Solitary Waves of the Schrödinger Lattice System with Nonlinear Hopping
This paper is concerned with the nonlinear Schrödinger lattice with nonlinear hopping. Via variation approach and the Nehari manifold argument, we obtain two types of solution: periodic ground state and localized ground state. Moreover, we consider the convergence of periodic solutions to the solitary waves.
In the last decades, a great deal of attention has been paid to study the existence of solitary waves for the lattice systems. They play a role in lots of physical models, such as nonlinear waves in crystals and arrays of coupled optical waveguides. The discrete nonlinear Schrödinger lattice is one of the most famous models in mathematics and physics. The existence and properties of discrete breathers (periodic in time and spatially localized) in discrete nonlinear Schrödinger lattice have been considered in a number of studies. One can see [1–9] and references therein.
In the present paper, we consider a variant of the discrete nonlinear Schrödinger lattice as follows: where , , and the nonlinear operator is defined by Here, denotes the set of the nearest neighbors of the point .
There has been a lot of interest in this equation as the modeling of waveguide arrays. Also, nonlinear hopping terms appear from Klein-Gordon and Fermi-Pasta-Ulam chains of anharmonic oscillators coupled with anharmonic intersite potentials or mixed FPU/KG chains. The generalized DNLS system with the nonlinear hopping terms has been derived as a perturbation of the integrable Ablowitz-Ladik system, by the rotating wave approximation on the FPU chain.
Karachalios et al. discuss the energy thresholds in the setting of DNLS lattice with nonlinear hopping terms by using fixed point method. The numerical results have also been obtained in their paper . However, the Dirichlet boundary condition in Section 2 of their paper is not suitable. Since , we have from the equation. With similar argument, the solution of lattice system under the Dirichlet boundary conditions is trivial. Our aim is to investigate the existence of nontrivial solitary waves for the infinite dimensional lattice (1). We firstly consider the -periodic problem. In the paper of Karachalios et al., they “expect that the variational approach can be applied in the case of periodic boundary conditions, but the details have to be checked.” We obtain the nontrivial periodic solution by Nehari manifolds argument . In Section 3, we obtain the solitary waves for the infinite dimensional lattice (1) by the concentration compactness method .
Here, we only consider the case of one dimension. That is, . The case of is similar. It notes that, for classical nonlinear Schrödinger lattice, Weinstein discusses a connection among the dimensionality, the degree of the nonlinearity, and the existence of the excitation threshold . Weinstein prove that if the degree of the nonlinearity satisfies where is the dimension, then there exists a ground state for the total power which is greater than the excitation threshold and there is no ground state for the total power which is less than the excitation threshold. It is interesting that the power of solitary waves for the infinite dimensional lattice (1) always has a lower bound from our arguments.
The paper is organized as follows. In Section 2, we firstly consider the -periodic problem. Note that the dimension in space variable is finite. We obtain the nontrivial periodic solution by Nehari manifolds argument. The existence of solitary waves is more complex. In Section 3, we follow the idea of [13–16] to obtain the solitary waves. The key point is to show the norms of periodic ground state are bounded. It is based on the concentration compactness. In Section 4, we concern the convergence of periodic ground states to solitary waves.
2. Periodic Ground State
In this paper, we consider the following equation: where .
To obtain breather, we seek the solution The equation of is Actually, we give the proofs only in the focusing case with and . For the defocusing case with and , the argument is similar. Here, we omit the details.
In this section, we prove the existence of -periodic solution which satisfies where is an integer.
Let Consider the Banach space with norm: We mention that Denote that is natural inner product in .
Define the functional and Nehari manifold Then, the minimizer of the constrained variational problem is the nontrivial periodic solution of (5). We mention that the minimizer is called a periodic ground state.
Note that We want to obtain the periodic solution with prescribed frequency . With the Nehari manifold approach, we have one of the main results.
Theorem 1. Assume that the frequency and . There exists a positive -periodic ground state for (5).
Lemma 2. Under the assumptions of Theorem 1, the Nehari manifold is nonempty.
Proof. For and , define Then, It holds that for small enough.
Observe that Therefore, admits a unique zero point . This implies . It completes the proof.
Lemma 3. Under the assumptions of Theorem 1, for , the function has a unique critical point at , which is a global maximum.
Proof. For and , we get Then, It holds that for small enough.
