Table of Contents Author Guidelines Submit a Manuscript
The Scientific World Journal
Volume 2014 (2014), Article ID 214821, 11 pages
http://dx.doi.org/10.1155/2014/214821
Review Article

Recent Progress on Nonlinear Schrödinger Systems with Quadratic Interactions

1Department of Mathematics, College of Science, Yanbian University, No.977 Gongyuan Road, Yanji, Jilin 133002, China
2Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan

Received 20 January 2014; Accepted 2 March 2014; Published 31 March 2014

Academic Editors: D. Baleanu and C. M. Khalique

Copyright © 2014 Chunhua Li and Nakao Hayashi. 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.

Abstract

The study of nonlinear Schrödinger systems with quadratic interactions has attracted much attention in the recent years. In this paper, we summarize time decay estimates of small solutions to the systems under the mass resonance condition in 2-dimensional space. We show the existence of wave operators and modified wave operators of the systems under some mass conditions in -dimensional space, where . The existence of scattering operators and finite time blow-up of the solutions for the systems in higher space dimensions is also shown.

1. Introduction

In this paper we survey recent progress on asymptotic behavior of solutions to nonlinear Schrödinger system, based on papers [19], where , is the complex conjugate of , is a mass of particle, and nonlinearities have the form with

Nonlinear Schrödinger systems with quadratic interactions are physically important subjects (see, e.g., [10] and references cited therein). The quadratic nonlinearities of nonlinear Schrödinger systems in two space dimensions are interesting mathematical problems since they are regarded as the borderline between short range and long range interactions. In this case, asymptotic behavior of solutions to nonlinear systems is different from that to linear systems under mass resonance conditions and is the same as that to linear systems under mass nonresonance conditions. Namely, it is impossible to find solutions of nonlinear systems in the neighborhood of those of linear systems under mass resonance conditions.

If , then we have a single nonlinear Schrödinger equation: There are a lot of works on this subject since the work by Ginibre and Velo [11] which is considered a milestone of the field. We refer the text book by Cazenave [12] concerning the development on studies of (4) for details.

It is interesting to compare (1) with the system of nonlinear Klein-Gordon equations in for , under the gauge invariant condition for any , where is the speed of light. If we let in (5), then by the condition (6) we find that satisfies with for . Therefore nonrelativistic version of (5) can be obtained by letting in (7) formally, which is (1). The first breakthrough on asymptotic behavior of solutions to (5) with was made by Klainerman [13] and Shatah [14] independently when . Their result was improved by a paper [15].

It is natural to require the conservation law of solutions to (1) from the point of view of quantum mechanics. A sufficient condition is where for ; then we have conservation law and as a result global existence in time of solutions to (1) is obtained by combining the conserved identity and the Strichartz estimate for (see [16] in which a single equation was considered and the proof used in [16] works for the system).

We now introduce some function spaces to present exact statements of our results. For any and , weighted Sobolev space is defined by where We write and for simplicity. denotes the usual Lebesgue space with the norm if and By we denote the homogeneous Besov space with the seminorm where , , ,  , , and is the largest integer less than . It is known that (see [17]). We let be the space of continuous functions from an interval to a Banach space . Different positive constants might be denoted by the same letter . The homogeneous Sobolev spaces and are defined by respectively. We define the dilation operator by for and and define and for .

Evolution operator is written as where , are the Fourier transform and the inverse Fourier transform, respectively. We also have The operator , where and is an important tool to study time decay of solutions to nonlinear Schrödinger equations satisfying the gauge invariant condition (6) since it acts as a differential operator. Fractional power of is defined as which is also represented as (see [18]) for . Moreover, for , we have commutation relations such that

Remark 1. The system (1) includes some important nonlinear Schrödinger systems from the physical point of view. For example, the following system appears in a physical model (see e.g., [10, 19]): in , where is a mass of particle for . In [10], the system (21) has been derived as a model describing nonlinear interactions between a laser beam and a plasma. In [19], the stability of solitary waves for the system (21) was investigated.

This paper is organized as follows. Section 2 is devoted to present our recent works and some remarks. From Section 2.1 to Section 2.5, we consider the asymptotic behavior of solutions to nonlinear Schrödinger systems. In Section 2.1, we survey the results on the time decay estimates of solutions to nonlinear Schrödinger systems for shown in [47, 9]. In Section 2.2, wave operators of nonlinear Schrödinger systems are investigated for based on a paper [1]. Section 2.3 is concerned with the study of modified wave operators for from a paper [1]. In the last two subsections, we survey the results in [2, 3]. In the last section we consider the related and open problems.

