It is well known that the form of the coupled nonlinear Korteweg-de Vries equations is as follows
where r is a real constant, are real-valued functions of x and t, is a coupling parameter and P, Q satisfy for a small function H. Model represents the physical problem of describing the strong interaction of two-dimensional long internal gravity waves propagating on neighboring pycnoclines in a stratified fluid. In this paper we consider a special case of (1.1), namely the following system of nonlinear evolution equations:
We always look for solutions of (1.2) of the form
where . Through calculating, if we get
Now we consider it in higher dimensional cases, as follows:
where , and , is a coupling parameter.
The system of (1.1) has been analysed many times. For example, see the recently derived model by Gear and Grimshaw , considering
where are constants. Moreover, the system (1.2) has been extensively studied in recent years and is also a special case of a general class of nonlinear evolution equations considered in  in the inverse scattering context. More properties of the system (1.2) have been proved. Alarcon and Montenegro proved the local and global well-posedness results for the initial-value problem for (1.2) with in  and . Panthee and Scialom improved the well-posedness results obtained in the case when in .
As we know, many analyses about higher order equation have been done many years ago including the third and fifth order KdV equation. Firstly, it is already well known that the third order KdV equation describes the evolution of weakly nonlinear and weakly dispersive shallow waves in physical contexts such as plasma, ion-acoustic waves, stratified internal, and atmospheric waves and it has been analysed during the last decades. For the fifth order equation, the results are less than the third. But it has attracted increasing attentions (see  -  ) and is used to model many physical phenomena such as gravity-capillary waves on a shallow layer and magnetosome propagation in plasmas. For example, Baker took the work and published in 1903; Li Xiaofeng proved the existence of solitary wave solutions of fifth order KdV equations in recent years. Santosh Bhattarai proved the existence of travelling-wave solutions of coupled KdV equations when it loses the compactness, using the method of concentrate compactness principle of Lions in 2015.
We know that the system of higher order equations is rare. We only can find other similar fourth-order systems studying the interaction of the long and short waves have appeared. P. Lvarez-Caudevilla and E. Colorado researched the coupled nonlinear Schrodinger Equations (1.7) and the system of Schrodinger and Korteweg-de Vries Equations (1.8).
They proved the existence of equations using the variation approach and minimization techniques on Nehari manifold and the multiplicity of the equations by fibering map.
But we know that there is not previous mathematics work analyzing a higher order system as (1.3) and we get the multiplicity of the equations by a bifurcation theory which is not founded in other higher order equations article.
We organize the paper as follows. In Section 2, we introduce the notation, establish the functional framework, define the Nehari manifold and give the main theorem. In Section 3, we construct semi-trivial solutions and show the properties depending on the coupling parameter. Moreover, we devoted to proving the main results of the paper by the variation principle and mountain-pass theorem. In Section 4, using the Crandall-Rabinowitz local bifurcation theory, we show the multiplicity of the ground state solutions.
2. Preliminaries and Main Theorems
In , we define the following equivalent norm and scalar product:
Accordingly, the inner product and induced norm on are given by
We define the radially symmetric functions in and . In addition, we define the energy functional associated with system (1.5) by
are the energy functionals of the uncoupled equations. Then, we define
Now, we restrict the Nehari Manifold to the setting, denoting it as
Remark 2.1. (see    )
Then we have the following Sobolev embedding:
Proposition 2.1. We are going to prove some properties for and on .
1) is a locally smooth manifold.
2) is a complete metric space.
3) is a critical point of if and only if is a critical point of constrained on .
4) is bounded from below on .
Proof. 1) Differentiating expression (2.4) yields
and because of , we have the fact that .
Then, we obtain
Then, is a locally smooth manifold near any point with .
2) Let be a sequence such that as . By the embedding theorem, we have and for . It is clear that
Since and , applying Holder's inequality, we get
So we have . Because of , we get . Using and , we get . Hence and is a complete metric space.
Taking the derivative of the functional in the direction , we find
The second derivative of is given by
So, we have
which is positive definite so that is a strict minimum critical point of . As a consequence, we have that is a smooth complete manifold, and there exists a constant such that
3) Assume that is a critical point of and with , then there is a Lagrange multiplier such that
Apply both sides to and we can get
Combining (2.6) and (2.11), we get . Now (2.10) gives , i.e. , is a critical point of .
4) The functional constrained on takes the form combining (2.3) and (2.4)
using (2.9) and (2.12), we can get
So, is bounded from below on . □
Lemma 2.1. For every , there is a number such that .
