Nonlinear Mobility of Power Type

• Nonlinear mobility of power type is defined by a response function with power-law characteristics that shape soliton propagation and energy landscapes.
• Constrained Newton–Raphson methods map Peierls–Nabarro energy surfaces, revealing stability regimes and directional mobility windows in discrete lattices.
• Applications in nonlinear optics and photonic devices leverage tunable mobility regimes to control energy transport and oscillatory soliton dynamics.

Nonlinear mobility of power type characterizes systems in which the transport, propagation, or motion of excitations or densities is fundamentally governed by a response function that exhibits a nonlinear dependence on the relevant state variable, often exhibiting power-law growth or saturation. Within discrete, continuum, and hybrid frameworks, this concept is closely linked to energy landscapes, stability regimes, and the interplay between nonlinearity, discreteness, and geometric constraints.

1. Peierls–Nabarro Energy Surfaces and Pseudopotential Landscapes

In two-dimensional discrete nonlinear Schrödinger (DNLS) lattices with saturable nonlinearity, the motion of solitons is determined by Peierls–Nabarro (PN) energy surfaces, which encode the effective energy required for a soliton to traverse the lattice at fixed power and nonlinearity constant (Naether et al., 2010). These surfaces reflect a pseudopotential landscape as a function of the soliton’s center-of-mass coordinates $(X,Y)$ at constant conserved power $P$, with local minima corresponding to stationary coherent excitations (one-site, two-site, four-site solutions) and saddle points representing intermediate asymmetric states (IS1, IS2, IS3).

Regions of the PN surface that are smooth and flat allow for enhanced mobility of solitons, minimizing radiative loss as the soliton moves through the lattice. Conversely, steep gradients or pronounced barriers restrict motion and promote localization. The topology of these surfaces shifts dramatically with variations in system parameters (nonlinearity constant $\gamma$ and power $P$), toggling the system between mobility-favoring and trapping regimes.

2. Role of Saturable and Power-Type Nonlinearities

Nonlinear mobility is substantially modified in lattices with saturable on-site nonlinearity compared to standard cubic (Kerr-type) DNLS systems. The saturable response, typified by a nonlinear term such as $\gamma U_{n,m}/(1 + |U_{n,m}|^2)$, “tames” the nonlinearity at high intensity and admits the existence of transparent regimes or “mobility windows” at selected power intervals (Naether et al., 2010). The efficacy of soliton mobility depends critically on these regimes, as saturable nonlinearity can induce multiple stability exchanges among stationary modes and can support nearly radiationless sliding motion.

Power-type mobility is generalized as (but not limited to) nonlinear terms scaling like $|u|^p$, with $p>1$, in equations governing solitary wave propagation or density evolution. In such systems, the exponent $p$ controls the saturating or focusing-defocusing character of the transport, with higher powers producing steeper barriers and reduced mobility, whereas lower powers or saturable forms can mitigate barrier effects and increase the likelihood of traveling excitations.

3. Numerical Methods: Constrained Newton–Raphson Computation

The PN energy surfaces are computed using a constrained Newton–Raphson technique, which precisely navigates the solution manifold at fixed power by controlling the amplitudes at designated lattice sites corresponding to the soliton’s center-of-mass position. Defining

$$X = \frac{\sum_{n,m} n|u_{n,m}|^2}{P}, \quad Y = \frac{\sum_{n,m} m|u_{n,m}|^2}{P},\quad P = \sum_{n,m} |u_{n,m}|^2,$$

the method constructs the Hamiltonian $H$ as a function $H(X,Y)$, thus mapping the system’s energetic landscape. Flatter regions in $H(X,Y)$ predict directions along which solitons can propagate with minimal loss. The same method exposes coupling between translational and transverse oscillatory modes, as the curvature of the surface in the transverse direction determines oscillation frequencies and amplitudes for kicked solitons.

4. Directional Mobility and Regime Classification

Directional soliton mobility is governed by the structure of the PN surface, which varies with $\gamma$ and $P$. For low power, stationary solutions (one-site) are stable minima—mobility is limited and primarily axial. At intermediate power, bifurcations introduce intermediate asymmetric states connecting the stationary modes, resulting in mobility windows along specific lattice axes or diagonals where the PN surface is nearly flat. High-power regimes reverse the stability roles (four-site solutions become minima), changing the mobility landscape. Mobility is favored in directions associated with energy surface valleys, as these minimize radiative hindrance during motion.

5. Effects of Lattice Anisotropy

Introducing anisotropy (parameter $\alpha < 1$) in coupling affects the mobility landscape by tilting PN surfaces and shifting the stability domains of fundamental modes. Anisotropy creates preferential mobility channels and modifies the orderings and configurations of stationary states but does not produce new mobility valleys. Thus, optimal transport remains along lattice directions, though these may be biased according to anisotropic coupling strengths.

6. Mathematical Formulation

The key governing equations and definitions are:

• DNLS with saturable nonlinearity:

$$i\frac{\partial U_{n,m}}{\partial z} + \left(U_{n+1,m}+U_{n-1,m}+\alpha(U_{n,m+1}+U_{n,m-1})\right) - \frac{\gamma U_{n,m}}{1+|U_{n,m}|^2} = 0,$$

• Conserved power:

$P = \sum_{n,m} |U_{n,m}|^2,$

• Hamiltonian:

$$H = -\sum_{n,m} \left( (U_{n+1,m} + \alpha U_{n,m+1})U_{n,m}^* - \frac{1}{2}\gamma\ln(1 + |U_{n,m}|^2) + c.c. \right),$$

• Center-of-mass coordinates:

$$X = \frac{\sum_{n,m} n |u_{n,m}|^2}{P}, \quad Y = \frac{\sum_{n,m} m |u_{n,m}|^2}{P}.$$

These formal tools construct the landscape upon which nonlinear mobility is interpreted and analyzed.

7. Implications, Applications, and Future Directions

Understanding nonlinear mobility of power type via PN energy surfaces enables targeted engineering of propagation characteristics in discrete and photonic systems. The ability to tune power and nonlinearity constants to optimize mobility regimes offers prospects for controllable transport in optical waveguide arrays and discrete nonlinear devices. The coupling between translational and transverse oscillations, revealed by the curvature of the energy surface, may lead to engineered mobility features such as directional switching, “locking,” or oscillatory steering of solitons.

Additionally, the numerical and analytical mapping of full PN surfaces provides a template for exploring higher-dimensional nonlinear lattices, anisotropic systems, and alternative nonlinearities. The general framework applies broadly—from nonlinear optics and photonic crystals to condensed matter systems and analogues in ultracold atomic lattices—where quantification and control of nonlinear mobility is critical for device function, energy transport, and information routing.

In conclusion, nonlinear mobility of power type is dictated by the global properties of multidimensional energy landscapes sculpted by power-dependent nonlinearities and geometric constraints. These landscapes are accessible through sophisticated numerical methods and provide a predictive basis for the design and analysis of advanced nonlinear transport systems.