Integrability properties and multi-kink solutions of a generalised Fokker-Planck equation

TL;DR

This work analyzes a generalised Fokker-Planck equation that is linearisable by a Cole-Hopf transformation, revealing rich integrability structures and shock dynamics in the small-viscosity regime. By distinguishing two parameter regimes Δ ≠ 0 and Δ = 0, the authors derive travelling-wave and multi-kink solutions, and introduce a geometric, tropical-like construction of moving shock lines that encode kink interactions as shock–particle scatterings with conserved mass and momentum. They establish a comprehensive framework, including one- and two-parameter Bäcklund transformations, to generate new traveling components and nonlinear superpositions from seed solutions. The results illuminate the role of viscous regularisation in nonlinear conservation laws and suggest broad implications for C-integrable models, tropical limits, and potential extensions to higher dimensions and Lax-pair formulations.

Abstract

We analyse a generalised Fokker-Planck equation by making essential use of its linearisability through a Cole-Hopf transformation. We determine solutions of travelling wave and multi-kink type by resorting to a geometric construction in the regime of small viscosity. The resulting asymptotic solutions are time-dependent Heaviside step functions representing classical (viscous) shock waves. As a result, line segments in the space of independent variables arise as resonance conditions of exponentials and represent shock trajectories. We then discuss fusion and fission dynamics exhibited by the multi-kinks by drawing parallels in terms of shock collisions and scattering processes between particles, which preserve total mass and momentum. Finally, we propose Bäcklund transformations and examine their action on the solutions to the equation under study.

Figures (11)

• Figure 1: Travelling wave solutions to Eqs. \ref{['eqscaled']} and \ref{['eqscaled2']}. (Left) Solution \ref{['eq:TW scaled']} at different times for $K_1^2+1/c>0$. Parameters are $K_1=1$, $K_2=0$ and $c=1/3$. (Centre) Solution \ref{['eq:TW tangent']} at different times obtained for parameters satisfying $K_1^2+1/c<0$. In particular, parameters are chosen to be $K_1=1$, $K_2=0$ and $c=-1/3$. (Right) Solution \ref{['eq:TW eq scaled 2']} at different times. Parameters are $K_1=1$, $K_2=0$ and $c=1/3$. Notice that, at equal values of $K_1$ the kink in the case $\Delta \neq 0$ is steeper than the one in the case $\Delta =0$, due to dependence of the amplitude on the velocity.
• Figure 2: A pair of $\widetilde{\mathcal{PT}}$--connected solutions representing fission and fusion of two shocks. Parameters are set to $k_1=-2, \, k_2=-1,\, k_3=3$, $\delta_1=0,\, \delta_2=8,\, \delta_3=20$ and $\eta=1/2$. (Left) The fission process of a shock wave decaying in two shocks. (Right) The $\widetilde{\mathcal{PT}}$-- transformed solution representing two shocks merging into a single shock. The dashed and dash--dotted semi-lines of resonance are obtained in both processes as conditions $\theta_i(X,T; k_i,\delta_i)=\theta_j(X,T; k_j,\delta_j)$ for all pairs of distinct indices $i,j=1,2,3$.
• Figure 3: Multi-kink solution \ref{['eq:weight decomposition']} with $N=5$. Parameters are set to $k_1=-1, \, k_2=1,\, k_3=2,\, k_4=3\,, k_5=4$, $\delta_1=-227/3,\, \delta_2=1,\, \delta_3=-20/3,\,\delta_4=-5/9\,, \delta_5=0$ and $\eta=0.5$. (Left) Solution for $u(X,T)$. (Right) Geometric construction identifying the asymptotic, discontinuous solution \ref{['eq:geometric construction']}.
• Figure 4: Multi-kink solution \ref{['eq:weight decomposition']} with $N=8$. Parameters are set to $k_1=-4, \, k_2=-3,\, k_3=-2,\, k_4=-1 \,, k_5=1\,,k_6=2\,,k_7=3\,,k_8=4$, $\delta_1=-29/2,\, \delta_2=0,\, \delta_3=9,\,\delta_4=-115/24\,, \delta_5=0,\,\delta_6=15,\,\delta_7= 107/8,\,\delta_8=3$ and $\eta=1$. (Left) Solution for $u(X,T)$. (Right) Geometric construction identifying the asymptotic, discontinuous solution, $u(X,T)\simeq k_j$, for $(X,T)\in \mathcal{R}_j$. Multiple fusion and fission of shocks are displayed, including the simultaneous fusion and fission (red circle) of two initial shocks to give two new ones.
• Figure 5: Mass density $u_{X}(X,T)$ for the multi-kink solution \ref{['eq:weight decomposition']} with $N=8$. Parameters are the same as in Fig \ref{['fig: 8coll']}, $k_1=-4, \, k_2=-3,\, k_3=-2,\, k_4=-1 \,, k_5=1\,,k_6=2\,,k_7=3\,,k_8=4$, $\delta_1=-29/2,\, \delta_2=0,\, \delta_3=9,\,\delta_4=-115/24\,, \delta_5=0,\,\delta_6=15,\,\delta_7= 107/8,\,\delta_8=3$ and $\eta=1$. Both figures show the mass density $u_{X}(X,T)$ and its soliton behaviour. Multiple fission and fusion of solitons are displayed in the Left panel. The density plot on the Right aims instead at showing the effective phase shift due to the interaction. The black dashed lines are constructed along the centres of the trajectories of the incoming shocks, while the yellow dashed lines are instead obtained from the trajectories of the outgoing shocks. The black and yellow circles highlight the intersections between ingoing and outgoing lines.

Theorems & Definitions (10)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Proposition 1
• Remark 6
• Remark 7
• Proposition 2
• Remark 4