arXiv subjects

Ailton C. Nascimento

Publications and source records attributed to Ailton C. Nascimento.

3 recordsLinked to original sources

arXiv2026

Explicit Analytical Representations for the Schwarzschild Radial Equation via Hypergeometric and Frobenius Expansions

The massive Klein-Gordon equation on the Schwarzschild exterior reduces to a radial differential equation of confluent-Heun type. We present its exact reduction by rigorously retaining a subleading centrifugal term that preserves the horizon indicial exponents but critically shifts the accessory parameters. Within the Svartholm--Schmidt hypergeometric-expansion framework, we derive the three-term recurrence relation and establish the corrected continued-fraction compatibility condition matching the minimal tail to the lower-end recurrence. Crucially, we demonstrate that the physical compactification coordinate $z=(r-1)/r$ shifts the irregular singular point to $z=1$, inherently transforming the hypergeometric expansion into a five-term recurrence that cannot degenerate to three terms. To circumvent this obstruction, we construct the horizon-normalized physical branch via a direct Frobenius series at $z=0$. This representation exhibits high-order convergence and near-precision-floor residuals against the exact radial ODE, while clarifying why quasinormal-mode spectral selection remains a connection problem at the irregular singular endpoint.

gr-qc↗

arXiv2026

On the local well-posedness of the Benjamin-Ono-Zakharov-Kuznetsov equation

We study the Cauchy problem for the Benjamin--Ono--Zakharov--Kuznetsov equation on $\mathbb R^2$. Following the strategy of Kenig and Ziesler, we establish new maximal-function estimates adapted to the BO--ZK equation and use them to implement the Kenig--Koenig method. As a result, we improve the best previously known isotropic result of Nascimento (2020), lowering the local well-posedness threshold from $s>5/4$ to $s>19/16$. At the BO--ZK endpoint, the resulting isotropic data class also contains the anisotropic $E^{5/4+}$ class of the preceding theory. On bounded subsets of $H^s(\mathbb R^2)$, the lifespan may be chosen so that $ T\gtrsim_s \bigl(1+\norm{u_0}_{H^s}\bigr)^{-8}. $ The proof combines a sharp dyadic mixed maximal-function estimate with an anisotropic local-smoothing mechanism that exploits the complementary behavior of the longitudinal and transverse group velocities. In particular, transverse dispersion compensates for the degeneration of longitudinal smoothing near the characteristic region. Together with refined short-time Strichartz estimates and a modified energy argument, these ingredients close the nonlinear estimates at the stated regularity. Existence, uniqueness, and continuous dependence on the initial data are then established in the corresponding solution class. The resulting threshold reflects the present optimization of the method and is not claimed to be sharp.

math.AP↗

arXiv2026

Unique Continuation for Fifth-Order KP Equation and its application to control problems

We develop a framework for the fifth-order Kadomtsev--Petviashvili equation on $\mathbb{T}_x \times \mathbb{R}_y$ within a mean-zero KP-adapted Sobolev scale. A localized high-order feedback acting on the periodic variable yields a $5/2$--derivative gain in suitable space--time norms, leading to propagation of regularity and a unique continuation property for the linear dynamics. As a consequence, we derive an observability inequality for the adjoint system and establish exponential stabilization of the nonlinear closed-loop equation: for small initial data in $X_{s,0}$, $s>2$, solutions are global and decay exponentially in $X_s$. Combining observability with the Hilbert Uniqueness Method and a fixed-point argument, we obtain local exact controllability near the origin, with $L^2$ controls supported in the feedback region and cost linear in the data size. The analysis relies on a novel combination of unique continuation, frequency grouping, and the one-sided Fourier vanishing mechanism introduced for the Benjamin--Ono equation by Linares and Rosier in \textit{Trans. Amer. Math. Soc.} (2015)~\cite{LR}, here extended to the fifth-order Kadomtsev--Petviashvili equation.

math.AP↗