follows that
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.
A simple calculation gives
By using , we obtain
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.
Proof We assume that , are solutions of Eq. (13) corresponding to , , respectively. Without loss of generality, let be , then we have
This shows that m is monotonically increasing in c, so that by (18), d must be monotonically increasing as well.
A tubular neighborhood around the orbital is defined by
It follows from (18) and the fact that is monotonically increasing in c that
The following lemma is helpful in order to prove the stability of solitary waves.
Lemma 4.3 If , then there exists such that for any , we have
Proof By using and Taylor’s formula, we have the expansion
for near c. Using the continuity of and choosing ε sufficiently small, we get that
It follows from (18) and (19) that . Since is a minimizer of subject to the constraint , we then have
Proof of Theorem 4.1 Suppose that is unstable. Then there exists and a sequence of initial data satisfying
where is a solution of Eq. (12) with initial datum . Then, by Theorem 2.1, is continuous in t, and there exist times such that
When , , E and F are invariants of (12) so that
by Lemma 4.3, we have
This implies that , as , since is uniformly bounded for k.
The continuity of d implies that
Using (17) and the fact that , we have
so it follows from (21) and (22) that
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. □
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This work was supported by Grant No. 13-FF-46 of the Dicle University of Scientific Research Projects Coordination (DUBAP), Diyarbakir, Turkey.
The authors declare that they have no competing interests.
All authors typed, read and approved the final manuscript.