arXiv subjects

Claudio Muñoz

Publications and source records attributed to Claudio Muñoz.

At least 19 recordsLinked to original sources

arXiv2026

Decay of solutions of nonlinear Dirac equations: the 2D case

We study the long-time behavior of small solutions for a broad class of 2D Dirac-type equations with suitable nonlinearities. First, we prove that for nonlinearities with power $p\geq 5$ (massless case) and $p\geq7$ (massive case), any small globally bounded radial solution with vorticity $S\ne -1,0$ decays to zero locally in $L^2_{loc}$, as time tends to infinity. For solutions uniformly bounded in time in a weighted $H^1$ space, this decay result extends to lower powers $p\geq 3$ (massless) and $p\geq5$ (massive). Our main results apply to several physical models of current interest, such as the 2D Dirac equation with a honeycomb potential described by Fefferman and Weinstein. Finally, we rule out the existence of small, localized structures such as standing breathers or solitary waves in the 2D regimes considered. To prove these results, we introduce new virial identities with a particular algebra that are applied directly to the Dirac model, and without resorting to the nonlinear Klein-Gordon equation.

math.AP↗

arXiv2026

Pushing the Primordial Frontier: Exact Linear Solutions in Multifield Inflation

We present exact analytic solutions for the linear dynamics of a two-field inflationary system in which the primordial curvature perturbation $ζ$ is coupled to an isocurvature perturbation $σ$ of entropy mass $μ$. The solutions are valid for arbitrary values of $μ$ and the dimensionless interaction strength $λ$, within a quasi-de Sitter background. They therefore provide analytic control over the strong mixing regime in which $ζ$ interacts with light isocurvature fields, commonly associated with rapid-turn inflation. As a first application, we derive the amplitude of the primordial power spectrum in closed form, obtaining an expression that interpolates between the weakly mixed, strongly mixed, light-field, and heavy-field regimes. These results open the way to analytic studies of multifield observables beyond the power spectrum, including non-Gaussianity, particle production, and loop corrections.

astro-ph.CO↗

arXiv2026

Pushing the Primordial Frontier: Cosmological Collider Signatures at Strong Mixing

We develop an analytic treatment of primordial non-Gaussianity in multifield inflation that is nonperturbative in the constant curvature--isocurvature mixing strength $λ$. Using exact linear solutions for the coupled curvature perturbation $ζ$ and isocurvature perturbation $σ$, we construct dressed propagators and derive exact integral representations for the tree-level bispectra generated by the interactions $\dotζ^2σ$, $\dotζ\,σ^2$, and $σ^3$. This formalism resums curvature--isocurvature transfer to all orders in $λ$ and applies for arbitrary values of the entropy mass $μ$. We obtain closed-form expressions for the leading squeezed limit of the bispectrum contributions and recover the well-known cosmological-collider and quasi-single-field results in the weak-mixing limit. In the strong-mixing regime, where conventional transfer perturbation theory breaks down, we find that both the power spectrum and the bispectrum can be dramatically enhanced, giving rise to distinctive nonperturbative scaling laws for the reduced non-Gaussian amplitude. Together, these results open a new analytic window onto multifield inflation beyond the weak-coupling approximation and establish a framework for studying cosmological-collider signals, primordial-black-hole production, and loop corrections in strongly mixed inflationary dynamics.

astro-ph.CO↗

arXiv2026

Kink dynamics for the Yang-Mills field in an extremal Reissner-Nordström black hole

Considered in this work is the Yang-Mills field in an extremal Reissner-Nordström black hole, a physically motivated mathematical model introduced by Bizoń and Kahl. The kink is a fundamental, strongly unstable stationary solution in this non-perturbative, variable coefficients model, with a polynomial tail and no explicit form. In this paper, we introduce and extend several virial techniques, adapt them to the inhomogeneous medium setting, and construct a finite codimensional manifold of the energy space where the kink is asymptotically stable. In particular, we handle, using virial techniques, the emergence of a weak threshold resonance in the description of the stable manifold.