Note that We can see that is the unique maximum point of . This implies the proof.
Assume that is the -periodic solution of (5); we have Therefore where is the unique positive solution of equation Observe that is independent of .
Thus, we get a lower bound of the power of the periodic solutions.
Theorem 4. The power of the periodic solution must be greater than .
Lemma 5. Under the assumptions of Theorem 1, is bounded below for all .
Lemma 6. Under the assumptions of Theorem 1, the minimizer of the constrained variational problem could be attained.
Proof. Assume that is a minimizing sequence. We can see that there exists a constant such that Thus, It holds that is bounded.
Note that is finite dimensional space. Passing to a subsequence, there exists such that in . Since the set is closed and the functional is continuous, we obtain that and .
By Lagrange multiplier method, there exists some constant such that Choose . Note that ; it holds that We have . It implies that is a nontrivial solution of (3).
Now, we prove that is positive. Observe that Since that is the nontrivial solution, then, there exists such that . It is obvious thatHence . We can assume that .
3. Localized Ground State
Here, we give some notations. Define the functional and Nehari manifold where is natural inner product in . Thus, we can see that the minimizer of the constrained variational problem is the nontrivial solitary waves of (5). We call this minimizer a localized ground state. Similar to Lemmas 2 and 3, the results are obtained by replacing , with , .
In this section, to obtain the localized ground state satisfying we follow the idea of . We want to pass to the limit as . The key point is the following result.
Lemma 7. Under the assumptions of Theorem 1, let be the -periodic solution. Therefore, the sequences and are bounded.
Proof. First, we concern the sequences which are bounded. From similar argument of Lemma 2, this holds that, for any given , there exists such . Since the sequences with finite support are dense in , therefore, there exists with finite support such that . It obtains that there exists such that . For large enough, we have . We can get such that for . This holds that . And is bounded.
Second, we prove that is uniformly bounded. Assume that is unbounded. Passing to a subsequence which is still denoted by itself, we have for . Let . One of the following should hold:(i) is vanishing; that is, ;(ii) is not vanishing; passing to a subsequence which is still denoted by itself, there exists and such that for all .
Now, we rule out case (i). This holds that Hence, Assume that where is a constant which is defined below.
Let be small enough such that Combine with (37), we have From the argument above, there exists a constant such that Hence, is uniformly bounded.
By Hölder’s inequality, we have Since is vanishing, we can see that It concludes that It contradicts with (40).
Let us rule out the nonvanishing case. By the discrete translation invariance, we can assume that . Since , there exists such that for all . It is obvious that , , and .
Since , then , as . On the other hand, we have It is a contradiction.
Theorem 8. Assume that the frequency and . There exists a positive localized ground state for (5).
Proof. Let be a periodic ground state. From Lemma 7, the sequence is bounded. Therefore, is either vanishing or nonvanishing. In the case of vanishing, we have , for . This holds that It is a contradiction.
Thus, the sequence is nonvanishing. By the discrete translation invariance, we assume that . There exists such that for all . It is obvious that and . Also, we obtain that is a nontrivial solution for (5) by pointwise limits. Now, we want to prove that is a localized ground state.
Let be a positive integer such that Let ; it obtains that For any given , let such that Choose such that From density argument, there exists a finite supported sequence sufficiently close to in such thatThus, there exists such that and .
Choose large enough such that contains the support of . Let such that for . It concludes that It implies Combining with (48), we have . It completes the proof.
4. Global Convergence
Theorem 10. Let be the periodic ground state to (5). Then, there exists a ground state such that is strongly convergent to in after some discrete translation.
Proof. Let be the periodic ground state and . Now, we consider a translation From the argument above, we can assume that for all where is a ground state. We want to prove that is convergent to as . First, it concludes that Indeed,Similar to the argument in , it obtains that and , as .
Since and are bounded. For any given , there exists such that . Therefore, we have for large enough.
Also, we havefor large enough.
On the other hand, from the point limits, we have that for large enough.
Combine with Hölder inequality, holds.
With similar argument, we obtain . Therefore, Since , we have . From Lemma 7, it is known that for . Hence, It completes the proof.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author sincerely thanks Professor Yong Li for many useful suggestions. The author also thanks the referees for their comments that improved this paper. This work was supported by NSF of China (NSFC) Grant no. 11401250.
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