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Keywords = inhomogeneous conservation laws

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15 pages, 2360 KiB  
Article
Analytic Investigation of a Generalized Variable-Coefficient KdV Equation with External-Force Term
by Gongxun Li, Zhiyan Wang, Ke Wang, Nianqin Jiang and Guangmei Wei
Mathematics 2025, 13(10), 1642; https://doi.org/10.3390/math13101642 - 17 May 2025
Viewed by 320
Abstract
This paper investigates integrable properties of a generalized variable-coefficient Korteweg–de Vries (gvcKdV) equation incorporating dissipation, inhomogeneous media, and an external-force term. Based on Painlevé analysis, sufficient and necessary conditions for the equation’s Painlevé integrability are obtained. Under specific integrability conditions, the Lax pair [...] Read more.
This paper investigates integrable properties of a generalized variable-coefficient Korteweg–de Vries (gvcKdV) equation incorporating dissipation, inhomogeneous media, and an external-force term. Based on Painlevé analysis, sufficient and necessary conditions for the equation’s Painlevé integrability are obtained. Under specific integrability conditions, the Lax pair for this equation is successfully constructed using the extended Ablowitz–Kaup–Newell–Segur system (AKNS system). Furthermore, the Riccati-type Bäcklund transformation (R-BT), Wahlquist–Estabrook-type Bäcklund transformation (WE-BT), and the nonlinear superposition formula are derived. In utilizing these transformations and the formula, explicit one-soliton-like and two-soliton-like solutions are constructed from a seed solution. Moreover, the infinite conservation laws of the equation are systematically derived. Finally, the influence of variable coefficients and the external-force term on the propagation characteristics of a solitory wave is discussed, and soliton interaction is illustrated graphically. Full article
(This article belongs to the Special Issue Research on Applied Partial Differential Equations)
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19 pages, 340 KiB  
Article
On the Convergence of α Schemes with Source Terms for Scalar Convex Conservation Laws
by Nan Jiang
Mathematics 2023, 11(15), 3267; https://doi.org/10.3390/math11153267 - 25 Jul 2023
Viewed by 998
Abstract
In this study, we use an extension of Yang’s convergence criterion [N. Jiang, On the wavewise entropy inequality for high-resolution schemes with source terms II: the fully discrete case] to show the entropy convergence of a class of fully discrete α schemes, [...] Read more.
In this study, we use an extension of Yang’s convergence criterion [N. Jiang, On the wavewise entropy inequality for high-resolution schemes with source terms II: the fully discrete case] to show the entropy convergence of a class of fully discrete α schemes, now with source terms, for non-homogeneous scalar convex conservation laws in the one-dimensional case. The homogeneous counterparts (HCPs) of these schemes were constructed by S. Osher and S. Chakravarthy in the mid-1980s [A New Class of High Accuracy TVD Schemes for Hyperbolic Conservation Laws (1985), Very High Order Accurate TVD Schemes (1986)], and the entropy convergence of these methods, when m=2, was settled by the author [N. Jiang, The Convergence of α Schemes for Conservation Laws II: Fully-Discrete]. For semi-discrete α schemes, with or without source terms, the entropy convergence of these schemes was previously established (for m=2) by the author [N. Jiang, The Convergence of α Schemes for Conservation Laws I: Semi-Discrete Case]. Full article
(This article belongs to the Special Issue Numerical Approaches for Solving Nonlinear Equations and Systems)
26 pages, 4874 KiB  
Article
Increasing Extractable Work in Small Qubit Landscapes
by Unnati Akhouri, Sarah Shandera and Gaukhar Yesmurzayeva
Entropy 2023, 25(6), 947; https://doi.org/10.3390/e25060947 - 16 Jun 2023
Viewed by 1659
Abstract
An interesting class of physical systems, including those associated with life, demonstrates the ability to hold thermalization at bay and perpetuate states of high free-energy compared to a local environment. In this work we study quantum systems with no external sources or sinks [...] Read more.
An interesting class of physical systems, including those associated with life, demonstrates the ability to hold thermalization at bay and perpetuate states of high free-energy compared to a local environment. In this work we study quantum systems with no external sources or sinks for energy, heat, work, or entropy that allow for high free-energy subsystems to form and persist. We initialize systems of qubits in mixed, uncorrelated states and evolve them subject to a conservation law. We find that four qubits make up the minimal system for which these restricted dynamics and initial conditions allow an increase in extractable work for a subsystem. On landscapes of eight co-evolving qubits, interacting in randomly selected subsystems at each step, we demonstrate that restricted connectivity and an inhomogeneous distribution of initial temperatures both lead to landscapes with longer intervals of increasing extractable work for individual qubits. We demonstrate the role of correlations that develop on the landscape in enabling a positive change in extractable work. Full article
(This article belongs to the Section Non-equilibrium Phenomena)
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14 pages, 434 KiB  
Article
Quantum Potentiality in Inhomogeneous Cosmology
by Andronikos Paliathanasis
Universe 2021, 7(3), 52; https://doi.org/10.3390/universe7030052 - 28 Feb 2021
Cited by 5 | Viewed by 1990
Abstract
For the Szekeres system which describes inhomogeneous and anisotropic spacetimes we make use of a point-like Lagrangian, which describes the evolution of the physical variables of the Szekeres model, in order to perform a canonical quantization and to study the quantum potentiality of [...] Read more.
For the Szekeres system which describes inhomogeneous and anisotropic spacetimes we make use of a point-like Lagrangian, which describes the evolution of the physical variables of the Szekeres model, in order to perform a canonical quantization and to study the quantum potentiality of the Szekeres system in the content of de Broglie–Bohm theory. We revise previous results on the subject and we find that for a specific family of trajectories with initial conditions which satisfy a constraint equation, there exists additional conservation laws for the classical Szekeres system which are used to define differential operators and to solve the Wheeler–DeWitt equation. From the new conservation laws we construct a wave function which provides a nonzero quantum potential term that modifies the Szekeres system. The quantum potential corresponds to new terms in the dynamical system such that new asymptotic solutions with a nonzero energy momentum tensor of an anisotropic fluid exist. Therefore, the silent property of the Szekeres spacetimes is violated by quantum correction terms, which results in the quantum potential adding pressure to the solution. Full article
(This article belongs to the Section Gravitation)
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