I study nonlinear partial differential equations (PDEs) with an emphasis on PDEs that model surface water waves. My work currently focuses on effects such as bathymetry, dispersion, viscosity, surface tension and vorticity. Many of these effects require the use of nonlocal PDEs such as the Whitham, fractional KdV, and Dysthe equations. My research interests include stability analysis, fast numerical methods for nonlinear evolution equations, and mathematical physics.

        If you are a Seattle University student interested in these problems, please contact me regarding research opportunities.

34. H. Potgieter, J.D. Carter, and D.M. Henderson. Modeling the second harmonic in surface water waves using generalizations of NLS. Water Waves, 4(1): 23-47, 2022.   .pdf
33. J.D. Carter, H. Kalisch, C. Kharif, and M. Abid, The cubic-vortical Whitham equation. Wave Motion, 110: 102883, 2022.   .pdf
32. C.R. Zaug and J.D. Carter. Dissipative models of swell propagation across the Pacific. Studies in Applied Mathematics, 147(4): 1519-1537, 2021.   .pdf
31. D.M. Henderson, J.D. Carter, and M.E. Catalano. On the variation of bi-periodic waves in the transverse direction. Studies in Applied Mathematics, 147(4): 1388-1408, 2021.   .pdf
30. J.D. Carter and B. Deconinck. An interview with Harvey Segur. Studies in Applied Mathematics, 147(4): 1199-1208, 2021.   .pdf
29. J.D. Carter, E. Dinvay, and H. Kalisch. Fully dispersive Boussinesq models with uneven bathymetry. Journal of Engineering Mathematics, 127: 10, 2021.   .pdf
28. J.D. Carter, C.W. Curtis, and H. Kalisch. Particle trajectories in nonlinear Schrodinger models. Water Waves, 2: 31-57, 2020.   .pdf
27. C. Kharif, M. Abid, J.D. Carter, and H. Kalisch, Stability of periodic progressive gravity wave solutions of the Whitham equation in the presence of vorticity, Physics Letters A, 384(2), 126060, 2020.   .pdf
26. J.D. Carter and M. Rozman, Stability of periodic, traveling-wave solutions to the capillary-Whitham equation. Fluids, 4(1): 58, 2019.   .pdf
25. J.D. Carter, D. Henderson, and I. Butterfield. A comparison of frequency downshift models of wave trains on deep water. Physics of Fluids, 31: 013103, 2019.   .pdf
24. C.W. Curtis, J.D. Carter, and H. Kalisch. Deep water particle paths in the presence of currents. Journal of Fluid Mechanics, 855: 322-350, 2018.   .pdf
23. J.D. Carter. Bidirectional Whitham equations as models of waves on shallow water. Wave Motion, 82: 51-61, 2018.   .pdf
22. D. Eeltink, A. Lemoine, H. Branger, O. Kimmoun, C. Kharif, J.D. Carter, A. Chabchoub, M. Brunetti, and J. Kasparian. Spectral up- and downshifting of Akhmediev breathers under wind forcing. Physics of Fluids, 29: 107103, 2017.   .pdf
21. D. Mitsotakis, D. Dutykh, and J.D. Carter. On the nonlinear dynamics of the traveling-wave solutions of the Serre system. Wave Motion, 70: 166-182, 2017.   .pdf
20. J.D. Carter and A. Govan. Frequency downshift in a viscous fluid. European Journal of Mechanics B: Fluids, 59: 177-185, 2016.   .pdf
19. J.D. Carter, D. Helliwell, A. Henrich, M. Principe, and J.M. Sloughter. Improving student success in calculus at Seattle University. PRIMUS, 26(2): 105-124, 2016.
18. N. Sanford, K. Kodama, J.D. Carter, and H. Kalisch. Stability of traveling wave solutions to the Whitham equation. Physics Letters A, 378: 2100-2107, 2014.   .pdf
17. J.D. Carter. Plane wave solutions of a dissipative generalization of the vector nonlinear Schrodinger equation. Mathematics and Computers in Simulation, 82: 1038-1046, 2012.   .pdf
16. J.D. Carter. A Review of Mathematica: A Problem-Centered Approach. SIAM Review, 53: 583-585, 2011.   .pdf
15. J.D. Carter and R. Cienfuegos. The kinematics and stability of solitary and cnoidal wave solutions of the Serre equations. European Journal of Mechanics B: Fluids, 30: 259-268, 2011.   .pdf
14. D.M. Henderson, H. Segur and J.D. Carter. Experimental evidence of stable wave patterns on deep water. Journal of Fluid Mechanics, 658, 247-278, 2010.   .pdf
13. J.D. Carter and C.C. Contreras. Stability of plane-wave solutions of a dissipative generalization of the nonlinear Schrodinger equation. Physica D, 237: 3292-3296, 2008.   .pdf
12. J.D. Carter. A Review of Maple and Mathematica: A Problem Solving Approach.   .pdf
11. J.D. Carter. A Review of Mathematica 6. SIAM Review, 50: 149-152, 2008.   .pdf
10. B. Deconinck, F. Kiyak and J.D. Carter. SpectrUW version 2.0, April 2007.
9. N. Canney and J.D. Carter. Stability of plane waves on deep water with dissipation. Mathematics and Computers in Simulation, 74: 159-167, 2007.   .pdf
8. B. Deconinck, F. Kiyak, J.D. Carter and J.N. Kutz. SpectrUW: a laboratory for the numerical exploration of spectra of linear operators.Mathematics and Computers in Simulation, 74: 370-378, 2007.   .pdf
7. J.D. Carter and B. Deconinck. Instabilities of one-dimensional trivial-phase solutions of the two-dimensional cubic nonlinear Schrodinger equation. Physica D, 214: 42-54, 2006.   .pdf
6. B. Deconinck, D.E. Pelinovsky and J.D. Carter. Transverse instabilities of deep-water solitary waves. Proceedings of the Royal Society A, 462: 2039-2061, 2006.   .pdf
5. R.J. Thelwell, J.D. Carter and B. Deconinck. Instabilities of one-dimensional stationary solutions of the cubic nonlinear Schrodinger equation. Journal of Physics A: Mathematical and General, 39: 73-84, 2006.   .pdf
4. H. Segur, D. Henderson, J.D. Carter, J. Hammack, C. Li, D. Pheiff and K. Socha. Stabilizing the Benjamin-Feir instability. Journal of Fluid Mechanics, 539: 229-271, 2005.   .pdf
3. J.D. Carter. A Review of Mathematica 5.0. SIAM Review, 46: 564-568, 2004.   .pdf
2. J.D. Carter and H. Segur. Instabilities in the two-dimensional cubic nonlinear Schrodinger equation. Physical Review E, 68: 045601, 2003.   .pdf
1. J.D. Carter (Ph.D. thesis). Stability and existence of traveling-wave solutions of the two-dimensional nonlinear Schrodinger equation and its higher-order generalizations. University of Colorado at Boulder, 2001.   .pdf

Curriculum Vitae

JCarterCV.pdf

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