Stabilization of Unsteady Nonlinear Waves by Phase-Space Manipulation

We introduce a dynamic stabilization scheme universally applicable to unidirectional nonlinear coherent waves. By abruptly changing the waveguiding properties, the breathing of wave packets subject to modulation instability can be stabilized as a result of the abrupt expansion a homoclinic orbit and its fall into an elliptic fixed point (center). We apply this concept to the nonlinear Schrödinger equation framework and show that an Akhmediev breather envelope, which is at the core of Fermi-Pasta-Ulam-Tsingou recurrence and extreme wave events, can be frozen into a steady periodic (dnoidal) wave by a suitable variation of a single external physical parameter. We experimentally demonstrate this general approach in the particular case of surface gravity water waves propagating in a wave flume with an abrupt bathymetry change. Our results highlight the influence of topography and waveguide properties on the lifetime of nonlinear waves and confirm the possibility to control them.

Figure 1

Principle of AB conversion into a dnoidal solution of the NLSE. (a) Space-time evolution of the AB for a normalized detuning ${\mathrm{\Omega }}^0=1.67$. (b) Dnoidal solution for normalized detuning ${\mathrm{\Omega }}^{\infty }=1.34$. (c) Best matching of Fourier coefficients of the AB [at the peak distance, blue shading in (a)] to the dnoidal solution. Inset: superimposed time profiles. (d) Phase plane trajectories of AB and dnoidal for ${\mathrm{\Omega }}^0$ (blue solid line and circle) and ${\mathrm{\Omega }}^{\infty }$ (red dashed line and cross). Arrows show the effect of separatrix dilation.

Figure 2

(a) Water wave flume with artificial floor setup. One end shows the piston-type wave maker and the other end an inclined wave absorber with an artificial grass layer. Top: positions of the wave gauges. (b) Wave height at each recorded position for the experiment with variable bathymetry, multiplied by a factor 20; the gray stripe indicates the position of the step. (c) Wave height at each recorded position for the experiment with constant bathymetry, multiplied by a factor 20. (d)–(k) Sideband evolution of the AB-type surface water wave over the adopted bathymetry with the depth step (d),(f),(h),(j), and the constant flat bottom $h^0$ (e),(g),(i),(k). (d)–(g) Sideband dynamics as identified from the eight gauge measurements, connected by a linear interpolation; (h)–(k) corresponding NLSE-simulated evolution. (d),(e),(h),(i) Sideband fractions ${\eta }_0$, ${\eta }_1$, ${\eta }_2$ of modes at frequencies 0 (carrier), $\pm \mathrm{\Omega }$, and $\pm 2\mathrm{\Omega }$, respectively. (f),(g),(j),(k) Phase $\psi$ of first-order sidebands (modes at $\pm \mathrm{\Omega }$) relative to the carrier frequency, i.e., ${\eta }_0\equiv |\widehat{V}(\xi ,0){|}^2/N$, ${\eta }_1\equiv [|\widehat{V}(\xi ,\mathrm{\Omega }){|}^2+|\widehat{V}(\xi ,-\mathrm{\Omega }){|}^2]/N$, and ${\eta }_2\equiv [|\widehat{V}(\xi ,2\mathrm{\Omega }){|}^2+|\widehat{V}(\xi ,-2\mathrm{\Omega }){|}^2]/N$, and $\psi \equiv [({\varphi }_1+{\varphi }_{-1})/2]-{\varphi }_0$, with ${\varphi }_n\equiv \mathrm{Arg}[\widehat{V}(\xi ,n\mathrm{\Omega })]$, where $\widehat{V}$ denotes the Fourier transform of $V$.

Figure 3

(a) Propagation of a surface water wave AB over the depth step followed by flat bottom, displayed in the phase plane of Fig. 1, where $\eta \equiv {\eta }_1$ of Fig. 2. The respective NLSE simulations last up to 37.7 m. (b) Corresponding envelope contrast $C\equiv 1-[\min |U|/\max |U|]$.