Research Article | Open Access
On the Travelling Waves for the Generalized Nonlinear Schrödinger Equation
This paper is devoted to the analysis of the travelling waves for a class of generalized nonlinear Schrödinger equations in a cylindric domain. Searching for travelling waves reduces the problem to the multiparameter eigenvalue problems for a class of perturbed p-Laplacians. We study dispersion relations between the eigenparameters, quantitative analysis of eigenfunctions and discuss some variational principles for eigenvalues of perturbed p-Laplacians. In this paper we analyze the Dirichlet, Neumann, No-flux, Robin and Steklov boundary value problems. Particularly, a “duality principle” between the Robin and the Steklov problems is presented.
The main concerns of the paper are the travelling waves for the generalized nonlinear Schrödinger (NLS) equation with the free initial condition in the following form (see  for generalized NLS): where and is a parameter. is a cylinder, is the lateral boundary of , , , and . Assume that is a bounded domain in with the smooth boundary. Particularly, in the case of we get which is a nonlinear Scrödinger equation (see ). On the other hand problem (1.1) can be considered as an evolution -Laplacian equation. Different aspects of such kind of problems, with some initial conditions, have been studied in . Thus problem (1.1) models the linear Schrödinger equations (), NLS (), evolution -Laplacians, and generalized NLS. This definitely means that we have a good motivation for problem (1.1).
In this paper by the travelling waves we mean the solutions of (1.1) in the form , where and is a real-valued function. A simple computation yields , , and , . Hence, and . By using the notation we obtain . Finally, by setting all of these into (1.1) we can obtain the following nonstandard multiparameter eigenvalue problems for perturbed -Laplacians: In what follows, by shifting the variables , we have used the following notations: , , and .
At this point we have to note that by searching for the standing waves for the NLS equation we obtain the following eigenvalue problem: On the other hand by setting the travelling wave solutions of the form into the NLS equation we obtain Thus we obtain the same type eigenvalue problems for both standing and travelling waves for the NLS equation (see  and references therein for standing waves for NLS). However, standing and travelling waves for generalized NLS are associated with quite different type of eigenvalue problems. Particularly, the eigenvalue problem associated to the travelling wave solutions is problem (1.3), which is clearly a nonstandard multiparameter eigenvalue problem in the nonlinear analysis, and we are not aware of any known result for this problem.
A solution of (1.3) is a weak solution, defined in the following way.
Definition 1.1. is a solution of (1.3) if and only if
holds for all , where is the Sobolev space (for Sobolev spaces, see ). In this case, we say that is an eigenfunction, corresponding to the eigenpair and , where is a frequency, is a wave number, and the parameter comes from the initial equation (1.1). We prefer to denote the test functions in (1.7) by , which is clearly different from the notation that is used in (1.1). Let us define We set . Then, is a solution of (1.7) if and only if is a free critical point for , that is, , where is the Fréchet derivative of , is the dual space, and denotes the value of the functional at . Indeed, the existence of Fréschet derivative implies the existence of directional (Gateaux) derivative. Using the definition of Gateaux derivative, we can obtain which is enough to see that (1.7) is the variational equation for the functional .
As , by Sobolev embedding theorems (see ) the functional can be well defined if(i) or(ii) and .
In the next section two cases and will be studied separately. If , then we may rewrite (1.7) in the form which is the equation for free critical points of the functional Evidently, there are not nontrivial solutions of (1.10) in the case of . Thus, , and by the scaling property, we obtain that if and (1.10) has a nontrivial solution for some , then it has nontrivial solutions for all .
In what follows, denotes the standard norm in and , which is equivalent to the norm .
This paper consists of an introduction (Section 1) and two sections. In Section 2 we study the structure of the eigenparameters , , and the eigenfunctions for problem (1.3), including the dispersion relations between , , and and variational principles in some special cases. We consider separately two cases: and. In the case of we have estimated bounds for the set of eigenfunctions, proved the existence of infinitely many eigenfunctions, corresponding to an eigenpair , , and demonstrated that, in the case of , the set is compact. For the general case we study the existence of positive solutions and variational principles in some special cases. The proofs are based on the Sobolev imbedding theorems, the Palais-Smale condition, variational techniques, and the Ljusternik-Schnirelman critical point theory. Various boundary problems and some relations between them are studied in Section 3.
