Learning Traveling Solitary Waves Using Separable Gaussian Neural Networks.

In this paper, we apply a machine-learning approach to learn traveling solitary waves across various physical systems that are described by families of partial differential equations (PDEs). Our approach integrates a novel interpretable neural network (NN) architecture, called Separable Gaussian Neural Networks (SGNN) into the framework of Physics-Informed Neural Networks (PINNs). Unlike the traditional PINNs that treat spatial and temporal data as independent inputs, the present method leverages wave characteristics to transform data into the so-called co-traveling wave frame. This reformulation effectively addresses the issue of propagation failure in PINNs when applied to large computational domains. Here, the SGNN architecture demonstrates robust approximation capabilities for single-peakon, multi-peakon, and stationary solutions (known as "leftons") within the (1+1)-dimensional, b-family of PDEs. In addition, we expand our investigations, and explore not only peakon solutions in the ab-family but also compacton solutions in (2+1)-dimensional, Rosenau-Hyman family of PDEs. A comparative analysis with multi-layer perceptron (MLP) reveals that SGNN achieves comparable accuracy with fewer than a tenth of the neurons, underscoring its efficiency and potential for broader application in solving complex nonlinear PDEs.

The application of the "inverse problem" method for constructing confining potentials that make N-soliton waveforms exact solutions in the Gross-Pitaevskii equation.

In this work, we discuss an application of the "inverse problem" method to find the external trapping potential, which has particular N trapped soliton-like solutions of the Gross-Pitaevskii equation (GPE) also known as the cubic nonlinear Schrödinger equation (NLSE). This inverse method assumes particular forms for the trapped soliton wave function, which then determines the (unique) external (confining) potential. The latter renders these assumed waveforms exact solutions of the GPE (NLSE) for both attractive (g<0) and repulsive (g>0) self-interactions. For both signs of g, we discuss the stability with respect to self-similar deformations and translations. For g<0, a critical mass Mc or equivalently the number of particles for instabilities to arise can often be found analytically. On the other hand, for the case with g>0 corresponding to repulsive self-interactions which is often discussed in the atomic physics realm of Bose-Einstein condensates, the bound solutions are found to be always stable. For g<0, we also determine the critical mass numerically by using linear stability or Bogoliubov-de Gennes analysis, and compare these results with our analytic estimates. Various analytic forms for the trapped N-soliton solutions in one, two, and three spatial dimensions are discussed, including sums of Gaussians or higher-order eigenfunctions of the harmonic oscillator Hamiltonian.

Uniform-density Bose-Einstein condensates of the Gross-Pitaevskii equation found by solving the inverse problem for the confining potential.

In this work, we study the existence and stability of constant density (flat-top) solutions to the Gross-Pitaevskii equation (GPE) in confining potentials. These are constructed by using the "inverse problem" approach which corresponds to the identification of confining potentials that make flat-top waveforms exact solutions to the GPE. In the one-dimensional case, the exact solution is the sum of stationary kink and antikink solutions, and in the overlapping region, the density is constant. In higher spatial dimensions, the exact solutions are generalizations of this wave function. In the absence of self-interactions, the confining potential is similar to a smoothed-out finite square well with minima also at the edges. When self-interactions are added, terms proportional to ±gψ^{*}ψ and ±gM with M representing the mass or number of particles in Bose-Einstein condensates get added to the confining potential and total energy, respectively. In the realm of stability analysis, we find (linearly) stable solutions in the case with repulsive self-interactions which also are stable to self-similar deformations. For attractive interactions, however, the minima at the edges of the potential get deeper and a barrier in the center forms as we increase the norm. This leads to instabilities at a critical value of M. Comparing the stability criteria from Derrick's theorem with Bogoliubov-de Gennes (BdG) analysis stability results, we find that both predict stability for repulsive self-interactions and instability at a critical mass M for attractive interactions. However, the numerical analysis gives a much lower critical mass. This is due to the emergence of symmetry-breaking instabilities that were detected by the BdG analysis and violate the symmetry x→-x assumed by Derrick's theorem.

Wave manipulation using a bistable chain with reversible impurities.

We systematically study linear and nonlinear wave propagation in a chain composed of piecewise-linear bistable springs. Such bistable systems are ideal test beds for supporting nonlinear wave dynamical features including transition and (supersonic) solitary waves. We show that bistable chains can support the propagation of subsonic wave packets which in turn can be trapped by a low-energy phase to induce energy localization. The spatial distribution of these energy foci strongly affects the propagation of linear waves, typically causing scattering, but, in special cases, leading to a reflectionless mode analogous to the Ramsauer-Townsend effect. Furthermore, we show that the propagation of nonlinear waves can spontaneously generate or remove additional foci, which act as effective "impurities." This behavior serves as a new mechanism for reversibly programming the dynamic response of bistable chains.

Origami-based impact mitigation via rarefaction solitary wave creation.

The principles underlying the art of origami paper folding can be applied to design sophisticated metamaterials with unique mechanical properties. By exploiting the flat crease patterns that determine the dynamic folding and unfolding motion of origami, we are able to design an origami-based metamaterial that can form rarefaction solitary waves. Our analytical, numerical, and experimental results demonstrate that this rarefaction solitary wave overtakes initial compressive strain waves, thereby causing the latter part of the origami structure to feel tension first instead of compression under impact. This counterintuitive dynamic mechanism can be used to create a highly efficient-yet reusable-impact mitigating system without relying on material damping, plasticity, or fracture.