المؤلفون: Traore, Diata¹ (AUTHOR) dtraore@lct.jussieu.fr, Giner, Emmanuel¹ (AUTHOR) emmanuel.giner@lct.jussieu.fr, Toulouse, Julien^1,2 (AUTHOR) toulouse@lct.jussieu.fr

المصدر: Journal of Chemical Physics. 6/21/2023, Vol. 158 Issue 23, p1-9. 9p.

مصطلحات موضوعية: *ELECTRON-electron interactions, *WAVE equation, *PROOF of concept, *ELECTRON configuration

مستخلص: The basis-set correction method based on density-functional theory consists in correcting the energy calculated by a wave-function method with a given basis set by a density functional. This basis-set correction density functional incorporates the short-range electron correlation effects missing in the basis set. This results in accelerated basis convergences of ground-state energies to the complete-basis-set limit. In this work, we extend the basis-set correction method to a linear-response formalism for calculating excited-state energies. We give the general linear-response equations as well as the more specific equations for configuration-interaction wave functions. As a proof of concept, we apply this approach to the calculations of excited-state energies in a one-dimensional two-electron model system with harmonic potential and a Dirac-delta electron–electron interaction. The results obtained with full-configuration-interaction wave functions expanded in a basis of Hermite functions and a local-density-approximation basis-set correction functional show that the present approach does not help in accelerating the basis convergence of excitation energies. However, we show that it significantly accelerates basis convergences of excited-state total energies. [ABSTRACT FROM AUTHOR]

المؤلفون: Li, Guopeng^1,2 (AUTHOR) guopeng.li@ed.ac.uk

المصدر: Nonlinearity. Jul2024, Vol. 37 Issue 7, p1-44. 44p.

مصطلحات موضوعية: *WAVE equation, *WATER depth, *CIRCLE, *KORTEWEG-de Vries equation

مستخلص: In this paper, we study the low regularity convergence problem for the intermediate long wave equation (ILW), with respect to the depth parameter δ > 0, on the real line and the circle. As a natural bridge between the Korteweg–de Vries (KdV) and the Benjamin–Ono (BO) equations, the ILW equation is of physical interest. We prove that the solutions of ILW converge in the Hs -Sobolev space for s > 1 2 , to those of BO in the deep-water limit (as δ → ∞ ), and to those of KdV in the shallow-water limit (as δ → 0). This improves previous convergence results by Abdelouhab et al (1989 Physica D 40 360–92), which required s > 3 2 in the deep-water limit and s ⩾ 2 in the shallow-water limit. Moreover, the convergence results also apply to the generalised ILW equation, i.e. with nonlinearity ∂ x (u k) for k ⩾ 2 . Furthermore, this work gives the first convergence results of generalised ILW solutions on the circle with regularity s ⩾ 3 4 . Overall, this study provides mathematical insights for the behaviour of the ILW equation and its solutions in different water depths, and has implications for predicting and modelling wave behaviour in various environments. [ABSTRACT FROM AUTHOR]

المؤلفون: Mohammed, Aili¹ (AUTHOR), Khemmoudj, Ammar¹ (AUTHOR) akhemmoudj@usthb.dz

المصدر: International Journal of Control. Jul2024, Vol. 97 Issue 7, p1612-1626. 15p.

مصطلحات موضوعية: *WAVE equation, *CONVEX functions, *EQUATIONS, *DELAY differential equations

مستخلص: In this paper, we have analysed the influence of viscoelastic and frictional damping on the decay rate of solutions for a Kirchhoff-type viscoelastic wave equation with a distributed delay acting on nonlinear internal damping. Taking the relaxation function of a fairly large class and using the method of energy in which we introduce an adapted Lyapunov functional and by exploiting certain properties of convex functions, under certain assumptions on the constants of system, we obtain the optimal decay rate of energy in the sense that it is compatible with the decay rate of the relaxation function. [ABSTRACT FROM AUTHOR]

المؤلفون: Ye, Zilong^1,2 (AUTHOR), Huang, Jianping^1,2 (AUTHOR) jphuang@upc.edu.cn, Mu, Xinru^1,2 (AUTHOR), Mao, Qiang^1,2 (AUTHOR)

المصدر: Geophysical Prospecting. Jul2024, Vol. 72 Issue 6, p2109-2122. 14p.

