arXiv subjects

Abrari Noor Hasmi

Publications and source records attributed to Abrari Noor Hasmi.

4 recordsLinked to original sources

arXiv2026

Phase space integrity in neural network models of Hamiltonian dynamics: A Lagrangian descriptor approach

We propose Lagrangian Descriptors (LDs) as a diagnostic framework for evaluating neural network models of Hamiltonian systems beyond conventional trajectory-based metrics. Standard error measures quantify short-term predictive accuracy but provide little insight into global geometric structures such as orbits and separatrices. Existing evaluation tools in dissipative systems are inadequate for Hamiltonian dynamics due to fundamental differences in the systems. By constructing probability density functions weighted by LD values, we embed geometric information into a statistical framework suitable for information-theoretic comparison. We benchmark physically constrained architectures (SympNet, HénonNet, Generalized Hamiltonian Neural Networks) against data-driven Reservoir Computing across two canonical systems. For the Duffing oscillator, all models recover the homoclinic orbit geometry with modest data requirements, though their accuracy near critical structures varies. For the three-mode nonlinear Schrödinger equation, however, clear differences emerge: symplectic architectures preserve energy but distort phase-space topology, while Reservoir Computing, despite lacking explicit physical constraints, reproduces the homoclinic structure with high fidelity. These results demonstrate the value of LD-based diagnostics for assessing not only predictive performance but also the global dynamical integrity of learned Hamiltonian models.

cs.LG↗

arXiv2025

Exponential asymptotics of dark and bright solitons in the discrete nonlinear Schrödinger equation

We investigate the existence and linear stability of solitons in the nonlinear Schrödinger lattices in the strong coupling regime. Focusing and defocusing nonlinearities are considered, giving rise to bright and dark solitons. In this regime, the effects of lattice discreteness become exponentially small, requiring a beyond-all-orders analysis. To this end, we employ exponential asymptotics to derive soliton solutions and examine their stability systematically. We show that only two symmetry-related soliton configurations are permissible: onsite solitons centered at lattice sites and intersite solitons positioned between adjacent sites. Although the instability of intersite solitons due to real eigenvalue pairs is known numerically, a rigorous analytical account, particularly for dark solitons, has been lacking. Our work fills this gap, yielding analytical predictions that match numerical computations with high accuracy. We also establish the linear stability of onsite bright solitons. While the method cannot directly resolve the quartet eigenvalue-induced instability of onsite dark solitons due to the continuous spectrum covering the entire imaginary axis, we conjecture an eigenvalue-counting argument that supports their instability. Overall, our application of the exponential asymptotics method shows the versatility of this approach for addressing multiscale problems in discrete nonlinear systems.

nlin.PS↗

arXiv2025

Model-free Forecasting of Rogue Waves using Reservoir Computing

Recent research has demonstrated Reservoir Computing's capability to model various chaotic dynamical systems, yet its application to Hamiltonian systems remains relatively unexplored. This paper investigates the effectiveness of Reservoir Computing in capturing rogue wave dynamics from the nonlinear Schrödinger equation, a challenging Hamiltonian system with modulation instability. The model-free approach learns from breather simulations with five unstable modes. A properly tuned parallel Echo State Network can predict dynamics from two distinct testing datasets. The first set is a continuation of the training data, whereas the second set involves a higher-order breather. An investigation of the one-step prediction capability shows remarkable agreement between the testing data and the models. Furthermore, we show that the trained reservoir can predict the propagation of rogue waves over a relatively long prediction horizon, despite facing unseen dynamics. Finally, we introduce a method to significantly improve the Reservoir Computing prediction in autonomous mode, enhancing its long-term forecasting ability. These results advance the application of Reservoir Computing to spatio-temporal Hamiltonian systems and highlight the critical importance of phase space coverage in the design of training data.

cs.CE↗

arXiv2025

Supratransmission of rogue wave-like breathing solitons

Supratransmission is a fascinating and counterintuitive nonlinear wave phenomenon that enables energy transmission through frequency band gaps. Recent studies have suggested that supratransmission in a damped-driven Klein-Gordon equation can lead to the formation of rogue waves. In this work, we reduce the same model to a damped and driven discrete nonlinear Schrödinger equation and demonstrate that the claimed rogue waves are unstable solitons exhibiting breathing dynamics. Through extensive numerical simulations, we thoroughly investigate and analyze the behavior of solitons and supratransmission in these lattices. Additionally, we perform asymptotic analysis to explain the underlying mechanism responsible for supratransmission. Our findings provide new insights into the dynamics of nonlinear waves and their stability in such systems.

nlin.PS↗