2. Nonlinear Schrödinger Systems

2.1. Time Decay of Solutions to Nonlinear Schrödinger Systems in Two Space Dimensions

To state time decay of solutions to nonlinear Schrödinger systems for , we start with time decay estimates of solutions to linear Schrödinger systems.

For (1), the corresponding linear system is written as where .

As we know the solution of (22) is represented as . It is known that is decomposed into a main term and a remainder one as for , where decays rapidly in time; indeed we have the estimate for . By and time decay estimate for , we have the following time decay estimates through the interpolation theorem (see [12]).

Theorem 2. Let , and let be conjugate indices, . Then we have which are bounded operators and satisfy for .

Now we consider a special type of the system (1), in , where and are the masses of particles and . If we let and in the above system, then we obtain the system as below in , where . Therefore we survey the results on time decay of solutions to the system (28). The first result was obtained in [4].

Theorem 3 (see [4]). Assume that and . Then there exists such that (28) with the initial data has a unique global solution for any satisfying Moreover the time decay estimate is true for all .

In Theorem 3, the main result is time decay estimates of solutions of (28) and which is the same rate as that of the corresponding free solutions.

When , (28) satisfies the condition (8). Under the condition, , (28) obeys the conservation law such that

In the case of the mass resonance condition , (28) satisfies the condition (6). Global existence of small solutions for (28) is obtained from the conservation law and the Strichartz estimate. time decay of small solutions for (28) is proved through a priori estimates of local solutions in the norm . The similar idea has been used for construction of solutions to a single nonlinear Schrödinger equation by a paper [20]. Theorem 3 extends this idea to (28). The main point in the proof of the result is to derive the ordinary differential equation under the condition by using the factorization formulas of Schrödinger evolution group stated in Section 1, where for and . Asymptotic behavior in time of solutions of (28) is determined by that of the ordinary differential equations (34). The main task is to show that remainder terms are estimated from above by which is integrable in time. This is the reason why we use the condition such that the data must be in .

The system (1) is a generalization of (28). For the system (1), we have global existence theorem and time decay estimates as follows.

Theorem 4 (see [5]). One assumes that and satisfies the conditions (6) and (8) for each . Then there exists such that (1) has a unique global solution for any satisfying Moreover the time decay estimate is true for all .

Theorem 4 was improved in [6] by replacing the condition such that by with .

Now we focus on the following system: in , where are the masses of particles and , are constants.

Time decay problem of solutions to (38) is considered in [7]. By using the similar method as [4, 5], we have global existence in time and time decay estimates of small solutions for (38) as below.

Theorem 5 (see [7]). Assume that the mass resonance condition is satisfied. One also assumes that for and with some . Then there exists such that (38) has a unique global solution for any satisfying where . Moreover, the time decay estimate is true for all .

If the nonlinear term acts as a dissipation one which requires logarithmic correction in time of solutions and the negative time is not considered. We have the following theorem.

Theorem 6 (see [7]). Suppose that the assumptions of Theorem 5 are fulfilled. Let be the solution to the system (38) constructed in Theorem 5. If is satisfied, then the time decay estimate is true for all .

This phenomenon was found in [21] for a single equation. We note here that the method presented in [7] is different from the one in [21]. It seems that the proof in [21] does not work for the system.

Define the scaled function by ; then satisfies the system (1) with the initial data . We have which implies that is one of the so-called invariant spaces for the problem (1). Other invariant spaces of (1) are given by . We consider the large time asymptotics of solutions to the system (1) for in the function space , where satisfy and can be taken to be close to . Therefore our function space has relation with the invariant space and the considered data are not necessarily in .

Theorem 7 (see [9]). Assume that (6) and (8) hold. One also assumes that , where . Then there exists such that (1) has a unique global solution such that for any satisfying Moreover, the time decay estimate is true for .

We note that in [8] we consider the problem (38) under the initial condition and the dissipation condition , where . Then we have the time decay estimate such that for all .

In the final part of this subsection, we will show that the assumption (8) is important for obtaining the time decay estimates of solutions. Let us consider the following system: in , where are the masses of particles. It is obvious that the nonlinearities of (50) do not satisfy the assumption (8). Since the first equation of this system with the initial condition can be considered the Cauchy problem for the linear Schrödinger equation, we find the value of explicitly by . Therefore, we have For the system (51), we obtain the following result.

