Symmetry

The Drinfeld–Sokolov–Wilson (DSW) system originates in the theory of affine Lie algebras and the Korteweg–de Vries hierarchy, so that its integrability reflects an underlying algebraic symmetry that persists in rescaled form under fractionalization, and it also models dispersive shallow-water waves. This work applies the hybrid Aboodh–Temimi–Ansari technique to the time-fractional DSW system under the Caputo derivative, a scheme that requires neither Adomian nor He polynomials. We show that its iterates are exactly the Picard iterates of an equivalent fractional Volterra integral equation. Under a Lipschitz condition on the nonlinear operator, this yields the existence and uniqueness of a local solution by a Banach fixed-point argument, together with explicit bounds on the truncation error. We also show that the dispersive term of the DSW system prevents this condition from holding in the classical spaces of bounded smooth functions, so that the well-posedness of the fractional system itself remains open. Two initial value problems built from known kink and soliton profiles are solved, recovering the exact classical solution when the fractional order equals one. After two corrections, the approximations agree with the exact profiles to several significant figures, with the residual error concentrated where the nonlinear coupling is largest.

Autonomous underwater vehicles (AUVs) have been extensively employed in diverse marine applications, where depth-tracking control remains a critical technical challenge. Although previous studies have investigated various guidance law designs for underactuated AUVs, compensation for the attack-angle effect induced by heave velocity remains underexplored, particularly in the presence of extreme time-varying external disturbances and sharp path curvature variations. Existing line-of-sight (LOS) based methods, including integral LOS (ILOS) and adaptive LOS (ALOS), rely solely on the integral action of the pitch-tracking error to estimate the attack angle, which inevitably introduces inherent time-delay characteristics and induces overshoot and chattering in the depth-tracking response. Given that a system’s position and its velocity are uniquely determined by its acceleration, a response-tunable adaptive LOS (RTALOS) guidance design is proposed for AUV depth-tracking control task. This guidance law incorporates both the depth-tracking error and its velocity term as boundary constraints, generating an acceleration reference through a time-based planning polynomial to achieve accurate attack-angle estimation. The proposed method redefines attack-angle estimation as a new paradigm based on a second-order depth-tracking kinematic system, where the introduced velocity term acts as an adjustable damping effect to effectively suppress the depth-tracking overshoot and oscillation. Simulations were conducted to verify that the proposed guidance design could make the depth-tracking motion converge within a given time duration if the state-constrained acceleration was promptly executed. Compared to the selected ILOS and ALOS methods, the proposed RTALOS method can rapidly and accurately compensate for time-varying attack-angle effects via the estimation of attack angle, resulting in precise depth-tracking control performance, even under severe time-varying disturbances and when the pitch-tracking control performance is unsatisfactory.

This paper investigates the explicit solvability of the congruence equation . By exploiting properties of repdigit Niven numbers, we introduce several infinite families of primes with the square-free property and memory, thereby providing infinitely many explicit solutions to the congruence. Furthermore, we construct infinitely many new explicit Niven numbers. The results can be extended to the most general congruence equations for arbitrary coprime integers a and b. The results reveal a recursive structure linking the equation to prime factorizations of repunits, and we propose several conjectures concerning the square-free property of the arising prime sets. Beyond pure number-theoretic value, the derived hierarchical prime families and modular periodic structures may have broad potential physical application including discrete quantum circuit design, quasi-crystal diffraction spectrum modeling, periodic photonic grating engineering and so on.