I am an Associate Professor in the math department at the University of Missouri. I received my Ph.D. in Applied Mathematics from Brown University in 2010; my thesis advisor was Walter Strauss. My undergraduate degree is a B.S. in Mathematical Sciences from Carnegie Mellon University. Prior to coming to MU, I was a Courant Instructor at NYU.
A summary of my teaching and research experience can be found in my CV [pdf].
Contact Information
Email: walshsa "at" missouri.edu
Office: 307 Math Sciences Building
Phone: (573) 882-4426
Research Interests
My research is in the area of nonlinear partial differential equations, particularly those pertaining to water waves. A large part of my work has been devoted to the study of steady waves. Steady waves are a special class of solution to a time-dependent PDE which, when viewed in an appropriately chosen moving reference frame, become time-independent. Steady waves have been an object of fascination for hundreds of years (Cauchy wrote one of the original treatises on the subject), but fundamental questions about them remain. For instance, the existence of large-amplitude traveling waves, potentially even overhanging waves, is not yet established in a number of physically important regimes. Still less is understood about the qualitative features of steady waves, or even whether or not they are stable in many instances.
I am also interested in the broader topic of dispersive nonlinear PDEs. A dispersive PDE is one for which a solution that is localized in frequency will tend to propagate in space with a speed and direction determined by that frequency. Water waves are one example of this phenomenon, but it is found in many physical settings, e.g., quantum mechanics and nonlinear optics.
My work has been supported in part by the National Science Foundation through DMS-1514950 and DMS-1812436.
I am on organizing committee for the annual KUMUNU+ISU Conference on PDE, Dynamical Systems, and Applications. The next meeting will take place on the campus of the University of Nebraska-Lincoln. Due to the ongoing health crisis, it has been postponed until Fall 2021; please see the website for more details. MU hosted the 2019 and 2016 meetings.
Publications and Preprints
Orbital stability of internal waves, (with R. M. Chen),
submitted [arXiv].Global bifurcation for monotone fronts of elliptic equations, (with R. M. Chen and M. H. Wheeler),
submitted [arXiv].Center manifolds without a phase space for quasilinear problems in elasticity, biology, and hydrodynamics, (with R. M. Chen and M. H. Wheeler),
submitted [arXiv].Smooth stationary water waves with exponentially localized vorticity, (with M. Ehrnström and C. Zeng),
J. Eur. Math. Soc., to appear [arXiv].Global bifurcation of anti-plane shear fronts, (with R. M. Chen and M. H. Wheeler),
J. Nonlinear Sci., to appear [article, arXiv].Large-amplitude internal fronts in two-fluid systems, (with R. M. Chen and M. H. Wheeler),
C. R. Acad. Sci. Paris, Ser. I, vol. 358(9-10) (2020), pp. 1073--1083 [article, arXiv].On the stability of solitary water waves with a point vortex, (with K. Varholm and E. Wahlén),
Comm. Pure Appl. Math., vol. 73(12) (2020), pp. 2634--2684. [article, arXiv].Existence, nonexistence, and asymptotics of deep water solitary waves with localized vorticity, (with R. M. Chen and M. H. Wheeler),
Arch. Rational Mech. Anal., vol. 234(2) (2019), pp. 595--633 [article, arXiv].Solitary water waves with discontinuous vorticity, (with A. Akers),
J. Math. Pures Appl., vol. 124 (2019), pp. 220--272 [article, arXiv].Existence and qualitative theory for stratified solitary water waves, (with R. M. Chen and M. H. Wheeler),
Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 25(2) (2018), pp. 517--576 [article, arXiv].Unique determination of stratified steady water waves from pressure, (with R. M. Chen),
J. Differential Equations, vol. 264(1) (2018), pp. 115--133 [article, arXiv].Pressure transfer functions for interfacial fluid problems, (with R. M. Chen and V. M. Hur),
J. Math. Fluid Mech., vol. 19(1) (2017), pp. 59--76 [article, arXiv].On the wind generation of water waves, (with O. Bühler, J. Shatah, and C. Zeng),
Arch. Rational Mech. Anal., vol 222(2) (2016), pp. 827--878 [article, arXiv].On the existence and qualitative theory for stratified solitary water waves, (with R. M. Chen and M. H. Wheeler),
C. R. Acad. Sci. Paris, Ser. I, vol. 354(6) (2016), pp. 601--605 [article].Continuous dependence on the density for stratified steady water waves, (with R. M. Chen),
Arch. Rational Mech. Anal., vol. 219(2) (2016), pp 741--792 [article, arXiv].Nonlinear resonances with a potential: multilinear estimates and an application to NLS (with P. Germain and Z. Hani),
Internat. Math. Res. Notices, vol. 2015(18) (2015), pp. 8484--8544 [article, arXiv].Steady stratified periodic gravity waves with surface tension I: Local bifurcation,
Discrete Cont. Dyn. Syst. Ser. A, no. 8 (2014), pp. 3287--3315 [article].Steady stratified periodic gravity waves with surface tension II: Global bifurcation,
Discrete Cont. Dyn. Syst. Ser. A, no. 8 (2014), pp. 3241--3285 [article].Travelling water waves with compactly supported vorticity (with J. Shatah and C. Zeng),
Nonlinearity, 26 (2013), pp. 1529--1564 [article, arXiv].Steady water waves in the presence of wind (with O. Bühler and J. Shatah),
SIAM J. Math. Anal., 45 (2013), pp. 2182--2227 [article, arXiv].Some criteria for the symmetry of stratified water waves,
Wave Motion, 46 (2009), pp. 350--362 [article, arXiv].Stratified steady periodic water waves,
SIAM J. Math. Anal., 41 (2009), pp. 1054--1105 [article, arXiv].
Current Teaching
I am on research leave for the 2020--2021 academic year.
Advising
I currently have two PhD students at MU, Daniel Sinambela and Thomas Hogancamp.
My past students are:
Hung Le (PhD, 2019)
Max Highsmith (MS, 2018)
Adelaide Akers (PhD, 2017)
Jessie Bleile (MS, 2016)
Evan Datz (MS, 2016).
I am the Math Competition Advisor at MU. If you are an undergraduate at MU and are interested in taking the Putnam exam, please contact me.
I was a faculty mentor to three students participating in the 2012 Summer Undergraduate Research Experience (S.U.R.E.) program at Courant. You can see their report here.