By Wei Lian
NTNU Norwegian University of Science and Technology
In engineering, “instability” is often viewed as a failure—a bridge buckling in the wind or a circuit overheating. But in pattern formation, instability is a lighthouse. It is the guiding mechanism where a system breaks the spatial constraint in low dimension.
An everyday example of this is the Plateau-Rayleigh Instability: a cylindrical stream of water breaks into droplets. In the language of the bifurcation theory, this is “dimension-breaking.”
Our MSCA project, UNSTABLE, shall adopt this philosophy to 1D solitons. The idea behind it is simple: the instability of 1D solutions predicts the existence of 2D surface waves, also called dimension-breaking solutions.
A lighthouse maps to Existence
The difficulty in water wave theory is understanding how the solutions evolve from simple, uniform waves into complex, multidimensional surface wave. The UNSTABLE project attacks it with the idea above:
Instability: Many 1D water wave models admit solitary wave solutions that extend uniformly in transverse direction, which are immediate solutions to 2D surface wave equations. By analyzing governing dynamics, we aim to show that these waves are unstable, when subject to perturbations transverse to the moving direction.
Capturing the 2D Landscape: We believe when a 1D wave loses transverse stability, it bifurcates into 2D surface patterns. Using a model that captures the full dispersion of the water wave problem, we seek to identify dimension-breaking solutions that emerge from this instability.
The Geometry of the Extreme
Finally, we test the same idea in its extreme complexity in the water wave theory. This leads us to the Highest Wave. It is the extreme of steady waves—the precise point where a wave reaches its maximal amplitude and its smooth profile sharpens into a singular peak of 120 degree.
Seeking the 2D Highest Waves: We seek the dimension-breaking for 1D highest wave where it becomes unstable and bifurcates into a 2D object. While the geometry of the 1D highest wave has been found for over a century, the existence/nonexistence of such singular 2D surfaces remains an open problem.
Defining the Singular Waves: If it exists, our ultimate goal is to identify the conditions for these 2D singular waves to exist. By constructing these waves, we hope to bridge the gap between 1D theory and the water wave dynamics.
References
[1] D. Lannes. The water waves problem: mathematical analysis and asymptotics. Vol. 188. American Mathematical Soc., 2013.
[2] M. D. Groves and A. Mielke. A bifurcation theory for a class of Hamiltonian systems on a cylindrical domain. Arch. Ration. Mech. Anal. (2001).
[3] F. Rousset and N. Tzvetkov. Transverse instability of the line solitary water-waves. Invent. math. 184, 257–388 (2011).
[4] M. Ehrnström and E. Wahlén. ‘On Whitham’s conjecture of a highest cusped wave’. Ann. Inst. H. Poincaré (2019).
[5] R. M. Chen, W. Lian, D. Wang, R. Xu. A rigidity property for the Novikov equation and the asymptotic stability of peakons, Arch. Ration. Mech. Anal. 241, 497-533 (2021).