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Infinitely many nonlocal symmetries and nonlocal conservation laws of the integrable modified KdV-sine-Gordon equation

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Communications in Theoretical Physics 2021年第5期73卷 18-24页

作者： Zu-feng Liang Xiao-yan Tang Wei Ding Department of Physics Hangzhou Normal UniversityHangzhou 10036China School of Mathematical Sciences Shanghai Key Laboratory of PMMPEast China Normal UniversityShanghai 200062China School of Computer Science and Technology Shanghai Normal UniversityShanghai 200234China

nonlocal symmetries related to the B?cklund transformation(BT)for the modified KdV-sine-Gordon(mKdV-SG)equation are obtained by requiring the mKdV-SG equation and its BT form invariant under the infinitesimal *** through the parameter expansion procedure,an infinite number of new nonlocal symmetries and new nonlocal conservation laws related to the nonlocal symmetries are ***,several new finite and infinite dimensional nonlinear systems are presented by applying the nonlocal symmetries as symmetry constraint conditions on the BT.

关键词： nonlocal conservation law The modified KdV-sine-Gordon equation nonlocal symmetry

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THE ONE-DIMENSIONAL HUGHES MODEL FOR PEDESTRIAN FLOW:RIEMANN-TYPE SOLUTIONS

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Acta Mathematica Scientia 2012年第1期32卷 259-280页

作者： Debora Amadori M.Di Francesco Dipartimento di Matematica Pura e Applicata Universita degli Studi dell'Aquila Departament de Matematiques Universitat Autonoma de Barcelona

This paper deals with a coupled system consisting of a scalar conservation law and an eikonal equation, called the Hughes model. Introduced in [24], this model attempts to describe the motion of pedestrians in a densely crowded region, in which they are seen as a ＇thinking＇（continuum） fluid. The main mathematical difficulty is the discontinuous gradient of the solution to the eikonal equation appearing in the flux of the conservation law. On a one dimensional interval with zero Dirichlet conditions （the two edges of the interval are interpreted as ＇targets＇）, the model can be decoupled in a way to consider two classical conservation laws on two sub-domains separated by a turning point at which the pedestrians change their direction. We shall consider solutions with a possible jump discontinuity around the turning point. For simplicity, we shall assume they are locally constant on both sides of the discontinuity. We provide a detailed description of the local- in-time behavior of the solution in terms of a ＇global＇ qualitative property of the pedestrian density （that we call ＇relative evacuation rate＇）, which can be interpreted as the attitude of the pedestrians to direct towards the left or the right target. We complement our result with explicitly computable examples.

关键词： pedestrian flow nonlocal conservation law eikonal equation

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