- Open Access
Existence and stability of solitary waves for the generalized Korteweg-de Vries equations
© Hong; licensee Springer. 2013
- Received: 31 March 2013
- Accepted: 16 April 2013
- Published: 10 May 2013
In this paper, we consider the fractional Korteweg-de Vries equations with general nonlinearities. By studying constrained minimization problems and applying the method of concentration-compactness, we obtain the existence of solitary waves for the generalized Korteweg-de Vries equations under some assumptions of the nonlinear term. Moreover, we prove that the set of minimizers is a stable set for the initial value problem of the equations, in the sense that a solution which starts near the set will remain near it for all time.
- generalized Korteweg-de Vries equations
- constrained minimization problems
- (A), and for some ,
, the Fourier transform .
When and , equation (1.1) is the well-known Korteweg-de Vries equation, introduced by Korteweg and de Vries in 1895 (cf. ). The existence and stability of solitary waves of the Korteweg-de Vries equation is considered by Benjamin in . Recently, in , Pelinovsky obtained a Korteweg-de Vries equation with a forcing term, which is a simple analytical model of tsunami generation by submarine landslides.
Inspired by the methods used in [4, 5], by studying the problem (), we obtain the existence of solitary waves for equation (1.1) with some special nonlinearities , where , and general nonlinearities satisfying the assumption (A). Moreover, we prove that the set of minimizers is a stable set for the initial value problem of equation (1.1) in the sense that a solution which starts near the set will remain near it for all time. In order to obtain those results, we have to overcome one main difficulty: the minimization problem () is given in the unbounded domain ℝ which results in the loss of compactness. As is done in [4, 6], we overcome the difficulty of loss of compactness by the method of concentration-compactness introduced by Lions in [7, 8] for solving some minimization problems in unbounded domains.
Now we give our main results.
where is the norm of .
for any .
where is the norm of given in Section 5.
for any .
The paper is organized as follows. In Section 2, we give some preliminaries. In Section 3, we study the existence and stability of solitary waves of equation (1.1) with some special nonlinearities . Section 4 is devoted to studying equation (1.1) with general nonlinearities satisfying the assumption (A). We shall consider the existence and stability of solitary waves of equation (1.1) with in Section 5.
for some . Then in for .
where is a constant independent of u.
Since is bounded in and as , applying the above inequality, we know that in for . □
Next, we establish a convergence result that will be used in the proof of Theorem 1.3.
At first, we commence by studying some properties of the functional and the behavior of the minimizing sequences for the problem ().
If is a minimizing sequence for the problem (), there exists a constant such that for all n;
- (iii)If is a minimizing sequence for the problem (), there exist a positive constant δ and a sequence of real numbers such that
for sufficiently large n.
For , by taking sufficiently small, we get .
- (ii)Let be a minimizing sequence for the problem (). Then, by (3.1), we have
- (iii)Let be a minimizing sequence for the problem (). Then we claim: there exists a constant such that for all sufficiently large n. We argue by contradiction: if no such exists, then . Hence
which contradicts (i). So, the claim is achieved.
for sufficiently large n. The proof of Lemma 3.1 is completed. □
The next lemma will establish a subadditivity inequality which will be a crucial step in the proof of the existence minimizer for the problem ().
Lemma 3.2 For all , .
Now we formulate the following two theorems, which are special cases corresponding to Theorem 1.1 and Theorem 1.2, and give their proof with the aim of Lemma 3.1 and Lemma 3.2.
for any .
for sufficiently large n.
which contradicts (3.10). □
An immediate consequence of Theorem 3.1 is that forms a stable set for the initial-value problem for equation (1.1).
for all n, where solves equation (1.1) with .
for sufficiently large . (3.14) is a contradiction. Therefore, the set is -stable with respect to equation (1.1). □
In this section, we consider (1.1) with and more general nonlinearities f satisfying condition (A). At first, we study the properties of the functional and the minimizing sequence of the problem ().
For any , is finite and continuous on . Moreover, each minimizing sequence for () is bounded;
for any .
where is a positive constant dependent only on ε and η for given . Choosing and such that , we see that .
Since for , it is easy to check that is continuous on .
- (ii)For given such that , let for . We obtain that(4.7)
is non-increasing on and ;
- (ii)there exists such that
This inequality yields for .
where is dependent only on .
We observe that , which implies (ii). □
Then we establish a subadditivity inequality similar to Lemma 3.2 with the aim of Lemma 4.2.
Lemma 4.3 Suppose that for some . Then there exists such that for .
for all . □
Now we give the proof of Theorem 1.1 with the aim of Lemma 4.1, Lemma 4.2 and Lemma 4.3. Since the proof of Theorem 1.1 is similar to that of Theorem 3.1, we only give the sketch of the proof.
in for , or
- (b)there exists a sequence such that
which contradicts Lemma 4.1. Hence (b) holds. Then it follows from Lemma 2.3 and Lemma 4.3 that g is the minimizer for the problem () (i.e., ) and the result of Theorem 1.1 holds. The proof is similar to that of Theorem 3.1, we omit the details.
If , we repeat the previous argument in the case (b) to obtain the result of Theorem 1.1. □
Proof of Theorem 1.2 Theorem 1.2 is an immediate result of Theorem 1.1. We can prove it with an argument similar to that of Theorem 3.2. Here, we omit the details of the proof. □
Similar to Lemma 4.1 and Lemma 4.3, we obtain the following two lemmas.
and are finite and continuous on ; moreover, for any , each minimizing sequence for the problem () or () is bounded in ;
for any .
Applying the above two lemmas and commutator estimates [, Lemma 2.5], we prove Theorem 1.3 and Theorem 1.4 by similar steps to those given in Section 4. Here we omit the details of Theorem 1.3 and Theorem 1.4.
The work was supported in part by Special Fund of Fundamental Scientific Research Business Expense for Higher School of Central Government (projects for young teachers, No. ZY20110226) and the National Natural Science Foundation of China (No. 41276020).
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