Proposition 8 (see [4]). Suppose that ,  . Let be a global solution of (51). Then the following estimate is true: for .

This fact was pointed first in [22, 23] in the case of Klein-Gordon equations and in Remark 3 of [24] in the case of Schrödinger equations.

2.2. Wave Operators of Nonlinear Schrödinger Systems in Two Space Dimensions

First, we briefly explain the definition of the wave operator (see [25]). For a given function , we assume that there exists a unique solution of the system (1) satisfying the asymptotics where is the solution of linear problems with the initial data and is the norm of Banach space . Then we define the map and call it the wave operator. We also call the final state (or the final value) since it is considered the value of at infinity if is the unitary operator in . The same problem can be considered for negative time.

To study existence of wave operators for (28), we consider the following problem for given final data in . If there exists a nontrivial solution for the above system, then we say that there exists a usual wave operator.

We consider (28) under the mass nonresonance conditions and .

Theorem 9 (see [1]). Let and . Then there exists such that, for any with the norm the system (28) has a unique global solution Moreover, the following estimate holds for all , where .

The mass nonresonance conditions and are used to obtain better time decay of solutions. Oscillating properties of nonlinear terms and are different from those of solutions to linear problem which yield an additional time decay from nonlinear terms; namely, nonlinear interactions are not critical. By combining this fact and the Strichartz type estimates, the result of Theorem 9 is obtained.

We next consider (28) under the mass condition which is also the mass nonresonance case and the support conditions on the data.

Theorem 10 (see [1]). Let . Assume that Then there exists such that, for any with the norm there exists a unique solution for the system (28) satisfying the estimate for all , where .

From the result we have wave operators when the support of the Fourier transform of the Schrödinger data is restricted. Restriction on the support of the Fourier transform of the Schrödinger data was used to obtain an improved time decay estimate of the nonlinear term .

We turn to investigate existence of wave operators for (1). First, we give a necessary condition of existence of asymptotically free solutions.

Theorem 11 (see [5]). Let and let be global in time of solutions of (1) satisfying a priori estimates We assume that the gauge condition (6) holds for each . If there exists such that then for every , where .

If the support condition is satisfied, we have (66).

We give existence of wave operators of the system (1) for small final states by (66).

Theorem 12 (see [5]). Let satisfy the so-called support condition (66). Assume that satisfies the gauge condition (6) for each . Then for some there exists a unique global solution of the system (1) such that for large and any satisfying

Existence of wave operator for a single nonlinear Schrödinger equation was studied in [26, 27].

2.3. Modified Wave Operators of Nonlinear Schrödinger Systems in Two Space Dimensions

In Section 2.2, we discuss the existence of wave operators of the systems (1) and (28). However, if it is impossible to show existence of the wave operator, we have to modify the setting of the problem. We define the modified wave operator (see [25]) as follows. Let us construct a function from a suitable function space and define a function by . Then we try to find a unique solution of nonlinear problems under the asymptotic condition where is the norm of Banach space . Namely, the problem is solved if we can define the function satisfying the asymptotic condition (70) by taking the structure of nonlinear terms into consideration. If we have a positive answer, we can define the map instead of the wave operator. We call the modified wave operator since we modified the final states.

We consider (28) again which is written as in . In Section 2.1, we stated the time decay estimates of solutions to this system in the case of and . Since the nonlinearity is critical in this case, it is impossible to find a solution in the neighborhood of the free final state . Indeed we have the nonexistence of the usual scattering states.

Theorem 13 (see [4]). Let , , and let be a global solution obtained in Theorem 3. Then there does not exist any nontrivial scattering state such that and as .

From the result, we need to modify the final state with time dependence. We note here that the modified wave operator for nonlinear dispersive equation was first constructed in [28] for the cubic nonlinear Schrödinger equations in one space dimension and then constructed in [29] for the derivative nonlinear Schrödinger equation, by changing it via a suitable transformation (see [30]) to a system of cubic nonlinear Schrödinger equations without derivatives of unknown function; see also [31] for recent developments. Two-dimensional case was studied in [32].

In Section 2.2, from (34), we see that the asymptotic behavior of solutions of (28) under the mass resonance condition is determined by the solutions of the following system:

By calculation we find that the particular solutions of (74) are in the case of and the particular solutions of (74) are in the case of , where and is a real valued given function. We also find that are particular solutions of (74) when .

The following theorem shows existence of the modified wave operators of (28).

