Meaning of the Equation:
The KdV equation is a nonlinear, dispersive partial differential equation for a variable (electrostatic potential in the case of plasma) which is a function of space and time. The KdV equation is a mathematical model for waves in different mediums such as shallow water surfaces, plasma etc.
When someone asks you the question- "What is a wave?", what is your reply? Most of the students on being asked this question would make a wave-like motion with their hands. Have you ever wondered how a propagating wave can have a simple mathematical form? Does this ever strike you which laws might govern the simple physical phenomenon such as waves and ripples in water? Well, today's equation is going to introduce you to a concept that you might have a random thought about but never actually read about. So, I will explain all the intriguing elements about today's equation while beginning from scratch.
A wave is the disturbance of a physical quantity that propagates in a medium. The Korteweg-de Vries or the KdV equation as it is famously known as an equation that governs the dynamics of a wave propagating in a medium. The equation is in a variable that is a function of space and time which means that the equation tells the evolution of a physical quantity in space and time.
Meaning of different terms in the KdV equation :
- The first term in this equation depicts the derivative of a quantity \phi w.r.t. time.
- The second term of the equation shows the product of a quantity \phi with its own derivative with space coordinate and this term represents the nonlinear effects in the medium.
- The third term represents a third order derivative of the quantity \phi w.r.t. space and indicate the dispersive effects in the medium.
Now, we almost know what is the meaning of linearity- it means the linear variation of one quantity with a change in other just like in Ohm's law- where a change in voltage leads to a proportional change current resulting in a straight line graph. So this means that essentially if I am considering nonlinear effects this might mean a non-proportional change of a dependent physical quantity with a change in the independent physical quantity. Consider that y varies as x^2. This will be called a nonlinear effect in y.
The reason behind the formation of rainbows is the splitting of light into seven colors after passing through small droplets of water. This phenomenon is called dispersion. The cause of dispersion is that the different components of white light propagate with different speeds in a medium and hence, they get separated. Its just like when you are going to a fair with your family of four people. You all were together until you were moving in an almost vacant space, but when you four had to make your way through a crowded part of the fair-ground, you got separated because all of you chose different paths or one of you was slow and the other was fast. This is called dispersive effect when you all got dispersed in the fair.
In 1834, John Scott Russel observed a stable hump-like or a pulse-like structure in water while observing a boat which was rapidly drawn along a narrow channel by a pair of horses. The hump-like structure moved with large velocity and without changing its shape for a long time. The KdV equation was not studied much until it was discovered numerically that its solutions seemed to decompose at later times into a collection of hump-like localized structures called solitons.
A localized solution of the KdV equation: Soliton
The KdV equation indicates both the nonlinear and dispersive effects on a physical quantity or disturbance that propagates in a medium. Let us suppose that the dispersive effects reduce the value of the quantity 'y' by exactly the value it was increased, i.e, let us say the dispersive effect reduce the quantity 'y' by its square root. At this point, the nonlinear effects have exactly balanced the dispersive effects and the physical quantity propagates without changing its value. Such a waveform that does not change its shape over long distances is called a soliton. Solitons have various applications in the field of plasma, optics, material science as well as in the field of gravity waves.