Professor · IMECC, Unicamp
Argenis J. Méndez
On the well‑posedness, regularity, and long‑time behavior of solutions to nonlinear dispersive equations.
Soliton resolution: a wave packet sorting itself into an ordered train of solitons, tallest and fastest in front
Soliton Resolution Conjecture
Education
Doctorate
- Instituto Nacional de Matemática Pura e Aplicada (IMPA), Brazil, 2019
Master's
- Universidad Central de Venezuela, 2014
Licenciatura
- Universidad Central de Venezuela, 2008
Interlude · My Doctoral Thesis
Propagation of regularity
If the initial data is smoother than usual on the right half of the line, that extra smoothness is not lost; it propagates to the left as time evolves. Any singularities in the data can persist, but only ever ahead of an advancing front, never behind it.
Schematic: a rough signal (many frequencies, sharp peaks) turning smooth as the regularity front sweeps past · from my doctoral thesis and my 2020 paper in Analysis & PDE ↓
Research
I study qualitative properties of solutions to nonlinear dispersive equations, the Korteweg–de Vries, Benjamin–Ono, Zakharov–Kuznetsov and Schrödinger–Benjamin–Ono systems among them, with a focus on how regularity propagates, how solutions decay, and how solitary waves respond to perturbation.
Definition 1.1: Areas of interest
Figure 1.2: The three base equations
KdV / fractional KdV
Soliton
Benjamin–Ono / dgBO
Algebraic soliton
Zakharov–Kuznetsov
2D solitary wave
Professor at IMECC (Instituto de Matemática, Estatística e Computação Científica), Unicamp, Campinas, Brazil.
Interlude · Harmonic Analysis
Decay in frequency, regularity in space
A function's smoothness and the decay of its Fourier transform are two sides of the same coin: round off a corner in physical space, and the tail of its spectrum falls off that much faster.
Smoothness ⇔ Fourier decay
Left: a bump function growing smoother · Right: its Fourier transform, decaying faster in frequency as smoothness increases
Interlude · Local Smoothing
Kato's smoothing effect
Rough initial data can turn locally smooth the instant time moves forward; dispersion radiates the roughness away as an oscillatory tail, sharpening the profile it leaves behind.
KdV equation
Schematic: Kato's local smoothing effect for the KdV equation · a fractional analogue for the Zakharov–Kuznetsov equation is in Publications ↓
Interlude · Long-Time Dynamics
Local energy decay, away from the soliton
For a large global solution of the 2D Zakharov–Kuznetsov equation, the energy trapped in any bounded region around the origin that clears the soliton's path eventually thins to nothing, along a sequence of times.
Zakharov–Kuznetsov equation (2D) · local energy decay
Schematic: the dashed box Ω(t) grows far more slowly than the soliton (the warm dot) travels, so the glow trapped inside fades toward zero along a sequence of times · full statement in Publications ↓
Publications
Figure 2.1: Solutions from three specific papers below
Schrödinger–Benjamin–Ono
Coupled solitary wave
Kadomtsev–Petviashvili
Lump
Shrira equation
Interacting line solitons
2026
2025
2024
2023
2021
2020
Remark 2.1. Four collaborative projects currently in progress, with M. Nogueira, F. Linares, and O. Riaño, on related questions in regularity theory and dispersive equations.
Interlude · Soliton Resolution, Revisited
The modified KdV: solitons and breathers
For the modified KdV equation, resolution is richer than for KdV: generic data sorts into ordinary solitons together with breathers, oscillating bound states that pulse in place as they travel at their own speed.
mKdV equation and the breather solution
Schematic: the teal glow traces the breather, pulsing at its own frequency
Interlude · Instantaneous Analyticity
The surface u(t,x): from a jump to real analyticity
The initial datum can be as rough as a jump discontinuity, sgn(x)e^{-ax²}. Yet the solution surface u(t,x) it generates is real analytic in x — and in t — the instant t moves away from zero. Below, each slice of the surface at a fixed t is drawn in depth, from the jump at t=0 to the fully analytic profile at t=Tmax; by symmetry the picture for t<0 is its mirror image.