Theorem 14 (see [1]). Let and . Then there exists such that, for any with norm , (28) has a unique global solution Moreover, the following estimate holds for all , where .

Using the resonance condition, , we get the existence of modified wave operators by the contraction mapping principle. Since the identity is known for , we have the estimate from the above theorem for all , where .

Existence of modified wave operator for a single nonlinear Schrödinger equation was studied in [26, 27]. Asymptotic behavior of solutions to nonlinear wave systems was studied in [33]. It was shown that the asymptotic behavior of solutions of them depends on the corresponding ordinary differential equations which are related to (74). In [33], another special solution was presented and the method can be applicable to nonlinear Schrödinger systems with a slight modification.

2.4. Nonlinear Schrödinger Systems in Higher Space Dimensions I

In the case of higher space dimensions, , the scattering theory for (28) was studied in [2].

We will explain the scattering problem (See [25]) briefly. We may assume the existence of wave operator which maps a Banach space into itself. Namely, for any given , we assume that there exists a unique solution of the nonlinear system such that We consider the initial value problem with the data which are determined by the solution in the time interval . If the initial value problem has a unique global solution and we can find a unique from the solution satisfying then we can define the inverse wave operator . From this operator we can define . We call the operator the scattering operator.

In the case, , existence of the scattering operator was proved in the space which is close to the invariant space . In the case of , we have the results in the invariant space . In the case of , existence of the scattering operator was proved in the space , under the mass resonance condition , which is close to the invariant space . To state the following theorem, we introduce

Theorem 15 (see [2]). Let . Then there exist and such that with the following property.
For any with and any , (28) has a unique global solution . Moreover, there exist unique such that as .
For any with and any , (28) has a unique global solution such that , as .
For any with and any , (28) has a unique solution such that , as .

Corollary 16 (see [2]). The wave operators are defined as mappings from to   for any with . The scattering operator is defined as a mapping from to for any with .To state the following theorem, we introduce

Theorem 17 (see [2]). Let  . Then there exist and such that with the following property.
For any with and any , (28) has a unique solution with . Moreover, there exist unique such that as .
For any with and any , (28) has a unique solution with such that , as .
For any with and any , (28) has a unique solution with such that , as .

Corollary 18 (see [2]). The wave operators are defined as mappings from to   for any with . The scattering operator is defined as a mapping from to for any with .

2.5. Nonlinear Schrödinger Systems in Higher Space Dimensions II

In [3], the finite time blow-up of the negative energy solutions for the system (28) was discussed in the case of under the mass conditions and . To state the blow-up result we need local existence in time of solutions to (28).

Theorem 19 (see [3]). Let and . Then, for any there exists such that (28) has a unique solution with

From Theorem 19 we have the energy conservation law such that where We need the virial identity to prove the blow-up result.

Theorem 20 (see [3]). Let and . Let and let be the local solution constructed in Theorem 19. Then for all , where

By a standard argument, we have the following result

Theorem 21 (see [3]). Let . Let and . Let and be as in Theorem 20. Then the maximal existence time for is finite in the following cases: where and are as in Theorem 20.

3. Related and Open Problems

Asymptotic behavior in time of solutions to (28) is an open problem for one space dimension which is considered the subcritical case. Existence of scattering operator is also an open problem for . We now turn to the relativistic version of (28) We let in (100); then where . Asymptotic behavior of solutions to (101) when was studied in papers [22, 34, 35] with mass nonresonance condition . Scattering operator was constructed in a paper [36] if . However existence of scattering operator for (28) is an open problem even if . Global existence and time decay of small solutions were obtained in [37] for the resonance case , under some regularity and compactness conditions on the initial data. However the large time asymptotics and existence of modified scattering operators are not known for the case. The asymptotic behavior of solutions to (101) in one space dimension is also an open problem.

Small data blow-up for a system of nonlinear Schrodinger equations was studied in [38] under some conditions on nonlinearities, but it is an open problem for (28).

