Today, I am going to explain a famous equation widely used to study the nonlinear processes in physics-the KdV equation. As a plasma physicist, I have encountered this equation a lot of times during my research career so far. This equation gives lots of insights into the nonlinear physical processes going on in plasma. KdV equation when applied to space and astrophysical scenarios uncovers a lot of underlying physics of the phenomena observed in these plasma environments. Let us first try to understand the meaning of this equation:
Meaning of the KdV 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.
What is a Wave?
When someone asks you the question- “What is a wave?”, what is your reply? Most of the students make a wave-like motion with their hands when asked this question. 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? The KdV Equation is going to introduce you to a concept that you might have thought randomly but never actually read about it.
So, I will explain all the intriguing elements of 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 is 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. This 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 respect to space coordinate. 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. In Ohm’s law, a change in voltage leads to a proportional change current resulting in a straight line graph. 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 is 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 is dispersion. Dispersion is caused due to different components of white light propagating with different speeds in a medium and hence separate.
It is just like when you are going to a fair with your family of four people. You all are together until you were moving in an almost vacant space. When all of you have to make your way through a crowded part of the fair-ground, you get separated. This is because all of you choose different paths or one of you is slow and the other is fast. This is the dispersive effect as you all get dispersed in the fair.
In 1834, John Scott Russel was observing a boat which was rapidly drawn along a narrow channel by a pair of horses. Alongside this, he also observed a stable hump-like or a pulse-like structure in the water. 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. That physical quantity may be the 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. In other words, let us say the dispersive effect reduces the quantity ‘y’ by its square root.
At this point, the nonlinear effects exactly balance 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.
The KdV equation is one of the fundamental equations in fluid mechanics and plasma physics. I hope this article helps you to understand the physics behind some basic phenomena in nature and how their balancing leads to some wonderful observations in space and astrophysical regions. Happy reading!