Article
Automation & Control Systems
Ugo Boscain, Eugenio Pozzoli, Mario Sigalotti
Summary: This paper studies the controllability problem for a symmetric-top molecule in both classical and quantum rotational dynamics using three orthogonal electric fields interacting with its electric dipole. Different controllability characteristics are observed based on the dipole position, resulting in the emergence of quantum symmetry in quantum dynamics without a classical counterpart. The approximate controllability of the symmetric-top Schriidinger equation is established using a Lie-Galerkin method based on blockwise approximations.
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2021)
Article
Automation & Control Systems
Kais Ammari, Alessandro Duca
Summary: This work examines the bilinear Schrodinger equation in the Hilbert space with an infinite graph, focusing on the Laplacian with self-adjoint boundary conditions and a bounded symmetric operator. Despite the dispersive behavior of the equation, the study explores the well-posedness of the (BSE) in suitable subspaces preserved by the dynamics, as well as global exact controllability and 'energetic controllability'. The examples provided include scenarios involving infinite tadpole graphs.
INTERNATIONAL JOURNAL OF CONTROL
(2021)
Article
Mathematics, Applied
Kai Wang, Dun Zhao, Binhua Feng
Summary: We investigate the optimal bilinear control of the 3D logarithmic nonlinear Schrodinger equation arising from quantum physics. By using an approximate scheme to overcome the difficulty caused by the absence of Lipschitz continuity at the origin due to the logarithmic nonlinearity, we finally obtain the minimizer of the original optimal control problem.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Jonas Lampart
Summary: We discuss the set of wavefunctions that can be obtained by applying the flow of the Schrodinger operator and varying the potential. We show that this set has empty interior as a subset of the L-2 (R-d) sphere and as a set of trajectories.
EVOLUTION EQUATIONS AND CONTROL THEORY
(2021)
Article
Physics, Applied
Shu-Zhi Liu, Hua Wu
Summary: In this paper, solutions to the derivative nonlinear Schrodinger equation associated with real and complex discrete eigenvalues of the Kaup-Newell spectral problem are derived. These solutions are obtained by investigating double Wronskian solutions of the coupled Kaup-Newell equations and reducing them using the bilinear method and a reduction technique. The obtained solutions exhibit not only periodic behavior, but also solitons on periodic background, with dynamics illustrated.
MODERN PHYSICS LETTERS B
(2021)
Article
Physics, Mathematical
Ivan Beschastnyi, Ugo Boscain, Mario Sigalotti
Summary: This article discusses the preserved controllability properties of classical Hamiltonian systems after quantization and explores small-time controllability of both classical and quantum systems using the WKB method. The conjecture that lack of small-time controllability in classical systems implies the same for quantum systems is also investigated.
JOURNAL OF MATHEMATICAL PHYSICS
(2021)
Article
Mathematics, Applied
Xue-Wei Yan, Yong Chen
Summary: This study investigates a generalized nonlinear Schrodinger equation that can describe subpicosecond pulse propagation in optical fibers. By developing the Hirota method, bilinear forms and analytical soliton solutions are derived, and the dynamics of pulse solitons are analyzed based on these solutions. The results show that high-order dispersion terms can change the periodicity of propagation, and the interaction between two pulse solitons is an elastic collision. By selecting suitable parameter values, parallel solitons can be obtained, which can improve the transmission quality and capacity of information in optical fibers.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Yongki Lee
Summary: This work studies a Riccati system governing pressure-less two-dimensional EP equations. It is found that the vorticity accelerates divergence while other terms amplify the blow-up behavior of the flow. By constructing an auxiliary system and finding an invariant space, it is shown that the Riccati system can have global solutions and admit global smooth solutions for a large set of initial configurations.
Article
Mathematics, Applied
Alexei A. Deriglazov
Summary: This article deduces the equations of a rotating body with one point constrained to move freely on a plane (dancing top) from the Lagrangian variational problem, which formally resemble the Euler-Poisson equations of a heavy body with a fixed point in a fictitious gravity field. By using this analogy, examples of analytical solutions for a heavy symmetrical dancing top are found, describing motions where the center of mass maintains a fixed height above the supporting plane. The general solution to the equations of a dancing top in terms of the exponential of the Hamiltonian field is provided. An additional constraint that incorporates the reaction of the supporting plane modifies the canonical Poisson structure, raising questions about its integrability according to Liouville.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
Gyeongha Hwang, Haewon Yoon
Summary: In this article, we study the Cauchy problem of the one-dimensional nonlinear Schrödinger equation, which is L2-critical. We prove the local well-posedness of the problem in a probabilistic manner for the scaling supercritical regularity regime -10 < s < 0. A key ingredient is the establishment of a probabilistic bilinear Strichartz estimate.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2023)
Article
Mathematics, Applied
Hiroyuki Hirayama, Shinya Kinoshita, Mamoru Okamoto
Summary: This paper addresses the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations introduced by Colin and Colin (2004), determining an almost optimal Sobolev regularity where the smooth flow map of the Cauchy problem exists, except for the scaling critical case. This result fills a gap left open in papers of the first and second authors (2014, 2019).