The asymptotic behavior of solutions to quadratic derivative nonlinear Schrödinger systems has been considered recently in papers [39, 40] under some structural conditions on the nonlinearity. If the structural conditions are not satisfied, the problem is open.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

  1. N. Hayashi, C. Li, and P. I. Naumkin, “Modified wave operator for a system of nonlinear Schrödinger equations in 2D,” Communications in Partial Differential Equations, vol. 37, no. 6, pp. 947–968, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  2. N. Hayashi, C. Li, and T. Ozawa, “Small data scattering for a system of nonlinear Schrödinger equations,” Differential Equations & Applications, vol. 3, no. 3, pp. 415–426, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  3. N. Hayashi, T. Ozawa, and K. Tanaka, “On a system of nonlinear Schrödinger equations with quadratic interaction,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 30, no. 4, pp. 661–690, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  4. N. Hayashi, C. Li, and P. I. Naumkin, “On a system of nonlinear Schrödinger equations in 2D,” Differential and Integral Equations, vol. 24, no. 5-6, pp. 417–434, 2011. View at Google Scholar · View at MathSciNet
  5. C. Li, “Decay of solutions for a system of nonlinear Schrödinger equations in 2D,” Discrete and Continuous Dynamical Systems A, vol. 32, no. 12, pp. 4265–4285, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  6. C. Li, “Further decay results on the system of NLS equations in lower order Sobolev spaces,” in Proceedings of the 4th MSJ-SI Conference on Nonlinear Dynamics in Partial Differential Equations, Kyushu University, September 2011.
  7. S. Katayama, C. Li, and H. Sunagawa, “A remark on decay rates of solutions for a system of quadratic nonlinear Schrödinger equations in 2D,” Differential and Integral Equations, vol. 27, no. 3-4, pp. 301–312, 2014. View at Google Scholar · View at MathSciNet
  8. C. Li and N. Hayashi, “Critical nonlinear Schrödinger equations with data in homogeneous weighted L2 spaces,” Preprint.
  9. C. Li, “On a system of quadratic nonlinear Schrödinger equations and scale invariant spaces in 2D,” Preprint.
  10. M. Colin and T. Colin, “On a quasilinear Zakharov system describing laser-plasma interactions,” Differential and Integral Equations, vol. 17, no. 3-4, pp. 297–330, 2004. View at Google Scholar · View at MathSciNet
  11. J. Ginibre and G. Velo, “On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case ; II Scattering theory, general case,” Journal of Functional Analysis, vol. 32, no. 1, pp. 1–71, 1979. View at Google Scholar · View at Scopus
  12. T. Cazenave, Semilinear Schrödinger Equations, vol. 10, Courant Institute of Mathematical Sciences, New York, NY, USA; American Mathematical Society, Providence, RI, USA, 2003. View at MathSciNet
  13. S. Klainerman, “Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions,” Communications on Pure and Applied Mathematics, vol. 38, no. 5, pp. 631–641, 1985. View at Publisher · View at Google Scholar · View at MathSciNet
  14. J. Shatah, “Normal forms and quadratic nonlinear Klein-Gordon equations,” Communications on Pure and Applied Mathematics, vol. 38, no. 5, pp. 685–696, 1985. View at Publisher · View at Google Scholar · View at MathSciNet
  15. N. Hayashi and P. I. Naumkin, “Scattering operator for nonlinear Klein-Gordon equations in higher space dimensions,” Journal of Differential Equations, vol. 244, no. 1, pp. 188–199, 2008. View at Publisher · View at Google Scholar · View at Scopus
  16. Y. Tsutsumi, “L2-solutions for nonlinear Schrödinger equations and nonlinear groups,” Funkcialaj Ekvacioj, vol. 30, no. 1, pp. 115–125, 1987. View at Google Scholar · View at MathSciNet
  17. J. Bergh and J. Löfström, Interpolation Spaces, Springer, New York, NY, USA, 1976.
  18. N. Hayashi and T. Ozawa, “Scattering theory in the weighted L2(𝐑n) spaces for some Schrödinger equations,” Annales de l'Institut Henri Poincaré. Physique Théorique, vol. 48, no. 1, pp. 17–37, 1988. View at Google Scholar · View at MathSciNet
  19. M. Colin, T. Colin, and M. Ohta, “Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 26, no. 6, pp. 2211–2226, 2009. View at Publisher · View at Google Scholar · View at Scopus
  20. N. Hayashi and P. I. Naumkin, “Asymptotics in large time of solutions to the nonlinear Schrödinger and Hartree equations,” American Journal of Mathematics, vol. 120, no. 2, pp. 369–389, 1998. View at Google Scholar · View at Scopus
