Discrete Nonlinear Schrödinger Equation

• The discrete nonlinear Schrödinger equation is a lattice dynamical system with on-site nonlinearities and inter-site couplings that model dispersive and nonlinear phenomena.
• It supports coherent structures such as discrete solitons, traveling waves, and exhibits phenomena like the Peierls–Nabarro barrier affecting energy localization.
• Rigorous analyses of DNLS establish well-posedness, reveal thermal phase transitions, and highlight its relevance in optics, condensed matter physics, and Bose–Einstein condensates.

The discrete nonlinear Schrödinger equation (DNLS) is a fundamental class of lattice dynamical systems, modeling dispersive, nonlinear phenomena in a variety of settings. As a discrete analog of the nonlinear Schrödinger equation (NLS), DNLS features complex-valued degrees of freedom on spatial lattices with on-site nonlinearities and either local or nonlocal inter-site coupling. DNLS arises in nonlinear optics, condensed matter physics, Bose–Einstein condensates, and as a prototypical example of Hamiltonian lattice field theories. Its mathematical structure allows analysis of coherent structures, transport, thermodynamics, and integrable and non-integrable behavior.

1. Mathematical Formulations and Core Properties

The prototypical DNLS on a $d$-dimensional cubic lattice, with lattice spacing $h>0$, is

$$i\frac{d u_n}{dt} + h^{-2} (\Delta_d u)_n + \lambda |u_n|^2 u_n = 0, \qquad n \in h\mathbb{Z}^d$$

where $(\Delta_d u)_n = \sum_{|m-n|=1} (u_m - u_n)$ is the discrete Laplacian and $\lambda$ parameterizes the Kerr nonlinearity, with $\lambda > 0$ for defocusing and $\lambda < 0$ for focusing cases (Bernier et al., 2018, Jenkinson et al., 2014, Vuoksenmaa, 2023, Tsoy et al., 2019). The system is Hamiltonian, conserving mass $M[u]=\sum_n |u_n|^2$ and energy

$$H[u] = \frac{1}{h^2} \sum_{n \in h\mathbb{Z}^d} \sum_{j=1}^d |\delta_j u|^2_n - \frac{\lambda}{2} \sum_n |u_n|^4$$

where $(\delta_j u)_n = u_{n+e^{(j)}} - u_n$.

On finite one-dimensional lattices, periodic or open boundary conditions can be imposed. In the $h \to 0$ (continuous) limit, DNLS recovers the focusing or defocusing NLS, $i\partial_t u = \partial_x^2 u + \lambda |u|^2 u$.

Extensions include higher-order nonlinearities, general coupling range, point defects, and fractional Laplacian generalizations (long-range hopping) (Molina, 2020, Molina, 2019, Hennig, 13 Dec 2024).

2. Solitary Waves, Discrete Solitons, and Peierls–Nabarro Structure

DNLS supports a diverse range of coherent structures, including spatially localized standing and traveling waves ("discrete solitons"), periodic traveling waves, and multi-soliton constructs.

Standing and Traveling Solitons

For $d=1$, focusing nonlinearity generates solitary standing waves of the form $u_n(t) = e^{-i\omega t} g_n$ with $\omega < 0$, where $g_n$ satisfies a nonlinear eigenvalue problem (Jenkinson et al., 2014). These states can be on-site (vertex-centered) or off-site (bond-centered), with exponential localization in space, bifurcating from the continuum NLS soliton as the lattice parameter approaches zero.

Traveling wave solutions with nonzero velocity $c$ exist for DNLS in the vicinity of the continuum limit, constructed as

$$u_g(t) = \phi\left(g/h - c t; h\right) e^{i \omega t}$$

with the profile $\phi$ solving an advance–delay ODE. A full perturbative expansion in $h$ yields corrections to the continuum traveling wave. Rigorous analysis establishes existence and long-time stability of these solitary traveling waves (Bernier et al., 2018).

Peierls–Nabarro Barrier

Distinct soliton families exist (on-site, off-site, higher-dimensional cell-centered, etc.), with energy differences ("Peierls–Nabarro barrier") between discrete translations. In the small-parameter (continuum) limit, the PN barrier becomes exponentially small in the effective bifurcation parameter and is a key feature distinguishing discrete from continuous NLS models (Jenkinson et al., 2014).

3. Periodic, Generalized, and Nonlocal DNLS: Advanced Solution Classes

Periodic Traveling Waves

For general local and nonlocal DNLS, periodic traveling waves are constructed using fixed-point methods on Banach spaces. Existence theorems require the frequency to lie outside the linear phonon band (which is determined by the range and strength of coupling), ensuring truly nonlinear (anharmonic) dynamics (Hennig, 2017).