Proof. For and , we have
On the one hand, we have and for a small enough t. On the other hand, we have as . So there is a maximum point of t. Moreover we get and deduce . □
Lemma 2.2. Assume that , then satisfies the PS condition constrained on .
Proof. Suppose is a sequence i.e.
From (2.4) and (2.9), we can get is bounded, then we have a weakly convergent subsequence (for convenience denoted again by ). Since H is compactly embedded into for , we infer that
Moreover, using the fact that and (2.3), we have
which implies that . Letting
denotes the constrained gradient of on with . Taking the scalar product with and with we can get
then, taking into account (2.6) and (2.7), we deduce that as . We also have that is bounded, so with (2.13) and the fact as , we obtain
So we deduce that To finish the proof, since as , it follows that strongly. □
Theorem 2.1. Suppose . The infimum of on is attained at some with and both components .
1)Let be the principal eigenvalue of
and let be the corresponding positive eigenfunction. Then there exists such that when , (1.3) has solutions of the form
2). There exists such that when , (1.3) has solutions of the form
3. Existence Results of Semi-Trivial Solutions and Non-Trivial Solutions
System (1.5) admits two kinds of semi-trivial solutions of the form and . So we take and , where and are radially symmetric ground state solutions of the equation in . Moreover, if we denote w a radially symmetric ground state solution of (3.1)
then, by scaling, we can get
Hence, system (1.5) has two kinds of semi-trivial solutions and with lowest energy among all the semi-trivial solutions.
1) We define new Nehari manifold corresponding to the equations of (1.5)by
Let us define the tangent space to on and by
And define the tangent space to on and on by
We can prove the following equivalence:
If we denote by the second derivative of constrained on ,using that is a critical point of ,plainly we obtain that
2) We define the following Sobolev constants related to and ,
1) If then is a strict local minimum of constrained on .
2) If either ,then is a saddle point of constrained on . Moreover
Proof. 1) Suppose .
For one has that
For one thing, since and the definition of , there exists such that
For another thing, using (3.3) and the fact that is a minimum of on , there exists a constant such that
Hence, using (3.6) and (3.7) we get
proving that is a strict local minimum of on .
When , we can obtain the same result by using the same argument as above.
In this case, we choose an element , such that
Now, taking we get
And taking we get
Hence, is a saddle point of on .
When we can obtain the same result using the same argument as above and obviously inequality (3.5) holds. □
Next, we will give the proof of Theorem 2.1 and Theorem 2.2.
Proof. By the Ekelands variational principle , there exists a minimizing sequence , i.e.,
Due to the Lemma 2.2, there exists such that
so, is a minimum point of on . □
We have . Then there exists such that . So we get
we get . According to the definition of ,
with , we get is a nonnegative ground state solution of the system. We can conclude that both components of are non-trivial. In fact, if the second component , then . So is the non-trivial solution of the system (1.5), Hence, we have
However, this is a contradiction due to the fact that is a radial ground state solution of . We conclude the first component using the same way. Lastly, taking into account Proposition 3.1-(2) and we have
4. Bifurcation of Nontrivial Solutions
In this subsection, we prove the existence of nontrivial solution of (1.5) by using local bifurcation theory (see  ). The main results follow.
Proof. Consider the eigenvalue problem
It is well known that this problem admits a sequence of eigenvalues
Moreover, we infer from  that the first eigenvalue is simple and the principle eigenfunction is a positive function. Set , We shall consider the bifurcation of nontrivial solution of (1.5) from the semitrivial branch near . To accomplish this, we apply the bifurcation results of Crandall and Rabinowitz. First, we define F by
Clearly, for , one sees that
From (4.1) and (4.2), we get that the null space . The solution space of in is . Hence, the null space
is trivial. So the null space , and is the principal eigenfunction of (4.1). The range space of L is defined by
Thus, . Since , it follows from (5.6) that
Thus, we can apply the result of  to conclude that the set of positive solutions to (1.5) near is a smooth curve
such that , , , where is a small constant. Moreover, can be calculated as (see, for example,   )
where is a linear functional on defined as . Hence, we infer from (4.7) and (4.8) that for
Now, we give the proof of (2). As we know, is the unique positive solution of (1.5) with . Recalling the map defined in (4.3), we have
It is well known that and are both invertible; hence, is nondegenerate in X2r, i.e., exists. We infer from the implicit function theorem that there exists and such that for any , . Moreover, we can compute . Differentiating by at , because of , we get
This gives the expression of .
In the paper, we studied the positive radial solutions for a higher order coupled system of Korteweg-de Vries equations in Theorem 2.1. Moreover, we proved the multiplicity of the equations by a bifurcation theory in Theorem 2.2.
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