math.AP↗

arXiv2026

Neural Discovery of Strichartz Extremizers

Strichartz inequalities are a cornerstone of the modern theory of dispersive PDEs, but their extremizers are known explicitly only in a handful of sharp cases. The non-convexity of the underlying functional makes the problem hard, and to our knowledge no systematic numerical attack has been attempted. We propose a simple neural-network-based pipeline that searches for extremizers as critical points of the Strichartz ratio, and apply it in three settings. First, on the Schrödinger group we recover the Gaussian extremizers of Foschi and Hundertmark--Zharnitsky in dimensions $d=1,2$ to within $10^{-3}$ relative error, with no analytical prior. Second, on $59$ further admissible pairs in $d=1$ where the answer is conjectural, the method consistently finds Gaussians, supporting the conjecture that Gaussians are the universal extremizers in the admissible range. Third, on the critical Airy--Strichartz inequality at $γ=1/q$, where existence is open, the optimization does not converge to any $L^2$ profile: instead, the iterates organize themselves as mKdV breathers $B(0,\cdot;α,1,0,0)$ with growing internal frequency $α$, and the discovered ratio approaches the Frank--Sabin universal lower bound $\widetilde A_{q,r}$ from below with a power-law gap $\simα^{-0.9}$. We confirm the same picture with an independent Hermite-basis ansatz. We propose a precise conjecture: the supremum equals $\widetilde A_{q,r}$ and is approached, but not attained, along the breather family. The pipeline thus serves both as a validator on known cases and as a discovery tool when no extremizer exists.

math.AP↗

arXiv2026

Global Existence and Long Time Behavior in Einstein-Belinski-Zakharov Soliton Spacetimes

We consider the vacuum Einstein field equations under the Belinski-Zakharov symmetry, which leaves the problem as a 1+1-dimensional quasilinear system of PDEs. Depending on the chosen signature of the metric, these spacetimes contain most of the well-known special solutions in General Relativity. In this paper, {\color{blue} we consider the case of cosmological metrics, in the Belinsky-Zakharov notation}, and prove global existence of small Belinski-Zakharov spacetimes under a natural nondegeneracy condition. We also construct new energies and virial functionals to provide a description of the energy decay of smooth global cosmological metrics inside the light cone. Finally, some applications are presented in the case {\color{blue} of the particular metrics called} generalized Kasner solitons.

math.AP↗

arXiv2026

Nonlinear Stability of nonsingular solitons of the Principal Chiral Field equation

We consider the Principal Chiral Field model posed in 1+1 dimensions into the Lie group $\text{SL}(2,\mathbb R)$. In this work we show the nonlinear stability of small enough nonsingular solitons. The method of proof involves the use of vector field methods as in a previous work by the second and third authors dealing with the Einstein's field equations under the Belinski-Zakharov formalism, extending for all times the size of suitable null weighted norms of the perturbations at time zero.

math.AP↗

arXiv2026

Error bounds for Physics Informed Neural Networks in Generalized KdV Equations placed on unbounded domains

In this paper we study a rigorous setting for the numerical approximation via deep neural networks of the generalized Korteweg-de Vries (gKdV) model in one dimension, for subcritical and critical nonlinearities, and assuming that the domain is the unbounded real line. The fact that the model is posed on the real line makes the problem difficult from the point of view of learning techniques, since the setting required to model gKdV is structured on intricate oscillatory estimates dating from Kato, Bourgain and Kenig, Ponce and Vega, among others. Therefore, a first task is to adapt the setting of these techniques to the deep learning setting. We shall use a battery of Kenig-Ponce-Vega suitable norms and Physics Informed Neural Networks (PINNs) to describe this approximative scheme, proving rigorous bounds on the approximation for each critical and subcritical gKdV model. We shall use this results to provide clear approximation results in the case of several gKdV nonlinear patterns such as solitons, multi-solitons, breathers, among other solutions.

