Conserved quantities in the Korteweg-de Vries equation

In summary, the conversation discusses the Kortweg-de Vries Equation and finding the relation between the coefficients c and d in order for a certain quantity to be conserved. The attempt at a solution involves creating a local conservation law and manipulating the integrand to show that it is a perfect derivative, but the attempt is not successful. The Wikipedia page on the KdV equation provides the values of c and d as -1 and 2, respectively.
  • #1
Emil_M
46
2

Homework Statement


Consider the Kortweg-de Vires Equation in the form

$$\frac{\partial \psi}{\partial t}+\frac{\partial^3 \psi}{\partial x^3}+6\psi\frac{\partial \psi}{\partial x}=0$$

Find the relation between the coefficients ##c## and ##d## , such that the following quantity is conserved:

$$c\; \int_{-\infty}^\infty\left(\frac{\partial\psi}{\partial x}\right)^2 \mathrm{d}x+d\;\int_{-\infty}^{\infty}\psi ^3 \mathrm{d}x$$

Homework Equations


A local conservation law is of the form ##D_t+F_x=0##.

The Attempt at a Solution


Usually, I would try to create a local conservation law, s.t. the quantity in question is conserved. But I really don't know how to do this in the given case. Thanks!
 
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  • #2
Hey, I've attempted to do the following:

Since the quantity in question needs to be conserved, it follows that

$$\frac{\partial}{\partial t}\left( \; \int_{-\infty}^\infty c\left(\frac{\partial\psi}{\partial x}\right)^2 \mathrm{d}x+d\;\int_{-\infty}^{\infty}\psi ^3 \mathrm{d}x \right) =0$$.

Thus, I have computed the temporal derivation, which yields

$$-c\int_{-\infty}^\infty \mathrm{d}x \left( 2c\frac{\partial^4 \psi}{\partial x^4}\frac{\partial\psi}{\partial x}+12c\left(\frac{\partial \psi}{\partial x}\right)^2+12c \psi\frac{\partial\psi}{\partial x}\frac{\partial^2 \psi}{\partial x^2}+3d\psi^2\frac{\partial^3\psi}{\partial x^3}+18\psi ^3 \frac{\partial \psi}{\partial x} \right)=0 $$

if I define ##d:=-c##.

If I can show that the integrand is a perfect derivative, I am finished. However, I can't seem to show that this is the case. Can anyone help me?
 
  • #3
I don't think you should have the factor of c outside the integral, but you need a factor of d in the last term of the integrand. (But I'm also one of those that deplore using d as a constant in anything involving calculus.)

It looks as though the integrand should be expressible in terms of an operator polynomial; something like (a + D)(D2 + D)(ψ3), where a is a constant and D is ∂/∂x, but I haven't been able to make it work.

I can only suggest putting in trial functions like ψ = sinx and seeing what happens.
Good luck.
 
  • #4
Why reinvent the wheel when Google is your friend? The Wikipedia page on the KdV equation has c=-1, d=2. Check it out.
 
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Likes John Park

1. What is the Korteweg-de Vries equation?

The Korteweg-de Vries (KdV) equation is a mathematical model that describes the behavior of long, shallow water waves. It was first proposed by Dutch physicist Johannes Martinus Burgers and Dutch mathematician Diederik Korteweg in 1895, and later refined by physicist Gustav de Vries in 1898.

2. What are conserved quantities in the KdV equation?

Conserved quantities in the KdV equation are physical properties that remain constant throughout the evolution of the system. In the case of the KdV equation, there are two conserved quantities: the mass and the energy.

3. How are conserved quantities related to symmetries in the KdV equation?

Conserved quantities in the KdV equation are directly related to symmetries in the equation. In particular, the two conserved quantities (mass and energy) are associated with the time and space translations, respectively. This means that the KdV equation is invariant under these symmetries, and as a result, the conserved quantities remain constant.

4. Can conserved quantities be used to solve the KdV equation?

Yes, the conserved quantities can be used to solve the KdV equation. By using the symmetries associated with the conserved quantities, scientists can apply techniques such as the inverse scattering transform or the method of characteristics to find exact solutions of the KdV equation.

5. Why are conserved quantities important in the study of the KdV equation?

Conserved quantities are important in the study of the KdV equation because they provide valuable insights into the behavior of the system. By studying how the conserved quantities evolve over time, scientists can gain a better understanding of the underlying dynamics and predict the long-term behavior of the system.

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