Generalized and Nonlocal DNLS

The DNLS admits physically important generalizations:

• Generalized DNLS (GDNLS): Appearing in models such as protein folding, with higher-order or rational nonlinearities and additional fields (torsion, curvature), admitting dark-soliton solutions and complex geometric interpretations (Molkenthin et al., 2010).
• Nonlocal DNLS: Nonlocalities (e.g., $u_n^2 \overline{u}_{-n}$) inspired by PT-symmetric or reverse-space NLS lead to new stationary solution classes, linear instabilities, and modified conservation law structures. Both integrable and nonintegrable discretizations are studied, with soliton profiles obtained via discrete Fourier and iterative methods (Ji et al., 2019, Zhao et al., 22 Apr 2024).
• Fractional DNLS: Fractional Laplacian generalizations interpolate between nearest-neighbor and fully connected lattices, modifying the bandwidth, dispersion, and stability of localized modes, with consequential shifts in self-trapping thresholds and ballistic spreading (Molina, 2019, Molina, 2020).

4. Well-Posedness, Cauchy Problem, and Thermodynamic Behavior

Cauchy Problem and Well-Posedness

The DNLS Cauchy problem is globally well-posed in $\ell^2(\mathbb{Z}^d)$ for all initial data; mass and energy conservation control the global dynamics. Recent results extend global well-posedness to initial data with mild power-law growth away from the origin, so that, in particular, random initial conditions from common equilibrium measures yield almost sure global solutions in $d=1$ (Vuoksenmaa, 2023).

Statistical Mechanics and Phase Transitions

In $d \geq 3$, the focusing DNLS admits exact thermodynamic descriptions. The finite-volume Gibbs measure is normalizable for a large parameter range, and one can compute all partition function-related thermodynamic quantities analytically. There is a first-order phase transition in the free energy at a critical value of $\beta B^2$. Below threshold, wavefunctions are delocalized (Gaussian fluctuations); above threshold, macroscopic mass localizes at a single site ("discrete breather") supported for exponentially long times (Chatterjee et al., 2010).

These features contrast with continuous NLS, where the invariant Gibbs measure is either non-normalizable or trivial, and highlight the fundamental role of discreteness in energy localization and statistical phase structure.

5. Transport, Nonequilibrium, and Dynamical phenomena

Nonequilibrium Transport

DNLS chains display normal, finite Onsager-coefficient transport in the thermodynamic limit, with coupled energy and norm currents. Monte Carlo thermostats (imposing both temperature and chemical potential at boundaries) realize fully coupled nonequilibrium steady states, with linear-response theory capturing coupled particle and energy fluxes. Notably, the Seebeck coefficient (thermoelectric response) can change sign in parameter space, and large biases can drive non-monotonic density and temperature profiles, a consequence of the nonlinear, state-dependent Onsager matrix (Iubini et al., 2012).

Dispersive Hydrodynamics and Dam Breaks

The DNLS supports dispersive shock waves (DSW), rarefactions, and kink-type coherent structures under hydrodynamic initial data. The cross-over from anti-continuum to continuum limits is described with Whitham modulation theory. Discrete DSWs exhibit critical thresholds in lattice spacing separating "continuum-like" from "deep-discrete" regimes, while two-phase resonance phenomena can drive modulational instabilities unique to the discrete case, generating complex multi-phase wave trains (Mohapatra et al., 15 Jul 2025).

6. Numerical Methods, Continuum Limits, and Integrability

Numerical Approximations and Continuum Limits

Finite-difference and spectral discretization methods are rigorously justified for the DNLS as approximations to the continuous NLS, with strong $L^2$ convergence results as $h \to 0$ for both focusing and defocusing cubic NLS in two spatial dimensions (Hong et al., 2019).

Integrable and Non-integrable Discrete NLS

Special choices of discrete nonlinearity and coupling yield integrable models (e.g., the Ablowitz–Ladik equation, discrete generalized NLS equations with Lax pairs and recursion operators) (Li et al., 2013). Integrable discretizations admit infinite symmetries and conservation laws, and features such as explicit multi-soliton and Casoratian/Wronskian solutions, contrasting with the more generic non-integrable DNLS which may lack analytical soliton formulae, possess nontrivial Peierls–Nabarro barriers, and admit only numerically constructed solutions (Zhao et al., 22 Apr 2024).

7. Physical Applications and Relevance

The DNLS underpins theoretical descriptions in nonlinear photonic lattices, arrays of coupled optical waveguides, and deep-optical-lattice Bose–Einstein condensates, where it models light or matter-wave envelope evolution in tight-binding regimes (Tsoy et al., 2019, Mohapatra et al., 15 Jul 2025). In statistical mechanics, it serves as a minimal model for high-dimensional energy localization and nonlinear transport processes. In biological physics, generalized DNLS frameworks model protein backbone geometries with high-fidelity through soliton-based folding templates (Molkenthin et al., 2010).

The mathematical richness and physical generality of the DNLS frame a wide array of interdisciplinary phenomena, from thermodynamic phase transitions, energy self-trapping, and transport anomalies, to the integrable hierarchy theory of lattice soliton systems.