math.AP↗

arXiv2025

Dynamics of generalized abcd Boussinesq solitary waves under a slowly variable bottom

The Boussinesq $abcd$ system is a 4-parameter set of equations posed in $\mathbb R_t\times\mathbb R_x$, originally derived by Bona, Chen and Saut as first-order 2-wave approximations of the incompressible and irrotational, two-dimensional water wave equations in the shallow water wave regime, in the spirit of the original Boussinesq derivation. Among the various particular regimes, each determined by the values of the parameters $(a, b, c, d)$ appearing in the equations, the \emph{generic} regime is characterized by the conditions $b, d > 0$ and $a, c < 0$. If additionally $b=d$, the $abcd$ system is Hamiltonian. In this paper, we investigate the existence of generalized solitary waves and the corresponding collision problem in the physically relevant \emph{variable bottom regime}, introduced by M.\ Chen. More precisely, the bottom is represented by a smooth space-time dependent function $h=\varepsilon h_0(\varepsilon t,\varepsilon x)$, where $\varepsilon$ is a small parameter and $h_0$ is a fixed smooth profile. This formulation allows for a detailed description of weak long-range interactions and the evolution of the solitary wave without its destruction. We establish this result by constructing a new approximate solution that captures the interaction between the solitary wave and the slowly varying bottom.

math.AP↗

arXiv2025

On the asymptotic dynamics for the $L^2$-supercritical gKDV equation

We study the $L^2$-supercritical generalized Korteweg-de Vries equation (gKdV) with nonlinearities $p>5$. While local well-posedness in $H^1$ is classical, the long-time dynamics in the supercritical regime remains largely unexplored beyond small data global solutions, the construction of multi-solitons for any power and self-similar blow-up near the critical power $p=5$. We develop a unified description of the non-solitonic region for arbitrary $H^1$ solutions, both global and blowing up. Our analysis shows that the asymptotic $L^2$ and $L^p$ dynamics in this region is completely determined by the growth rate of the $L^2$ norm of the gradient (or, equivalently, the critical $H^{s_p}$ norm). In particular, we prove sharp far-field decay on both half-lines and establish normalized local vanishing along sequences of times, with improved estimates in the case of even-power nonlinearities. A key ingredient is a new virial method that compensates for the possible unboundedness of the $H^1$ norm by exploiting the conservation of mass and a careful localization of the nonlinear flux. This yields quantitative versions of decay phenomena previously known only in subcritical settings, and it applies without any smallness or proximity-to-soliton assumptions.

math.AP↗

arXiv2025

A note on two Collatz evolution flows

Two evolution models based on the generalized Collatz operator are introduced. These models are characterized by coefficients $α$ and $β$ in the Collatz dynamics, and are suitably defined. Here, $α=β=1$, and $α=3$, $β=1$ correspond to the Nollatz and classical Collatz operators, respectively. In general, the first evolution model is a continuum, Fourier side based, motivated by the Cubic Szegő operator of Gérard and Grellier. The second evolution considers discrete time derivatives of the Collatz orbits. In this paper we describe the evolution of both models, with particular emphasis on dynamical properties. For the first one, it is proved local and global existence in the space $L^2(\mathbb T)$, and a one-to-one characterization of the existence of nontrivial periodic and unbounded orbits of the Collatz mapping in terms of particular set of solutions of this continuous Collatz flow. For the discrete part, a sort of discrete energy is introduced. This energy has the property of being conserved by the discrete flow. An estimate of each term in this energy is given, proving suitable growth bounds. Finally, the meaning of the discrete time derivative for the generalized Collatz orbits is discussed. It is proved that, except for the Nollatz and Collatz operators, the sum of coefficients related to this discrete time derivative is an increasing sequence in $n$ as the iteration parameter $n$ evolves.

