Email:

Office:

Phone:

Office:

Phone:

walshsa "at" missouri.edu

307 Math Sciences Building

(573) 882-4426

307 Math Sciences Building

(573) 882-4426

A summary of my teaching and research experience can be found in my

**CV**[pdf].

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.

Another subject of my research is

*wind-driven water waves*. While it is an easily observed fact that wind blowing over a body of water can generate waves, it is notoriously hard to understand this process quantitatively. In an ongoing program, I have sought to give a mathematically rigorous examination of some of the predominant theories in the applied literature.

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 is supported in part by the National Science Foundation through DMS-1514950 and DMS-1812436.

I am co-organizing the 2019 KUMUNU Conference on PDE, Dynamical Systems, and Applications. Please see the site for more details.

Publications and Preprints

*On the stability of solitary water waves with a point vortex*, (with K. Varholm and E. Wahlén),

submitted [arXiv].*Existence, nonexistence, and asymptotics of deep water solitary waves with localized vorticity*, (with R. M. Chen and M. H. Wheeler),

submitted [arXiv].*Solitary water waves with discontinuous vorticity*, (with A. Akers),

J. Math. Pures Appl., to appear (online 2018) [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].*Reconstruction of stratified steady water waves from pressure readings on the ocean bed*, (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].

In Spring 2018, I am teaching MATH 4100-13 Differential Equations (syllabus: [pdf]) and MATH 8720 Advanced Topics in Applied Mathematics: Local and Gloabl Bifurcation Theoretic Methods (syllabus: [pdf]). My office hours are Wednesdays and Thursdays from 4:30-5:30PM in MSB 307, and by appointment.

Advising

I currently have two PhD students at MU, Hung Le and Daniel Sinambela.

My past students are:

- Max Highsmith (MS, 2018)
- Adelaide Akers (PhD, 2017)
- Jessie Bleile (MS, 2016)
- Evan Datz (MS, 2016).

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.

Past Teaching

Instructor for MATH 4540 Mathematical Modeling, Spring 2017, University of Missouri.

Syllabus [pdf]

Instructor for MATH 4100-06 Differential Equations, Spring 2017, University of Missouri.

Syllabus [pdf]

Instructor for MATH 4100-03 Differential Equations, Spring 2017, University of Missouri.

Syllabus [pdf]

Instructor for MATH 8702 Advanced Topics in Applied Mathematics: Advanced Topics in Partial Differential Equations, Spring 2017, University of Missouri.

Syllabus [pdf]

Instructor for MATH 4540 Mathematical Modeling, Fall 2016, University of Missouri.

Syllabus [pdf]

Instructor for MATH 8445 Partial Differential Equations I, Fall 2016, University of Missouri.

Syllabus [pdf]

Instructor for MATH 8702-02 Topics in Applied Math: Nonlinear Dispersive Equations II, Spring 2016, University of Missouri.

Syllabus [pdf]

Instructor for MATH 4540 Mathematical Modeling, Fall 2015, University of Missouri.

Syllabus [pdf]

Instructor for MATH 8702-03 Topics in Applied Math: Nonlinear Dispersive Equations, Fall 2015, University of Missouri.

Syllabus [pdf]

Instructor for MATH 4100-02 Differential Equations, Spring 2015, University of Missouri.

Syllabus [pdf]

Instructor for MATH 8445 Partial Differential Equations I, Spring 2015, University of Missouri.

Syllabus [pdf]

Instructor for MATH 8702-03 Topics in Applied Math: Mathematical Theory of Water Waves, Fall 2014, University of Missouri.

Syllabus [pdf]

Instructor for MATH 4100-07 Differential Equations, Fall 2014, University of Missouri.

Syllabus [pdf]

Instructor for MATH 8445 Partial Differential Equations I, Spring 2014, University of Missouri.

Syllabus [pdf]

Instructor for MATH 4100-001 and 4100-002 Differential Equations, Fall 2013, University of Missouri.

Syllabus [001][002]

Instructor for MATH-UA.325 (Section 3) Analysis I, Spring 2013, NYU.

Syllabus [pdf]

Instructor for MATH-UA 120 (Section 1) Discrete Mathematics, Fall 2012, NYU.

Syllabus [pdf]

Instructor for MATH-UA 263 Partial Differential Equations, Spring 2012, NYU.

Syllabus [pdf]

Instructor for MATH-UA 140 Linear Algebra, Fall 2011, NYU.

Instructor for MATH-UA 123: Calculus III, Spring 2011, NYU.

Instructor for V63.0120: Discrete Mathematics, Fall 2010, NYU.

Syllabus [pdf]

Instructor for MATH 0200: Intermediate Calculus (Physics/Engineering), Fall 2009, Brown.

Course website

TA for APMA 0650: Essential Statistics, Spring 2009, Brown.

Instructor: Stuart Geman.

TA for APMA 0330: Methods of Applied Mathematics I, Fall 2007, Brown.

Instructors: Bo Dong, Vladimir A. Dobrushkin.

TA for APMA 0340: Methods of Applied Mathematics II, Spring 2007, Brown.

Instructor: Suzanne Sindi.

TA for APMA 0330: Methods of Applied Mathematics I, Fall 2006, Brown.

Instructors: Suzanne Sindi, Natalie Kleinfelter Domelle.

TA for 21-127: Concepts of Mathematics, Spring 2005, CMU.

Instructor: John Mackey.

TA for 21-122: Integration, Differential Equations, and Approximation, Fall 2004, CMU.

Instructor: Katherine Thompson.