The KdV equation and Solitary Waves

Main Concept A solitary wave, or soliton, is a wave-packet that propagates through space without a change in its shape. Such a phenomenon can be observed on the surface of shallow water, as first described by John Scott Russell in 1844 in his "Report on Waves" (see e.g. "John Scott Russell and the solitary wave"). This phenomenon is not only studied in hydrodynamics, but also e.g. in fiber optics, neuroscience and particle physics. Furthermore, there are various realizations of solitary waves. KdV Equation and Soliton Solutions One realization of such a phenomenon can be achieved via the Korteweg-de Vries (KdV) equation. It is a nonlinear third-order partial differential equation for a function $u\left(t\,x\right)$ of the form >  $\mathrm{KdV}≔\frac{\partial }{\partial t} u\left(t\,x\right)+6 u\left(t\,x\right)\cdot \frac{\partial }{\partial x}\mathit{ }u\left(t\,x\right)\+\frac{{\partial }^3}{{\partial }^{}{x}^3}u\left(t\,x\right)=0$ $\mathrm{KdV}≔\frac{\partial }{\partial t}u\left(t\,x\right)+6u\left(t\,x\right)\left(\frac{\partial }{\partial x}u\left(t\,x\right)\right)+\frac{{\partial }^3}{\partial x^3}u\left(t\,x\right)=0$ (1.1) with $t$ being the time and $x$ the space coordinate. A specific type of solutions to this equation are solitary waves. A single soliton is described by the solution of (1.1) of the form >  $\mathrm{Soliton}≔v\cdot {\mathrm{sech}}^2\left(\sqrt{\frac{v}{2}}\cdot \left(x-2\cdot v\cdot t-\mathrm{x\_\_0}\right)\right)\:$ as can be easily verified by pdetest >  $\mathrm{pdetest}\left(u\left(t\,x\right)\=\mathrm{Soliton}\, \mathrm{KdV}\right)$ $0$ (1.2) returning $0$, i.e. the KdV-equation is annihilated by the soliton solution. This solution contains two constants: $v$ corresponding to the amplitude of this wave and $\mathrm{x\_\_0}$ that is related to a spatial shift (a third one is already fixed in such a way that $\mathrm{lim\_\_x\→\±\∞}\=0$). Note that the width and the velocity of this traveling wave are related to its amplitude $v$ and are not independent parameters. The following plot animates the propagation of a soliton with adjustable $v$.     $\mathrm{v\_\_}\:$ Note that the soliton maintains its shape while propagating. In many other models, wave packets are a superposition of plane waves of different wavelengths. If their propagation velocity depends on their frequency, then the wave packet disperses as it travels through space. In the KdV model non-linear effects exactly cancel these dispersive effects and the waves maintain their shape.   Due to the nonlinearity of the KdV equation, the superposition of two solitons is not necessarily another solution. Indeed, you can easily check using $\mathrm{pdetest}$ that, for example,  $2\cdot \mathrm{Soliton}$ is not a solution. Instead, there exists a method to obtain multi-soliton solutions called classical inverse scattering method, see, for example, Dunajski. Applying this method, the two-soliton solution turns out to be given by >  $\mathrm{TwoSolitons}≔-2\left(\mathrm{v\_\_1}-\mathrm{v\_\_2}\right)\cdot \frac{\mathrm{v\_\_1}\cdot {\mathrm{sech}}^2\left(\sqrt{\frac{\mathrm{v\_\_1}}{2}}\left(x-2\cdot \mathrm{v\_\_1}\cdot t-\mathrm{x\_\_1}\right)\right)\+\mathrm{v\_\_2}\cdot {\mathrm{csch}}^2\left(\sqrt{\frac{\mathrm{v\_\_2}}{2}}\left(x-2\cdot \mathrm{v\_\_2}\cdot t-\mathrm{x\_\_2}\right)\right)}{{\left(\sqrt{2\cdot \mathrm{v\_\_1}}\tanh \left(\sqrt{\frac{\mathrm{v\_\_1}}{2}}\left(x-2\cdot \mathrm{v\_\_1}\cdot t-\mathrm{x\_\_1}\right)\right)-\sqrt{2\cdot \mathrm{v\_\_2}}\coth \left(\sqrt{\frac{\mathrm{v\_\_2}}{2}}\left(x-2\cdot \mathrm{v\_\_2}\cdot t-\mathrm{x\_\_2}\right)\right)\right)}^2}\:$ which can easily be verified using $\mathrm{pdetest}$ (note that the evaluation takes a few seconds) >  $\mathrm{pdetest}\left(u\left(t\,x\right)\=\mathrm{TwoSolitons}\, \mathrm{KdV}\right)$ $0$ (1.3) In the following you can study this solution and identify important properties of solitary waves.

1) The following plot shows the two-soliton solution $\mathrm{TwoSolitons}$ at $t\=0$ for adjustable values of $\mathrm{v\_\_1}$, $\mathrm{v\_\_2}$, $\mathrm{x\_\_1}$ and $\mathrm{x\_\_2}$. Play with these values using the four sliders to see how they determine the amplitudes and position of the two well-separated solitons. Note that the value of $\mathrm{v\_\_2}$ must be bigger than that of $\mathrm{v\_\_1}$ for a soliton solution.

2) You may have realized in the previous plot that the two solitons are not completely independent of each other. Investigate their scattering in the following animation by adjusting all the parameters in such a way that two well-separated solitons in the initial state get closer as they propagate (click on Play), interact and then separate. Afterwards, compare the shapes of the initial (in gray) and final solitons. Investigate the shape of the wave during the interaction and realize that it is not a simple superposition of the two solitons.

3) Although the two solitons after the interaction look just as they did before, they experience a phase shift during the scattering. Investigate this in the following animation. The top plot contains an animation of a two-soliton interaction. The middle and bottom plot contain single solitons with the same amplitude as the solitons in the two-soliton plot and the two lines in the plots denote the position of these. By clicking on Play, observe that the solitons in the two-soliton solution behave just like single solitons as long as they are well-separated. During the interaction though, they experience a phase shift, which is also visible in the density and three-dimensional surface plot of the interaction. This confirms again that the two-soliton solution is not a simple superposition of two single solitons.

4) Alice and Bob are standing on two platforms over a canal and are sending each other solitary waves. There is a bridge for cars between the two of height $h$ which makes it impossible to send single solitons with amplitude $v$ bigger than $h$. Still Alice wants to send Bob a soliton $A$ with amplitude $\mathrm{v\_\_A}\>h$. How would she have to model a second soliton $B$ with parameters $\mathrm{v\_\_B}$ and $\mathrm{x\_\_B}$ such that soliton $A$ fits below the bridge and splashes over Bob? Check your solution by clicking on "Animate" and then wait for the Play button to appear and press it to start the animation.