مصطلحات موضوعية: *WAVE equation, *FINITE difference method, *TIME management, *SEISMIC waves, *DISPERSION relations

مستخلص: Seismic waves propagating through attenuating media induce amplitude loss and phase dispersion. Neglecting the attenuation effects during seismic processing results in the imaging profiles with weakened energy, mispositioned interfaces and reduced resolution. To obtain high‐quality imaging results, Q‐compensated reverse time migration is developed. The kernel of the Q‐compensated reverse time migration algorithm is a viscoacoustic wave equation with decoupled amplitude loss and phase dispersion terms. However, the majority of current decoupled viscoacoustic wave equations are solved using the computationally expensive pseudo‐spectral method. To enhance computational efficiency, we initiate our approach from the dispersion relation of a single standard linear solid model. Subsequently, we derive a novel decoupled viscoacoustic wave equation by separating the amplitude loss and phase dispersion terms, previously coupled in the memory variable. The newly derived decoupled viscoacoustic wave equation can be efficiently solved using the finite‐difference method. Then, we reverse the sign of the amplitude loss term of the newly derived viscoacoustic wave equation to implement high‐efficient Q‐compensated reverse time migration based on the finite‐difference method. In addition, we design a regularization term to suppress the high‐frequency noise for stabilizing the wavefield extrapolation. Forward modelling tests validate the decoupled amplitude loss and phase dispersion characteristics of the newly derived viscoacoustic wave equation. Numerical examples in both two‐dimensional and three‐dimensional confirm the effectiveness of the Q‐compensated reverse time migration based on the finite‐difference algorithm in mitigating the attenuation effects in subsurface media and providing high‐quality imaging results. [ABSTRACT FROM AUTHOR]

المؤلفون: Xu, Chengqiang¹ (AUTHOR), Wang, Yibo¹ (AUTHOR), Cao, Wanrong¹ (AUTHOR) wrcao@seu.edu.cn

المصدر: Numerical Methods for Partial Differential Equations. Jul2024, Vol. 40 Issue 4, p1-32. 32p.

مصطلحات موضوعية: *WAVE equation, *EULER method, *FINITE element method, *WHITE noise

مستخلص: This article develops an efficient fully discrete scheme for a stochastic strongly damped wave equation (SSDWE) driven by an additive noise and presents its error estimates in the strong sense. We use the truncated spectral expansion of the noise to get an approximate equation and prove its regularity. Then we establish a spatio‐temporal discretization of the approximate equation by a finite element method in space and an exponential trapezoidal scheme in time. We prove that the combination can derive higher strong convergence order in time than the use of the piecewise approximation of the noise and the exponential Euler scheme or the implicit Euler scheme in time. Particularly, the temporal strong convergence order of the fully discrete scheme reaches 5/4−ε$$ 5/4-\varepsilon $$ for the one‐dimensional space‐time white noise, which overcomes the order barrier one. Moreover, we allow the covariance operator of the noise to be noncommutative with the Dirichlet Laplacian, which weakens the common assumptions on the noise in the literature. Finally, some numerical experiments in different spatial dimensions are presented to support our theoretical findings. By means of the piecewise spectral approximation of the noise, a piecewise version of the fully discrete scheme is constructed to fulfill a long‐time simulation. [ABSTRACT FROM AUTHOR]

المؤلفون: Jiang, Hao¹, Qi, Xiao-Qiu¹ zjst_qi@163.com

المصدر: Modern Physics Letters A. Jun2024, p1. 10p. 2 Illustrations.

مصطلحات موضوعية: *HAWKING radiation, *WAVE equation, *ANALYTICAL solutions, *BLACK holes

مستخلص: In this paper, we investigate the properties of a five-dimensional Ricci-flat black hole within the framework of the Space–Time–Matter theory. By employing the Klein–Gordon equation, we derive the wave function equations for both the mass dimensional coordinate and the radial dimensional coordinate. We present the analytical solutions for the wave function on the mass dimensional coordinate under two distinct scenarios. Remarkably, we observe the absence of quantum effects on the mass dimensional coordinate, which aligns with the absence of a barrier in the wave function equation. For the radial component, we employ the B-spline method to numerically solve its wave function. Our findings reveal the existence of Hawking radiation between the event horizon and the cosmological horizon, with the transmission and reflection probabilities being determined by the presence of a barrier. Furthermore, we discover a significantly higher probability of particle detection near the cosmological horizon, while the probability near the event horizon approaches zero. [ABSTRACT FROM AUTHOR]

المؤلفون: Wang, Shuo¹ (AUTHOR) shuowang@mail.sdu.edu.cn, Zheng, Xiangcheng¹ (AUTHOR) xzheng@sdu.edu.cn, Du, Ning¹ (AUTHOR) duning@sdu.edu.cn

المصدر: Computers & Mathematics with Applications. Jun2024, Vol. 164, p45-66. 22p.

