# Wave Modulation & Stability in Applied Mathematics

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · University of Kansas Center for Research Inc (KS) · $220,000

## Abstract

This project investigates the stability and dynamics of spatially periodic solutions to nonlinear partial differential equations (PDEs) that arise across physics, engineering, and applied mathematics.  The principal investigator (PI) focuses on the dynamical stability of these patterns -- their ability to persist under small perturbations – which is crucial since unstable solutions are generally not observable in practical settings except as transient phenomena.  Specifically, the project aims to study the modulational stability of periodic wave patterns, examining how their fundamental wave characteristics evolve under slow spatial and temporal variations. Insights from this research have important implications for many applications, including optical signal propagation, fluid flows, and plasma physics.  The project also includes opportunities for both undergraduate and graduate students to participate in advanced research training.

Building on the PI’s prior success in studying modulated signals in one-dimensional media, this research extends to multi-dimensional nonlinear wave phenomena. The goal is to develop new techniques and methodologies for studying the behavior of spatially periodic patterns under small modulational perturbations.  Because periodic patterns can support multiple modulated signals simultaneously, their dynamics exhibit rich, multi-scale structures that are effectively infinite-dimensional. Analyzing these systems involves addressing many nonstandar

## Key facts

- **NSF award ID:** 2510069
- **Awardee organization:** University of Kansas Center for Research Inc (KS)
- **SAM.gov UEI:** SSUJB3GSH8A5
- **PI:** Mathew A Johnson
- **Primary program:** 01002526DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** EXP PROG TO STIM COMP RES
- **Estimated total:** $220,000
- **Funds obligated:** $220,000
- **Transaction type:** Standard Grant
- **Period:** 08/15/2025 → 07/31/2028

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2510069

## Citation

> US National Science Foundation, Award 2510069, Wave Modulation & Stability in Applied Mathematics. Retrieved via AI Analytics 2026-06-07 from https://api.ai-analytics.org/grant/nsf/2510069. Licensed CC0.

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