math.AP↗

arXiv2025

Decay of solutions of nonlinear Dirac equations

We study the long-time behavior of small and large solutions to a broad class of nonlinear Dirac-type equations. Our results are classified in 1D massless and massive cases, 3D general and $n$ dimensional in generality. In the 1D massless case we prove that any globally defined solution converges to zero as time tends to infinity, within a spatial region expanding at a rate proportional to $ t \log^{-2} t$. This result holds without assumptions on the smallness of initial data or specific power of nonlinearity, ruling out the existence of standing breather-like or solitary wave structures in this regime. In the 1D massive case, solitary waves are known to exist. Introducing new virial identities adapted to the Dirac's distinctive algebra, we prove that there are ``holomorphic'' odd nonlinearities under which globally defined small odd solutions decay to zero on spatial compact sets as time tends to infinity. This result is extended to the 3D case under boundedness of the $H^1$ norm but without requiring the parity condition on the data, giving decay proofs for an important class of nonlinear Dirac models, and opening the door to the future use of virial identities to prove asymptotic stability of well-chosen Dirac solitary waves. Finally, in higher dimensions $ n \geq 1$, we prove the $L^2$ decay for global solutions of nonlinear Dirac equations in the ``exterior light-cone'' region. This confirms the non-existence of breathers and other solutions propagating faster than the speed of light. Our proofs rely on carefully constructed weighted virial identities.

math.AP↗

arXiv2025

Decay of small energy solutions in the ABCD Boussinesq model under the influence of an uneven bottom

The $abcd$ Boussinesq system, introduced by Bona, Chen, and Saut, describes a four-parameter $(a,b,c,d)$ family of models formulated on the time-space domain $\mathbb{R}_t \times \mathbb{R}_x$. It serves as a first-order two-wave approximation to the two-dimensional incompressible, irrotational water wave equations in shallow water, inspired by Boussinesq's classical derivation. Within the different parameter regimes, the generic regime is described by $b,d>0$ and $a,c<0$ while the system becomes Hamiltonian when $b=d$. Previously, sharp local in space $H^1\times H^1$ decay properties were proved in the case of a large class of $abcd$ model under the small data assumption. In this paper, we generalize [C. Kwak, \emph{et. al.}, \emph{The scattering problem for Hamiltonian ABCD Boussinesq systems in the energy space}. J. Math. Pures Appl. (9) 127 (2019), 121--159] by considering the small data $abcd$ decay problem in the physically relevant \emph{variable bottom regime} described by M. Chen. The nontrivial bathymetry is represented by a smooth space-time dependent function $h=h(t,x)$, which obeys integrability in time and smallness in space. We prove first the existence of small global solutions in $H^1\times H^1$. Then, for a sharp set of dispersive $abcd$ systems (characterized only in terms of parameters $a, b$ and $c$), every $H^1\times H^1$ small solution must converges to zero inside of the light cone $|x|\leq |t|$.

math.AP↗

arXiv2025

On uniqueness of KP soliton structures

We consider the Kadomtsev-Petviashvili II (KP) model placed in $\mathbb R_t \times \mathbb R_{x,y}^2$, in the case of smooth data that are not necessarily in a Sobolev space. In this paper, the subclass of smooth solutions we study is of ``soliton type'', characterized by a phase $Θ=Θ(t,x,y)$ and a unidimensional profile $F$. In particular, every classical KP soliton and multi-soliton falls into this category with suitable $Θ$ and $F$. We establish concrete characterizations of KP solitons by means of a natural set of nonlinear differential equations and inclusions of functionals of Wronskian, Airy and Heat types, among others. These functional equations only depend on the new variables $Θ$ and $F$. A distinct characteristic of this set of functionals is its special and rigid structure tailored to the considered soliton. By analyzing $Θ$ and $F$, we establish the uniqueness of line-solitons, multi-solitons, and other degenerate solutions among a large class of KP solutions. Our results are also valid for other 2D dispersive models such as the quadratic and cubic Zakharov-Kuznetsov equations.