  21. A. Shimomura, “Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities,” Communications in Partial Differential Equations, vol. 31, no. 9, pp. 1407–1423, 2006. View at Publisher · View at Google Scholar · View at Scopus
  22. H. Sunagawa, “A note on the large time asymptotics for a system of Klein-Gordon equations,” Hokkaido Mathematical Journal, vol. 33, no. 2, pp. 457–472, 2004. View at Google Scholar · View at MathSciNet
  23. H. Sunagawa, “Large time asymptotics of solutions to nonlinear Klein-Gordon systems,” Osaka Journal of Mathematics, vol. 42, no. 1, pp. 65–83, 2005. View at Google Scholar · View at Scopus
  24. Y. Kawahara and H. Sunagawa, “Remarks on global behavior of solutions to nonlinear Schrödinger equations,” Proceedings of the Japan Academy A: Mathematical Sciences, vol. 82, no. 8, pp. 117–122, 2006. View at Publisher · View at Google Scholar · View at Scopus
  25. N. Hayashi, “Asymptotics of nonlinear dispersive-type evolution equations,” American Mathematical Society Translations, vol. 2, no. 230, pp. 21–43, 2010. View at Google Scholar
  26. N. Hayashi, P. I. Naumkin, A. Shimomura, and S. Tonegawa, “Modified wave operators for nonlinear Schrödinger equations in one and two dimensions,” Electronic Journal of Differential Equations, vol. 2004, no. 62, pp. 1–16, 2004. View at Google Scholar · View at Scopus
  27. N. Hayashi, H. Wang, and P. I. Naumkin, “Modified wave operators for nonlinear Schrödinger equations in lower order Sobolev spaces,” Journal of Hyperbolic Differential Equations, vol. 8, no. 4, pp. 759–775, 2011. View at Publisher · View at Google Scholar · View at Scopus
  28. T. Ozawa, “Long range scattering for nonlinear Schrödinger equations in one space dimension,” Communications in Mathematical Physics, vol. 139, no. 3, pp. 479–493, 1991. View at Publisher · View at Google Scholar · View at Scopus
  29. N. Hayashi and T. Ozawa, “Modified wave operators for the derivative nonlinear Schrödinger equation,” Mathematische Annalen, vol. 298, no. 1, pp. 557–576, 1994. View at Publisher · View at Google Scholar · View at Scopus
  30. N. Hayashi, “The initial value problem for the derivative nonlinear Schrödinger equation in the energy space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 20, no. 7, pp. 823–833, 1993. View at Google Scholar · View at Scopus
  31. Z. Guo, N. Hayashi, Y. Lin, and P. I. Naumkin, “Modified scattering operator for the derivative nonlinear Schrödinger equation,” SIAM Journal on Mathematical Analysis, vol. 45, no. 6, pp. 3854–3871, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  32. J. Ginibre and T. Ozawa, “Long range scattering for non-linear Schrödinger and Hartree equations in space dimension n ≥ 2,” Communications in Mathematical Physics, vol. 151, no. 3, pp. 619–645, 1993. View at Publisher · View at Google Scholar · View at Scopus
  33. S. Katayama, T. Matoba, and H. Sunagawa, “A semilinear hyperbolic system violating the null condition,” http://arxiv.org/abs/1206.0066.
  34. J.-M. Delort, D. Fang, and R. Xue, “Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions,” Journal of Functional Analysis, vol. 211, no. 2, pp. 288–323, 2004. View at Publisher · View at Google Scholar · View at Scopus
  35. H. Sunagawa, “On global small amplitude solutions to systems of cubic nonlinear Klein-Gordon equations with different mass terms in one space dimension,” Journal of Differential Equations, vol. 192, no. 2, pp. 308–325, 2003. View at Publisher · View at Google Scholar · View at Scopus
  36. N. Hayashi and P. I. Naumkin, “A system of quadratic nonlinear Klein-Gordon equations in 2D,” Journal of Differential Equations, vol. 254, no. 8, pp. 3615–3646, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  37. Y. Kawahara and H. Sunagawa, “Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance,” Journal of Differential Equations, vol. 251, no. 9, pp. 2549–2567, 2011. View at Publisher · View at Google Scholar · View at Scopus
  38. T. Ozawa and H. Sunagawa, “Small data blow-up for a system of nonlinear Schrödinger equations,” Journal of Mathematical Analysis and Applications, vol. 399, no. 1, pp. 147–155, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  39. H. Hirayama, “Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data,” http://arxiv-web3.library.cornell.edu/abs/1311.6102.
  40. M. Ikeda, S. Katayama, and H. Sunagawa, “Null structure in a system of quadratic derivative nonlinear Schrödinger equations,” http://arxiv.org/abs/1305.3662.