Compound Burgers-KdV Soliton Behaviour: Refraction, Reflection and Fusion

TL;DR

This work introduces a Burgers–swept KdV system that couples Burgers bore dynamics with KdV wave dynamics through the momentum $m = u - \tfrac{1}{2} v^2$, and derives the model via Dirac–Frenkel variational principles. It shows that Gardner's equation for the KdV component is recovered when $m=0$ and provides explicit compound soliton and traveling-wave solutions, framed within a Lie–Poisson Hamiltonian structure. Through pseudo-spectral simulations, the authors demonstrate rich wave–current interaction phenomena—refraction, reflection, and soliton fusion—driven by nonzero $m$ and the approach toward Gardner-like dynamics. The results extend Gardner dynamics to WCI settings, revealing intricate nonlinear structures and suggesting open problems on transitions between integrable and non-integrable regimes in an infinite-dimensional context.

Abstract

We consider a coupled PDE system between the Burgers equation and the KdV equation to model the interactions between `bore'-like structures and wave-like solitons in shallow water. Two derivations of the resulting Burgers-swept KdV system are presented, based on Lie group symmetry and reduced variational principles. Exact compound soliton solutions are obtained, and numerical simulations show that the Burgers and KdV momenta tend toward a balance at which the coupled system reduces to the integrable Gardner equation. The numerical simulations also reveal rich nonlinear solution behaviours that include refraction, reflection, and soliton fusion, before the balance is finally achieved.

Figures (14)

• Figure 1: Propagation and collisions of three compound solitons in the Burgers–swept KdV system \ref{['eq: Burgers–swept KdV']} that satisfies $u = v^2/2$. Left: waterfall plot of $u$ (red, Burgers velocity) and $v$ (black, KdV velocity). Right: contour plot of $v(x,t)$, showing the emergence and collision of three compound solitons, which retain their forms but reverse order after interaction.
• Figure 2: Comparison between the Burgers–swept KdV compound soliton (left) / solitary wave (right) and the KdV soliton $v_\text{KdV}(\xi)$ at the same wave speed $c=3.5$, under two boundary conditions $\lim _{x \rightarrow \pm \infty} u(x)=u_0$: $u_0=0$ (left) for \ref{['Gardner-redux']} and $u_0=2$ (right) for \ref{['eq: solitary_wave']}.
• Figure 3: $\operatorname{dn}[\beta \xi, m]$ travelling wave solution profiles in the Burgers–swept KdV system with elliptic parameter $m=1$. The Burgers component $u$ (mean flow velocity, blue) and KdV component $v$ (wave parameter, orange) are plotted for increasing values of the wave parameter $\beta$ (a) $\beta=0.7$, (b) $\beta=1$, (c) $\beta=\sqrt3$, when $\beta>1$ a W-shape structure appears in $u$ (d) $\beta=2$, (e) $\beta=\sqrt6$ is corresponding to the case $c=0$ (no propagation), the wave profile appears stationary, with a highly localized, peaked structure. This profile resembles the Peregrine soliton, a localized wave in the nonlinear Schrödinger equation.
• Figure 4: Formation of Burgers–swept KdV compound solitons from $m\ne0$ initializations. (a) Initial condition with $u=0$ and a KdV soliton $v= v_{\mathrm{KdV}}$ with $c=2$ (top), Gaussian packet $v=\exp (-x^2 / 32)$ (bottom). (b) The Burgers component develops an N-wave structure. (c) The right-propagating peak in $u$ aligns with the part in $v$, initiating compound structure formation. (d) A single (top)/ train (bottom) of compound solitons emerges and travels to the right, leaving residual waves to the left. For spacetime plot see appendix \ref{['appendix: spacetime']}.
• Figure 5: Refraction type interaction between bore and a compound wave. For spacetime plot see appendix \ref{['appendix: spacetime']}. Top: a compound soliton satisfying \ref{['Gardner-redux']} swept by the Burgers current and forms a solitary wave slower than the bore. (a) a compound soliton ($c=0.3$) out of the Burgers current ($u_0=3$). (b) - (c) the bore takeovers the soliton (d) a solitary wave remains in the current. Bottom: a compound solitary wave satisfying \ref{['eq: solitary_wave']} in the Burgers current exits current and forms a faster compound soliton. (a) a compound solitary wave with speed $c=3.5$ within a Burgers current ($u_0=2$) (bore front speed $c=1.5u_0$). The solitary wave approach the bore front at speed $c=0.5$ without much shape deformation. (b) the solitary wave arrives the bore front and starts to detach the bore, (c) the solitary wave carries away a piece of the bore and start to leave the bore front, (d) a compound soliton faster than the bore is formed with a 'train' of waves carried along in the 'current' behind the bore front.

Theorems & Definitions (10)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Proposition 2.1
• Lemma 2.1: Burgers–swept KdV travelling wave condition
• Theorem 2.1: Compound Soliton Solution
• proof
• Remark 2.4
• Remark 2.5: Compound Solitary Wave Solution with $u_0>0$
• Theorem 2.2: Periodic wave