2. The Structure of Eigenparameters , , and Related Eigenfunctions
As mentioned above the problem of the existence of travelling waves and a quantitative analysis for travelling waves is a multiparameter eigenvalue problem given by (1.7) or (1.10). This section is devoted to these problems, and the techniques we use in this section are partially close to that used in .
We study separately two cases: and.
Case 1 (). This subsection is devoted to the quantitative analysis of solutions of (1.10). We assume that(i) or (ii) and .
Our first observation for eigenvalue problem (1.10) is given in the following proposition.
Proposition 2.1. (a) Let . In this case, all the eigenpairs of problem (1.10) lie in the parabola , where is the first eigenvalue of in and . Moreover, for a fixed , there is a finite number of and for a fixed , there is a countable number of , such that as ,
(b) If , then for a fixed , the solutions of (1.10) are bounded and holds for some ,
(c) Let . In this case, one has
Proof. The proof easily follows from (1.10), by using the Courant-Weyl variational principle the bounded embedding , and the equivalence of the norms and .
In the following theorem, we have proved some properties of the functional in , which guarantee the existence of nontrivial solutions of (1.10) for a fixed , .
Theorem 2.2. Let and , . Then (i)for some .(ii) attains its infimum at a nontrivial vector .
Proof. One has . As , the Poincaré inequality yields . Hence,
Let . We have and as . This indicates that has a global minimum point, and clearly, this point is . Now, by putting the vectors with into , we obtain
Now, we will show that . Let be a vector such that . Subsequently, by setting in , we obtain , where . Thus if .
(ii) We have to show that is attained. Let . Then, there exists a sequence such that as . The sequence should be bounded, because and as , which means that is coercive. However, is a reflexive Banach space. Consequently, for some , where “” denotes the weak convergence in . Evidently, is sequentially lower semicontinuous, that is,
Indeed, , and it is known that the norm is sequentially lower semicontinuous. The second term of is , and this term is sequentially continuous, because the embedding is compact. Now, it follows from (2.8) that , which means . Finally, because by (i) and .
Corollary 2.3. In the case of , all pairs , , are eigenpairs of problem (1.10).
Proof. This immediately follows from (ii) of Theorem 2.2.
Now, we will prove that there are infinitely many solutions of (1.10) for all , . For this, our main component will be Proposition 2.5 (, p. 324 Proposition 44.18) about free critical points of a functional, that is, about the solutions of the operator equation
Definition 2.4. Let . satisfies the -condition at a point if each sequence , such that and in has a convergent subsequence.
Particularly, satisfies if and only if it satisfies the PS-condition for all .
Let us denote by the class of all compact, symmetric, and zero-free subsets of , such that . Here, is defined as the smallest natural number for which there exists an odd and continuous function . Let Suppose that is a real -space, is an even functional with , satisfies with respect to and .
As mentioned earlier, our main component will be the following proposition.
Proposition 2.5. If , , and hold and , then has a pair of critical points on such that , to which solutions of (2.9) correspond. Moreover, if , then , where .
Now, we are ready to prove the following theorem.
Theorem 2.6. Let . (a) For each , problem (1.10) has an infinite number of nontrivial solutions.
(b) The set is compact, where .
Proof. Our proof is based on the previous proposition. Clearly, conditions and are satisfied. The fact that satisfies is standard (see ). In our case satisfies -condition for all . It needs to be demonstrated that . By the definition of “”, it is adequate to show the existence of a set such that . We have
Let be an -dimensional subspace of and the unit sphere in . We can choose and define . As is compact, . Hence,
Moreover, and provided . Using this fact we obtain that for every there exist and such that for all . Clearly and . Now, set . Then . Consequently, . By Theorem 2.2 is bounded below. Hence,
and the statements of the theorem in (a) follow from Proposition 2.5.
