The modified Zakharov–Kuznetsov (mZK) equation governs weakly nonlinear ion-acoustic and dust-acoustic waves in a magnetized plasma whose quadratic nonlinearity is suppressed by the background composition. Writing B and C for its parallel and transverse dispersion coefficients, we show that in the obliquely propagating sector ζ =x+my , where m=tanθ measures the obliqueness of the wave normal, the anisotropic mZK equation collapses exactly onto the focusing modified Korteweg–de Vries equation. The two dispersion coefficients enter only through the single effective dispersion D=B+Cm^2 , and the magnetic field only through C/B=1+Ω^-2 , with Ω the ion-cyclotron frequency normalized to the ion plasma frequency. This places the inverse-scattering apparatus at our disposal. We construct the 2× 2 Ablowitz–Kaup–Newell–Segur Lax pair, verify the zero-curvature condition, and establish the analyticity and symmetry properties of the Jost solutions. A matrix Riemann–Hilbert problem is posed on the real spectral axis, its jump matrix derived explicitly and its solution shown to be unique by a vanishing lemma. In the reflectionless case the problem reduces to a finite algebraic system and the N -soliton family is written through a single complex τ -function. Purely imaginary eigenvalues give oblique line solitons, conjugate pairs give breathers, and the coalescence of a conjugate pair gives a degenerate double-pole structure whose peak transiently doubles the soliton amplitude. At fixed amplitude, obliqueness and weak magnetization broaden a structure as √(D) while leaving its front speed unchanged.
We examine the consistency requirements for static, spherically symmetric black holes in the Horndeski subclass G_2=η X+2Λ , G_4=(16π G)^-1+βϕ ^2 , and G_3=G_5=0 . A scalar field linear in time is compatible with a static metric in some shift-symmetric theories, but that construction cannot be transferred automatically to a coupling with explicit scalar dependence. Requiring the full metric equations to hold at every time forces a Ricci-flat metric when β q^2 0 ; the stated asymptotically flat scalar boundary conditions then conflict with a nonzero q for η 0 and Λ =0 . We correct the variational equations and horizon boundary conditions, formulate radial and non-radial null-convergence diagnostics, and distinguish a test-scalar wave equation from a coupled gravitational stability calculation. We also derive the photon-sphere and innermost stable circular orbit diagnostics and explain why percent-level geometric differences alone do not establish detectability. The previously reported hairy branch, mode frequencies, and shadow and orbital shifts require recalculation and are not used to claim existence, linear stability, or observational distinguishability.
In this paper, we present an analytical framework for variable-step discrete-time quantum walks (DTQWs), focusing on their complexity characteristics and resource scaling. Focusing on the exponentially decaying step sequence framework, we analyze how the walker’s probability distribution forms a binary tree (BT) topology in the position space, thereby realizing super-ballistic diffusion. Through analytical derivations and numerical simulations, we verify that the time complexity for covering a d-dimensional Hilbert space is reduced from the linear O(d) to the logarithmic O(log d) , and confirm that the resource consumption of non-identity coin operations follows the linear O(d) scaling. These results provide a mathematically solid verification for the efficiency of variable-step quantum walks, and also lay a clear theoretical foundation for their application in high-dimensional quantum information processing.
This work is devoted to study the topological classification of a Kerr-Newman-de Sitter (KNdS) black hole (BH) in bumblebee gravity (BBG) subject to changes in various parameters. Based on generalized free energy techniques coupled with Duan’s ϕ mapping topological current theory, we find that BH can be separated into different topological classes. Interestingly, our result demonstrates that the existence of a number of horizons in dS spacetime, in particular, we find that both the event horizon and the cosmological horizon carry two zero points with opposite winding numbers, so that each horizon has vanishing total topological number, W_h=W_c=0 . The winding-number assignment of the two cosmological-horizon defects is reversed relative to the event horizon, and this classification is unchanged by the electric charge Q and the Lorentz-violation parameter L.
An exact analytical solution of the nonstationary Schrödinger equation is presented for the three-atom multiphoton Tavis-Cummings model with a vacuum cavity field, and is extended to a thermal field of arbitrary intensity via thermal averaging. The solution is valid for an arbitrary pure initial atomic state and arbitrary photon transition multiplicity. Based on this solution, analytical formulas for two measures of quantum correlations, pairwise negativity and symmetric quantum discord, are derived for two-atom density matrices of X-form. A numerical analysis is performed for a wide class of initial states: biseparable, W- and GHZ-type states, in both vacuum and thermal cases. The photon transition multiplicity serves as an effective control parameter for selective generation of quantum correlations. Increasing this parameter can symmetrize quantum and classical correlation dynamics, stabilize their behavior, and suppress both sudden death of pairwise entanglement and switching of the optimal projective measurement. For certain initial states, analytical expressions for the sudden death of entanglement time intervals are obtained, in which the symmetric quantum discord remains nonzero. In the thermal case, the symmetric quantum discord either stabilizes, grows with mean photon number for the GHZ-state, or is suppressed for initially uncorrelated pairs, while the pairwise negativity vanishes. The thermal field can both simplify and complicate the optimal measurement strategy, depending on the initial atomic state. The results can be applied in quantum information processing protocols requiring controlled generation of quantum correlations, and establish limits on the times of three-atom operations in the considered cavity quantum electrodynamics model.
