Boussinesq-Klein-Gordon and Ostrovsky equations: evolution of cnoidal waves with local defects

Abstract

The Boussinesq-Klein-Gordon (BKG) equation has emerged in the studies of nonlinear bulk strain waves in layered solid waveguides. The developed bi-directional weakly-nonlinear solution leads to two copies of the Ostrovsky equation, for the right- and left-propagating waves. Importantly, the derivation avoids the so-called `zero-mean contradiction' between the type of initial conditions in the parent equation and in the reduced model. In this paper, we apply the solution to describe the evolution of cnoidal waves with local periodicity defects and generic localised perturbations, and compare the results with the direct numerical simulations for the full BKG equation. The cnoidal waves with the periodicity defects discussed in our work constitute generalised travelling waves of the Korteweg-de Vries equation, while the Ostrovsky equation leads to a strong burst (and may lead to a rogue wave), qualitatively similar to the wavepacket emerging from a soliton initial condition, but appearing much faster. We compare the weakly-nonlinear solution with the direct numerical simulations within the bi-directional setting of the BKG equation and show that the discussed uni-directional waves and evolution scenarios remain stable in the presence of counter-propagating perturbations.

Figures (10)

• Figure 1: Evolution of the weakly nonlinear solution $u$ given by \ref{['WNLFV']} for a cnoidal wave initial condition, showing view from above (left) and view from below (right) (first row). Numerical parameters are $\varepsilon = 0.005,\, \alpha_1 = -1.73,\, \beta_1 = 0.08,\, \gamma_1 =0, u_1 = -10^{-3},\, u_2=0,\, u_3=3$. Comparison of the direct numerical simulations (blue, solid) and weakly nonlinear solution (red, dashed) for a cnoidal wave initial condition when $\gamma=0$ at times $T=20$ (left), $T=35$ (middle) and $T=40$ (right), computed for $\varepsilon = 0.005$ (second row) and $\varepsilon = 0.01$ (third row).
• Figure 2: Least-squares fitting of error computed for $\varepsilon = 0.0001,\, 0.0005,\, 0.001,\, 0.005,\, 0.01$ at $T_{max} = 100$.
• Figure 3: Evolution of the weakly nonlinear solution $u$ given by \ref{['WNLFV']} for a pure cnoidal wave initial condition with non-zero $\gamma_1$, showing view from above (left) and view from below (right) (first row). Numerical parameters are $\varepsilon = 0.005,\, \alpha_1 = -1.73,\, \beta_1 = 0.08,\, \gamma_1 = 0.10, u_1 = -10^{-3},\, u_2=0,\, u_3=3$. Comparison of the profiles for a pure cnoidal wave initial condition with non-zero $\gamma_1$, on periodic boundary conditions. Direct computation (blue solid) and weakly nonlinear solution (red dash) are shown at times $T=20$ (left), $T=30$ (middle) and $T=40$ (right), computed with $\varepsilon = 0.005$ (second row) and $\varepsilon = 0.01$ (third row)..
• Figure 4: Schematic of construction of the generalised travelling wave of the KdV equation in the form of a cnoidal waves with an expansion defect.
• Figure 5: Evolution of the weakly nonlinear solution \ref{['WNLFV']} for a cnoidal wave initial condition with expansion defect, showing view from above (left) and view from below (right) for (a) $\gamma_1 = 0$, and (c) $\gamma_1 = 0.10$. Comparison of the direct numerical simulations (blue, solid) and weakly nonlinear solution (red, dashed) at times $T=20$ (left), $T=30$ (middle) and $T=40$ (right) for (b) $\gamma_1 = 0$ and (d) $\gamma_1 = 0.10$, computed for $\varepsilon = 0.005$. Numerical parameters are $\varepsilon = 0.005,\, \alpha_1 = -1.73,\, \beta_1 = 0.08, u_1 = -10^{-3},\, u_2=0,\, u_3=3$.