arXiv subjects

Sarswati Shah

Publications and source records attributed to Sarswati Shah.

4 recordsLinked to original sources

arXiv2026

QH-GEM: Quantum-Hydrodynamic Generative Modeling

In this paper, we develop a deterministic, physically constrained generative framework based on the Madelung formulation of the free-particle Schrödinger equation. A reference Born probability density and a controllable initial phase function serve as initial data for the free Madelung system, which couples the Born probability density and phase function through the Bohm quantum potential, while the phase function determines the hydrodynamic velocity field. Provided that the Born probability density remains positive and the hydrodynamic velocity field generates a unique global characteristic flow, samples drawn from the reference density and transported along the characteristic flow are distributed according to the evolving Born probability density at every time. As a consequence, randomness enters only through the initial sampling; the subsequent generation is deterministic and involves neither stochastic dynamics nor an independently parameterized time-dependent velocity field. We formulate terminal-time distribution matching as a PDE-constrained phase-identification problem and derive the underlying Hamiltonian and Fisher-information structure. For isotropic Gaussian wave packets, we obtain explicit dynamics and a necessary and sufficient condition for exact reachability of isotropic Gaussian targets by quadratic initial phase functions, together with the corresponding sampling map. For a smooth prescribed potential initial velocity field, we further establish that the characteristic flow approximates the associated first-order transport map with an O(T^2) error, both uniformly and in the 1- and 2-Wasserstein distances. A numerical Gaussian benchmark validates the fully discrete forward solver, while full-grid PDE-constrained phase identification is demonstrated for asymmetric bimodal targets.

math.NA↗

arXiv2026

Reduced Order Modeling of One-Dimensional Conservative PDEs via the Cumulative Distribution Transform

We propose a reduced order modeling (ROM) framework for 1D conservative PDEs based on the cumulative distribution transform (CDT). The CDT maps nonnegative, equal-mass states into a Hilbert space in which 1D Wasserstein distances become weighted $L^2$ distances and translations become affine shifts. This makes the transform especially suited for transport-dominated dynamics, where Eulerian linear-subspace ROMs often suffer from slow decay of Kolmogorov widths. We study this phenomenon for scalar conservative dynamics by analyzing the solution manifold in CDT coordinates. For linear transport, the transformed solution manifold is contained in the 2-dimensional space spanned by the transformed initial datum and the constant function, and has zero Kolmogorov $2$-width. For nonlinear hyperbolic conservation laws, we prove two complementary types of estimates: robust $O(n^{-1})$ bounds that rely only on the conservative transport structure and remain meaningful after shock formation, and sharper $O(n^{-2})$ bounds in smooth pre-shock regimes. For conservative advection-diffusion, we show that the CDT trajectory remains within distance $O(\sqrt{DT})$ of the pure-transport plane, and we also obtain sharper $O(D^2T^2)$ estimates under additional regularity or away from initial layers. In both cases, the zero 2-width behavior of linear transport is recovered as the diffusion coefficient tends to zero. Motivated by these estimates, we develop a CDT-POD numerical scheme: snapshots are mapped to CDT space, Proper Orthogonal Decomposition (POD) is performed in transformed coordinates, and the inverse CDT is used to reconstruct physical states. Numerical experiments for several transport-dominated dynamics show that CDT-POD can capture solution manifolds with substantially fewer modes than Eulerian POD.

math.AP↗

arXiv2026

Polarization-Induced Beam Bending: Mathematical Model, Discretization, and Algorithm

We study a reduced hydrodynamic formulation of paraxial vector beam propagation in which the beam intensity, optical phase, and spatially-dependent polarization are coupled through a nonlinear dispersive system. While prior analytical work derived a solution for the beam path valid for short propagation distances, a fully resolved numerical treatment of the model over long ranges has not previously been available. Here we present a conservative numerical scheme for the coupled system, combining a finite-volume discretization of the intensity equation with monotone Hamilton--Jacobi (H-J) solvers for the phase dynamics and upwind transport of polarization. The method preserves the nonnegativity of the intensity and remains stable under long-distance propagation. We perform large-scale simulations over propagation distances of tens of meters, while resolving millimeter-scale transverse structure. The numerical results reproduce the analytically predicted and experimentally observed quadratic beam bending at short distances and reveal systematic deviations beyond the asymptotic regime. These deviations arise from nonlinear phase accumulation and dispersive effects captured by the full model but are neglected in the short-distance approximation.

physics.optics↗

arXiv2026

A new model for two-layer liquid-gas stratified flows in pipes with general cross sections

In this work, we derive a new model for immiscible two-layer gas-liquid stratified flows in pipes with general cross sections. The bottom layer is occupied by an incompressible fluid in liquid phase with hydrodynamics based on a hydrostatic pressure, following a shallow water approximation. The top layer is occupied by a compressible gas, following an ideal gas law leading to conservation of mass, momentum and energy. The two subsystems are linked through non-conservative products, representing momentum and energy exchanges between layers. The hyperbolic properties of the resulting model are analyzed, including the derivation of entropy inequalities, and the approximations of eigenvalues of the corresponding coefficient matrix. Numerical tests are included to demonstrate the merits of the model and the numerical approximations, including well-balancedness, Riemann problems, and perturbations and convergence toward steady states at rest. Besides simulations of water and air where the density difference between layers is significant, a case where such difference is not so pronounced (like gas and liquid hydrogen) is also shown.

physics.comp-ph↗