On the solvability of the discrete nonlinear Schrödinger equation with subcubic potential

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Abstract

In this paper, we analyze the solvability of the discrete nonlinear Schrödinger equation

$i \beta\left(\Delta_{t}+\nabla_{t}\right) \varphi(t, k)+\gamma|\varphi(t, k)|^{2} \varphi(t, k)+\varepsilon \Delta_{k}^{2} \varphi(t, k-1)=g(t, \varphi(t, k)),$

where ∆_t and ∆_k denote the standard forward difference operators in the variables t and k, respectively, ∇_t denotes the standard backward difference operator in t, and

$\Delta_{k}^{2} \varphi(t, k-1)=\varphi(t, k+1)-2 \varphi(t, k)+\varphi(t, k-1)$

is the discrete Laplacian operator in the spatial variable k. Throughout, we will assume the parameters β and ε are positive real numbers, the parameter γ is a nonzero real number, and the potential function g : $\mathbb{Z} \times \mathbb{C} \rightarrow \mathbb{C}$ is continuous.

Keywords

Schrödinger’s equation / superlinear potentials / subcubic potentials / discrete boundary value problems / periodic solutions

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References

[1] M. J. Ablowitz, J. F. Ladik, Nonlinear differential-difference equations, J. Math. Phys., 16 (1975), 598-603. https://doi.org/10.1063/1.522558

[2] D. Hennig, G. P. Tsironis, Wave transmission in nonlinear lattices, Phys. Rep., 307 (1999), 333-432. https://doi.org/10.1016/S0370-1573(98)00025-8

[3] P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives , Springer, 2009.

[4] A. Vuoksenmaa, Dynamics of the infinite discrete nonlinear Schrödinger equation, J. Stat. Phys., 191 (2024), 160. https://doi.org/10.1007/s10955-024-03374-w

[5] C. Brunhuber, F. G. Mertens, Y. Gaididei, Persistent Breathers in long-ranged discrete nonlinear Schrödinger models, Phys. Rev. E, 73 (2006), 016614. https://doi.org/10.1103/PhysRevE.73.056610

[6] S. Flach, A. Gorbach, Discrete Breathers-Advances in theory and applications, Phys. Rep., 467 (2008), 1-116. https://doi.org/10.1016/j.physrep.2008.05.002

[7] S. Flach, K. Kladko, R. S. MacKay, Energy thresholds for discrete breathers in one-, two-, and three-dimensional lattices, Phys. Rev. Lett., 78 (1997), 1207-1210. https://doi.org/10.1103/PhysRevLett.78.1207

[8] G. Zhang, Breather solutions of the discrete nonlinear Schrödinger equations with unbounded potentials, J. Math. Phys., 50 (2009), 013505. https://doi.org/10.1063/1.3036182

[9] S. M. Bak, Existence of standing waves in discrete nonlinear Schrödinger equations with saturable nonlinearity, Mat. Stud., 33 (2010), 78-84.

[10] L. Zhang, S. Ma, Ground state solutions for periodic discrete nonlinear Schrödinger equations with saturable nonlinearities, Adv. Differ. Equations, 2018 (2018), 157. https://doi.org/10.1186/s13662-018-1574-2

[11] D. Hennig, Discrete nonlinear Schrödinger equations with double power nonlinearities, J. Differ. Equations Appl., 31 (2025), 1343-1364. https://doi.org/10.1080/10236198.2025.2554129

[12] C. Bereanu, J. Mawhin, Periodic solutions of nonlinear difference equations involving the φ-Laplacian, J. Math. Anal. Appl., 330 (2007), 1002-1015. https://doi.org/10.1016/j.jmaa.2006.07.104

[13] D. Maroncelli, J. Rodríguez, On the solvability of multipoint boundary value problems for discrete systems at resonance, J. Differ. Equations Appl., 20 (2014), 24-35. https://doi.org/10.1080/10236198.2013.805216

[14] D. Maroncelli, J. Rodríguez, Periodic behaviour of nonlinear, second-order discrete dynamical systems, J. Differ. Equations Appl., 22 (2016), 280-294. https://doi.org/10.1080/10236198.2015.1083016

[15] D. Maroncelli, Nonlinear scalar multipoint boundary value problems at resonance, J. Differ. Equations Appl., 24 (2018), 1935-1952. https://doi.org/10.1080/10236198.2018.1551378

[16] N. Rouche, J. Mawhin, Ordinary Differential Equations: Stability and Periodic Solutions , Pitman, 1980.

[17] J. B. Conway, Functions of One Complex Variable I , 2nd edition, Graduate Texts in Mathematics, Springer-Verlag, New York, 1978.

[18] K. Deimling, Nonlinear Functional Analysis , Springer, 1985. https://doi.org/10.1007/978-3-662-00547-7