Simulations

Disclaimer: Code for generating images of the Aztec diamond provided by Christophe Charlier through his website. Plots of the KdV and Toda solution were produced with the help of AI.

A two-soliton solution of the KdV equation lattice (see [2], [5]). Observe that the solitons retain their form after their nonlinear interaction and undergo a phase shift.

Typical numerically computed solution to the KdV equation with smooth steplike initial data. Observed the modulated wave region emerging from the initial step (see [2], [5]).

A three-soliton solution of the Toda lattice equation. Observe that the solitons retain their form after their nonlinear interaction. They also undergo a phase shift, but it is difficult to see as it is rather small.

Literature

[1] S. Chhita and K. Johansson, Domino statistics of the two-periodic Aztec diamond, Adv. Math. 294, 37-149 (2016), doi.org/10.1016/j.aim.2016.02.025 

[2] I. Egorova, M. Piorkowski and G. Teschl, Asymptotics of the Korteweg–de Vries shock waves via the Riemann–Hilbert approach, Indiana Univ. Math. J., 73 No. 2, 645-690 (2024), 

[3] W. Jockusch, J. Propp and P. Shor, Random Domino Tilings and the Arctic Circle Theorem, arXiv:math/9801068

[4] A. Perret, Z. Ristivojevic, P. Le Doussal, G. Schehr and K.J. Wiese, Super Rough Glassy Phase of the Random Field XY Model in Two Dimensions, Phys. Rev. Lett. 109, 157205 (2012), doi.org/10.1103/PhysRevLett.109.157205 

[5] M. Piorkowski, Parametrix problem for the Korteweg–de Vries equation with steplike initial data, J. of Differential Equations, 375, 280–314 (2023), doi.org/10.1016/j.jde.2023.06.052

[6] M. Piorkowski, Arctic curves of periodic dimer models and generalized discriminants,  Selecta Math. accepted,  32(99), 40 pp. (2026),

doi.org/10.1007/s00029-026-01210-5