math.AP↗

arXiv2025

Error bounds for Physics Informed Neural Networks in Nonlinear Schrödinger equations placed on unbounded domains

We consider the subcritical nonlinear Schrödinger (NLS) in dimension one posed on the unbounded real line. Several previous works have considered the deep neural network approximation of NLS solutions from the numerical and theoretical point of view in the case of bounded domains. In this paper, we introduce a new PINNs method to treat the case of unbounded domains and show rigorous bounds on the associated approximation error in terms of the energy and Strichartz norms, provided a reasonable integration scheme is available. Applications to traveling waves, breathers and solitons, as well as numerical experiments confirming the validity of the approximation are also presented as well.

math.AP↗

arXiv2025

On asymptotic stability of stable Good Boussinesq solitary waves

We consider the generalized Good-Boussinesq (GB) model in one dimension, with subcritical power nonlinearity $1 \frac{p-1}{4}$, Bona and Sachs showed the orbital stability of such waves. Previously, one of us proved that unstable GB standing waves can be perturbed with particular odd-even data in a suitable submanifold of the energy space, leading to the asymptotic stability property if $p\ge 2$. In this paper we prove that stable GB solitary waves are asymptotically stable in the case of general initial data placed in the energy space for any $p\ge 2$ and speeds $|c|>c_+(p)\geq \sqrt{\frac{p-1}{4}}$. The proof involves the introduction of a new set of virial estimates specifically adapted to the GB system in a moving setting. In particular, a new virial estimate with mixed variables is considered to treat arbitrary scaling and shift modulations. Another new ingredient is the understanding the corresponding linear matrix operator under mixed orthogonality conditions, a feature absent in our previous works.

math.AP↗

arXiv2024

Interaction of jamitons in second-order macroscopic traffic models

Jamitons are self-sustained traveling wave solutions that arise in certain second-order macroscopic models of vehicular traffic. A necessary condition for a jamiton to appear is that the local traffic density breaks the so-called sub-characteristic condition. This condition states that the characteristic velocity of the corresponding first-order Lighthill-Whitham-Richards (LWR) model formed with the same desired speed function is enclosed by the characteristic speeds of the corresponding second-order model. The phenomenon of collision of jamitons in second-order models of traffic flow is studied analytically and numerically for the particular case of the second-order Aw-Rascle-Zhang (ARZ) traffic model [A.\ Aw, M.\ Rascle, {\it SIAM J.\ Appl.\ Math.} {\bf 60} (2000) 916--938; H.\ M.\ Zhang, {\it Transp.\ Res.\ B} {\bf 36} (2002) 275--290]. A compatibility condition is first defined to select jamitons that can collide each other. The collision of jamitons produces a new jamiton with a velocity different from the initial ones. It is observed that the exit velocities smooth out the velocity of the test jamiton and the initial velocities of the jamitons that collide. Other properties such as the amplitude of the exit jamitons, lengths, and maximum density are also explored.In the cases of the amplitude and maximum exit density it turns out that over a wide range of sonic densities, the exit values exceed or equal the input values. On the other hand, the resulting jamiton has a greater length than the incoming ones. Finally, the behavior for various driver reaction times is explored. It is obtained that some properties do not depend on that time, such as the amplitude, exit velocity, or maximum density, while the exit length does depend on driver reaction time.

math.AP↗

arXiv2024

Bounds on the approximation error for deep neural networks applied to dispersive models: Nonlinear waves

We present a comprehensive framework for deriving rigorous and efficient bounds on the approximation error of deep neural networks in PDE models characterized by branching mechanisms, such as waves, Schrödinger equations, and other dispersive models. This framework utilizes the probabilistic setting established by Henry-Labordère and Touzi. We illustrate this approach by providing rigorous bounds on the approximation error for both linear and nonlinear waves in physical dimensions $d=1,2,3$, and analyze their respective computational costs starting from time zero. We investigate two key scenarios: one involving a linear perturbative source term, and another focusing on pure nonlinear internal interactions.

math.NA↗