(b) Let us prove that the set is compact. Let be a sequence. Then However, and by Theorem 2.2, is bounded below. Thus, is bounded, and consequently, it has a convergent subsequence (denoted again by ). We have and . Hence, by PS-condition, has a convergent subsequence. The limit points of belong to because, by PS-condition, the set is closed.
Case 2 (). We first look at the following problem: where and (2.16) holds for all . The main result is as follows.
Proof. (a) As a result of the compact imbedding and the fact that is a norm in , the functional is coercive and lower semicontinuous on the weakly closed set . From these properties, by using the condition we obtain the existence of a nonnegative solution. The positivity follows from the maximum principle.(b) This fact follows from (a) and the relation as .
Note 1. The case is the critical case: lack of compactness, which is a subject that deserves a separate study.
Now, our concern is the following typical eigenvalue problem:
Let us look at problem (2.18) with respect to for a fixed . This problem is a typical eigenvalue problem. If , then we get the -Laplacian eigenvalue problem, and these questions have been studied by many authors (see [8, 9] and the references therein). Particularly, it has been shown in  that there is a sequence of “variational eigenvalues” which can be described by the Ljusternik-Schnirelman type variational principles. Our aim is to get the similar results for perturbed -Laplacian eigenvalue problem (2.18). In our case , and we can apply two methods.
Method 1. For the Diriclet problem the norms: , which is the standard norm in , and are equivalent. Then it is enough to replace by the Banach space and follow the methods of [8, 9] to get the needed results.
Method 2. One can construct a Ljusternik-Schnirelman deformation (see [6, 7]) and check Palais-Smale condition for the functional on the manifold A such construction was given in our previous paper (see ). We follow our construction and just give the final result.
Theorem 2.8. For a fixed , there exists a sequence of eigenvalues of problem (2.18), depending on , which is given by where one denotes by the class of all compact, symmetric subsets of , such that .
3. On the Neumann, No-Flux, Robin, and Steklov Boundary Value Problems
At the end of the paper we briefly discuss the other boundary problems, such as Neumman, No-flux, Robin, and Steklov. We note that all of the above given results are related to problem (1.3) with the Diriclet boundary condition; however the similar results are valid for the following boundary conditions too:
Evidently, the energy space (the Banach space, we use in the critical point theory) for Dirichlet, Neuman, No-flux, Robin, and Steklov problems is , , , , and , respectively. In the case of and we obtain the standard eigenvalue problems for -Laplacians, which have been studied in detail in  for all of the above given boundary value problems. Many results for standard -Laplacians, including the regularity results, may be extended to the perturbed -Laplacians by the similar techniques that are used in . However, we omit these questions in this paper.
Our simple observation between Robin and Steklov problems is as follows: is an eigentriple for Steklov problem if and only if is an eigentriple for Robin problem at .
Finally, we use a similar connection between the typical Robin and Steklov eigenvalue problems to prove the existence of negative eigenvalues for the Robin problem. For sake of simplicity we choose and consider the following standard eigenvalue problems for -Laplacians: Problems (3.5) can be rewritten in the following variational forms: respectively.
It is known that (see )(I)if then the Robin problem has a sequence of positive eigenvalues such that as ;(II)the Steklov problem also has a sequence of positive eigenvalues such that as .
An Inverse Problem
Now let us be given . Our question is as follows: for what values of the given number will be an eigenvalue for Robin problem (3.6). To answer this question we use a “duality principle” between Robin and Steklov problems and give the final result in the following theorem.
Theorem 3.1. For a given there exists a sequence , such that the number will be an eigenvalue for the Robin problem at , . Moreover, and are the eigenvalues of the Steklov problem.
Proof. The proof is based on the relations between the Robin and Steklov problems. To answer this question we consider the Steklov problem in the form Then the variational problem (3.7) is replaced by By comparing (3.6) and (3.9) we obtain that is an eigenpair for the Steklov problem if and only if is an eigenpair for the Robin problem. We know that (see ) for a positive number Steklov problem (3.9) has a sequence of positive eigenvalues such that as . Thus , are eigenpairs for (3.9). Then it follows that , are eigenpairs for the Robin problem. To end the proof we notice that and .
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Copyright © 2011 Mahir Hasanov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.