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In this paper, we revisit the investigation of solitary-wave interactions in the nonlinear Schrödinger model, both in the presence and absence of a parabolic trapping potential. While approximate dynamics, based on variational—or similar—methods, governed by a system of ordinary differential equations (ODEs) for both bright and dark-soliton interactions have been well established in the literature based on physical expert considerations, this study focuses on a data-driven approach, the so-called sparse identification of nonlinear dynamics. Accordingly, our purpose is to use partial differential equation (PDE) time-series of select waveform diagnostics in order to numerically reconstruct such approximate dynamics, without prior knowledge thereof. The purpose is not only to verify the robustness of the dynamical approximated ODEs, but also to shed light on the application of such a data-driven methodology in the study of soliton interactions and to formulate a complementary approach, more reliant on the wealth of PDE data and less so on expert theoretical constructs. 

Interactions between solitary waves are crucial for understanding nonlinear phenomena in systems such as optics, fluid dynamics, and mechanical metamaterials. Rarefaction solitary waves, in particular, offer insight into nonlinear wave dynamics in strain-softening media. Despite their proposed applications in waveguides and energy harvesting, key characteristics—particularly solitonic collisions—are not yet fully understood due to energy dissipation and the need for high-precision measurement techniques. In this work, we introduce an experimental platform for studying pure rarefaction solitons in a strain-softening lattice. Our results show that both symmetric and asymmetric collisions display elastic interactions and amplitude-dependent phase shifts. The experimentally observed dynamics, including soliton speed and phase shifts, closely match numerical simulations and analytical predictions based on the Boussinesq approximation. These findings not only validate our platform but also highlight the potential of mechanical rarefaction solitons for probing nonlinear wave interactions and advancing wave-based computing. 

Abstract. We formulate the discovery of Lax integrability of Hamiltonian dynamical systems as a symbolic regression problem, which, loosely speaking, seeks to maximize the compatibility between a pair of Lax operators and the known Hamiltonian of the dynamical system. Our approach is first tested on the simple harmonic oscillator. We then move on to the Henon–Heiles system, i.e., a two-degree-of-freedom system of nonlinear oscillators. The integrability of the Henon–Heiles system is critically dependent on a set of three parameters within its Hamiltonian, a fact that we leverage to assess the robustness of our approach in detecting the integrability of this system with respect to the parameter dependence of the Hamiltonian. We then adapt our method to canonical examples of Hamiltonian partial differential equations, including the Korteweg–de Vries and cubic nonlinear Schrödinger equations, again testing robustness against nonintegrable perturbations of their respective Hamiltonians. In all examples, our approach reliably confirms or denies the integrability of the equations of interest. Moreover, by appropriately adjusting the loss function and applying thresholded [Formula: see text] regularization to enforce sparsity in the operator weights, we successfully recover accurate forms of the Lax pairs despite wide initial hypotheses on the operators. Some of the relevant Lax pairs, notably for the Henon–Heiles system and the Korteweg–de Vries equation, are distinct from the ones that are typically reported in the literature. The Lax pairs that our methodology discovers warrant further mathematical and computational investigation, and we discuss extensively the opportunities for further improvement of SILO as a viable tool for interpretable exploration of integrable Hamiltonian dynamical systems. 

We propose a data-driven framework for learning reduced-order moment dynamics from partial differential equation (PDE)-governed systems using neural Ordinary Differential Equations (neural ODEs). In contrast to derivative-based methods, such as SINDy (Sparse Identification of Nonlinear Dynamics), which necessitate densely sampled data and are sensitive to noise, our approach based on neural ODEs directly models moment trajectories, enabling robust learning from sparse and potentially irregular time series. Using as an application platform the nonlinear Schrödinger equation, the framework accurately recovers governing moment dynamics when closure is available, even from limited, irregular, and noisy observations. For systems without analytical closure, we introduce a data-driven coordinate transformation strategy based on Stiefel manifold optimization, enabling the discovery of low-dimensional representations in which the moment dynamics become closed, facilitating interpretable and reliable modeling. We also explore cases where a closure model is not known, such as a Fisher–Kolmogorov–Petrovskii–Piskounov reaction–diffusion system. Here, we demonstrate that neural ODEs can still effectively approximate the unclosed moment dynamics and achieve superior extrapolation accuracy compared to physical-expert-derived ODE models. This advantage remains robust even under sparse and irregular sampling, highlighting the method’s robustness in data-limited settings. Our results highlight the neural ODE framework as a powerful and flexible tool for learning interpretable, low-dimensional moment dynamics in complex PDE-governed systems. 

The sine-Gordon model is studied with model parameters that depend on both space and time. An effective model with one degree of freedom is constructed, allowing the description of the kink movement in both a temporally non-autonomous and spatially inhomogeneous setting. As a highly demanding test of the reduced model, the case of a temporal drive leading to a parametric instability is considered. Here, the boundaries of the instability region in the form of Arnold tongues were examined in both the simplified effective model and the full field model. Good agreement was obtained as concerns the regions of stability. It was observed that only in the bottom parts of Arnold’s tongues is the dynamics of the field model more complicated than in the approximate model.