مصطلحات موضوعية: *FINITE element method, *WAVE equation, *OPTIMAL control theory

مستخلص: In this paper, we consider a finite element approximation of an optimal control problem governed by a time fractional wave equation. We first prove the well-posedness and regularity of the optimal control problem. Furthermore, we discuss the fully discrete scheme and analyze its stability and a priori error estimate. Numerical experiments are given to illustrate the theoretical findings. • The weak solution to the time fractional wave equation is studied under low regular data. • The well-posedness and regularity of the optimal control problem are proven. • Numerical schemes exist the first-order and second-order accuracy in time and space respectively. [ABSTRACT FROM AUTHOR]

المؤلفون: Yamazaki, Yohei^1,2,3,4,5 (AUTHOR) yamazaki.yohei.557@m.kyushu-u.ac.jp

المصدر: Journal of Dynamics & Differential Equations. Jun2024, Vol. 36 Issue 2, p871-914. 44p.

مصطلحات موضوعية: *WAVE equation, *MATHEMATICS

مستخلص: In this paper, we construct center stable manifolds of unstable line solitary waves for the Zakharov–Kuznetsov equation on R × T L and show the orbital stability of the unstable line solitary waves on the center stable manifolds, which yields the asymptotic stability of unstable solitary waves on the center stable manifolds near by stable line solitary waves. The construction is based on the graph transform approach by Nakanishi and Schlag (SIAM J Math Anal 44:1175–1210, 2012). Applying the bilinear estimate on Fourier restriction spaces by Molinet and Pilod (Ann Inst H Poincaré Anal Non Lineaire 32:347–371, 2015) and modifying the mobile distance in Nakanishi and Schlag (2012), we construct a contraction map on the graph space. [ABSTRACT FROM AUTHOR]

المؤلفون: Goldberg, Gabriel¹ ggoldberg@berkeley.edu, Schlutzenberg, Farmer² schlutze@uni-muenster.de

المصدر: Journal of the European Mathematical Society (EMS Publishing). 2024, Vol. 26 Issue 6, p2091-2126. 36p.

مصطلحات موضوعية: *GIBBS' equation, *WAVE equation, *NONLINEAR analysis, *MATHEMATICAL analysis, *DIFFERENTIAL equations

مستخلص: We investigate the structure of rank-to-rank elementary embeddings at successor rank, working in ZF set theory without the Axiom of Choice. Recall that the set-theoretic universe is naturally stratified by the cumulative hierarchy, whose levels V α are defined via iterated application of the power set operation, starting from V 0 =∅, setting V α+1 =P(V α), and taking unions at limit stages. Assuming that j:V α+1 →V α+1 is a (non-trivial) elementary embedding, we show that V α is fundamentally different from V α+1: we show that j is definable from parameters over V α+1 iff α+1 is an odd ordinal. The definability is uniform in odd α+1 and j. We also give a characterization of elementary j:Vα+2 → Vα+2 in terms of ultrapower maps via certain ultrafilters. For limit ordinals λ, we prove that if j:Vλ→Vλ is Σ1-elementary, then j is not definable over Vλ from parameters, and if β < λ and j:Vβ → Vλ is fully elementary and ∈-cofinal, then j is likewise not definable. If there is a Reinhardt cardinal, then for all sufficiently large ordinals α, there is indeed an elementary j: Vα → Vα, and therefore the cumulative hierarchy is eventually periodic (with period 2). [ABSTRACT FROM AUTHOR]

المؤلفون: Bringmann, Bjoern¹ bjoern@ias.edu

المصدر: Journal of the European Mathematical Society (EMS Publishing). 2024, Vol. 26 Issue 6, p1933-2089. 157p.

مصطلحات موضوعية: *GIBBS' equation, *WAVE equation, *NONLINEAR analysis, *MATHEMATICAL analysis, *DIFFERENTIAL equations

مستخلص: In this two-paper series, we prove the invariance of the Gibbs measure for a threedimensional wave equation with a Hartree nonlinearity. The novelty lies in the singularity of the Gibbs measure with respect to the Gaussian free field. In this paper, we focus on the dynamical aspects of our main result. The local theory is based on a paracontrolled approach, which combines ingredients from dispersive equations, harmonic analysis, and random matrix theory. The main contribution, however, lies in the global theory. We develop a new globalization argument, which addresses the singularity of the Gibbs measure and its consequences. [ABSTRACT FROM AUTHOR]