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Automation & Control Systems
Alessandro Duca
Summary: The study focuses on the bilinear Schrodinger equation on a bounded one-dimensional domain, providing explicit times for global exact controllability and demonstrating methods for constructing controls for global approximate controllability.
INTERNATIONAL JOURNAL OF CONTROL
(2021)
Article
Mathematics
Xuan Liu, Ting Zhang
Summary: This paper investigates the well-posedness of the inhomogeneous nonlinear biharmonic Schrodinger equation with a spatial inhomogeneity coefficient K(x) behaving like vertical bar x vertical bar(-b) for 0 < b < min {N/2, 4}. The local well-posedness is demonstrated in the whole H-s-subcritical case, with 0 < s <= 2. The difficulties of the problem are addressed by deriving bilinear Strichartz's type estimates for the nonlinear biharmonic Schrodinger equations in Besov spaces.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Engineering, Mechanical
Xiang Chen, Dumitru Mihalache, Jiguang Rao
Summary: This paper studies the dynamics and collisions of degenerate and nondegenerate bright solitons in two-component nonlinear Schrodinger equations coupled to Boussinesq equation. Degenerate solitons with single-hump profiles exhibit elastic and inelastic collisions, while their velocities in the short wave components and long wave component are identical. Nondegenerate single solitons can have double- or single-hump profiles with identical velocities in all components or single-hump profiles with unequal velocities in the short wave components. The collisions of nondegenerate solitons do not result in the redistribution of soliton intensities. Three different types of collisions of nondegenerate two-soliton solutions are studied in detail.
NONLINEAR DYNAMICS
(2023)
Article
Physics, Mathematical
Renato Spigler
Summary: The quantum lattice Boltzmann (qlB) algorithm is used to approximate the 1D Dirac equations and the non-relativistic Schrodinger equation. This method provides accurate solutions in the non-relativistic limit, but with a small error.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Matteo Gallone, Alessandro Michelangeli, Eugenio Pozzoli
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
(2019)
Article
Automation & Control Systems
Ugo Boscain, Eugenio Pozzoli, Mario Sigalotti
Summary: This paper studies the controllability problem for a symmetric-top molecule in both classical and quantum rotational dynamics using three orthogonal electric fields interacting with its electric dipole. Different controllability characteristics are observed based on the dipole position, resulting in the emergence of quantum symmetry in quantum dynamics without a classical counterpart. The approximate controllability of the symmetric-top Schriidinger equation is established using a Lie-Galerkin method based on blockwise approximations.
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2021)
Article
Mathematics
Ivan Beschastnyi, Ugo Boscain, Eugenio Pozzoli
Summary: Two-dimension almost-Riemannian structures of step 2 are generalizations of the Grushin plane, where the singular set Z acts as a 1D embedded submanifold causing Riemannian quantities to diverge when approached. Geodesics can cross the singular set without singularities, unlike the heat and Schrodinger equation solutions. Quantum confinement occurs due to Laplace-Beltrami operator being essentially self-adjoint on a connected component without the singular set, which is not observed for the curvature Laplacian operator -Delta + cK in this study.
POTENTIAL ANALYSIS
(2021)
Article
Physics, Multidisciplinary
E. Pozzoli, M. Leibscher, M. Sigalotti, U. Boscain, C. P. Koch
Summary: We propose an analytical approach to construct the Lie algebra of finite-dimensional subsystems of the driven asymmetric top rotor. We determine the nested commutators between drift and drive Hamiltonians using a graph representation for a given rotational excitation, and generate the Lie algebra for subsystems with arbitrary rotational excitation using an inductive argument.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2022)
Article
Physics, Mathematical
Matteo Gallone, Alessandro Michelangeli, Eugenio Pozzoli
Summary: This article classifies the self-adjoint realizations of the Laplace-Beltrami operator minimally defined on an infinite cylinder equipped with an incomplete Riemannian metric of Grushin type. These realizations can be naturally interpreted as Hamiltonians governing the confinement or transmission of a Schrodinger quantum particle away from a singularity. The authors characterize all physically meaningful extensions with explicit local boundary conditions at the singularity, and identify the most confining and the most transmitting ones.
REVIEWS IN MATHEMATICAL PHYSICS
(2022)
Article
Physics, Multidisciplinary
Monika Leibscher, Eugenio Pozzoli, Cristobal Perez, Melanie Schnell, Mario Sigalotti, Ugo Boscain, Christiane P. Koch
Summary: This study presents a method for complete control of the rotational dynamics in chiral asymmetric top molecules using quantum control theory. It demonstrates the potential for solution of practical quantum control problems by leveraging controllability analysis.
COMMUNICATIONS PHYSICS
(2022)
Article
Automation & Control Systems
Thomas Chambrion, Eugenio Pozzoli
Summary: In this study, we investigated the Schrodinger partial differential equation of a rotating symmetric rigid molecule driven by an electric field. By classifying its approximate controllability and performing numerical simulations, we achieved an orientational selective transfer of rotational population.
IEEE CONTROL SYSTEMS LETTERS
(2022)
Proceedings Paper
Automation & Control Systems
U. Boscain, E. Pozzoli, M. Sigalotti