Modified KdV equation · rough data, analytic flow
Schematic: the two-variable surface u(t,x), stacked by t from the jump at x=0 (t=0) to a real-analytic profile (t=Tmax) · from my preprint with M. Nogueira, arXiv:2607.27115 ↓
Talks
Remark 3.1: Coming soon
A list of seminars, conference talks, and invited lectures will appear here.
Figure 3.1: Not from a paper of mine, just a favorite
Davey–Stewartson (illustrative)
Dromion
Teaching
Courses taught at IMECC, Unicamp, since 2023.
2026
- Calculus 2
- Harmonic Analysis
2025
- Linear Algebra
- Calculus 1
- Functional Analysis
2024
- Linear Algebra
- Analytic Geometry
Documents
Lecture notes, slides, and course materials, organized by course.
Remark 6.1: Coming soon
Downloadable materials for the courses above will be linked here.
Students
Supervision of undergraduate and graduate research: from undergraduate research through bachelor's and master's projects.
Undergraduate Research · Iniciação Científica
-
Vinicius de Oliveira Morais
Timoshenko system · infinite domain
Deflection w and shear strain γ = wₓ − φ
Análise de Fourier Aplicada a Modelos Dispersivos de Difusão e Vibração: Estrutura Espectral e Soluções Exatas em Sistemas Refinados
Projeto interdisciplinar entre Engenharia Mecânica e Matemática Aplicada
Abstract
This project explores the application of Fourier Analysis to solve and interpret linear partial differential equations governing heat and vibration phenomena in Mechanical Engineering. The central focus is to contrast classical models with refined formulations: in thermal conduction, the Fourier equation is compared with the Maxwell-Cattaneo-Vernotte (MCV) model; in mechanics, the Euler-Bernoulli beam theory is contrasted with Timoshenko's theory. The primary objective is to detail the mathematical development behind these models, employing Fourier series and transforms to derive exact analytical solutions.
The work prioritizes step-by-step derivation and the analysis of the spectral structure of the operators, investigating how each model describes wave propagation, modal decay, and energy dissipation. The computational phase will be used exclusively for the graphical representation of the derived solutions and to evaluate the behavior of the series regarding truncation and convergence. Thus, the project aims to consolidate the mastery of mathematical tools necessary for understanding refined physical models and their application limits in engineering.
Resumo
Este projeto estuda o uso da Análise de Fourier para resolver e interpretar equações diferenciais parciais lineares que descrevem fenômenos de calor e vibração na Engenharia Mecânica. O foco central é comparar modelos clássicos com formulações mais detalhadas: na condução térmica, contrasta-se a equação de Fourier com o modelo de Maxwell-Cattaneo-Vernotte (MCV); na mecânica, compara-se a viga de Euler-Bernoulli com a teoria de Timoshenko. O objetivo principal é detalhar o desenvolvimento matemático por trás desses modelos, utilizando séries e transformadas de Fourier para obter soluções analíticas exatas.
Além das soluções analíticas, o trabalho prioriza a dedução passo a passo e a análise da estrutura espectral dos operadores, investigando como cada modelo descreve a propagação de ondas, o decaimento modal e a dissipação de energia. A etapa computacional será utilizada exclusivamente para a representação gráfica das soluções deduzidas e para avaliar o comportamento das séries em termos de truncagem e convergência. Com isso, busca-se consolidar o domínio das ferramentas matemáticas necessárias para a compreensão de modelos físicos refinados e seus limites de aplicação na engenharia.
Bachelor's · Licenciatura
- Cristian Andrade
Master's · Maestría
-
Iván Felipe Salamanca Medina
Project: asymptotic behavior of solutions to certain nonlinear, nonlocal dispersive models.
Soliton resolution · moving window
Presolitonic region, gap, and soliton scale separating in time
Prospective students and postdoctoral researchers interested in working with me are welcome to write to me at ajmendez@ime.unicamp.br to discuss possible new research projects. My area of specialization is mathematical analysis, with an emphasis on partial differential equations.
Curriculum Vitae
Remark 8.1: Full CV
A complete academic CV, including publications, funded research projects, talks, and service, is available for download below.
Download CV (PDF) ↗Contact
Office hours
Tuesdays & Thursdays
11:00 – 17:00