Consider the two-dimensional non-linear Korteweg-de Vries (KdV) Equation:

Where *A, B, α, β, γ, δ* are real constants such that |α| + |β| + |γ| + |δ| >0, and |A| + |B| >0, and *f(u)* is a non-linear smooth function of u, such that *f(0) = 0*.

Suppose:

Let:

If we substitute in *f(u)* into the KdV Equation, we get:

Next, we integrate with respect to z:

Since we are looking for the general solution, we can set the integration constant equal to zero:

Which we can rearrange into:

A quick aside into solving the equation... Given an equation of the form:

Where *α > 0, β > 0, m > 0*.

If we multiply the ODE by *2φ'(z)* and integrate with respect to z, setting the integration constant equal to zero then:

Solving for *φ'(z)* yields:

Multiplying by yields:

Let:

Then:

Simplifying yields:

Let:

Then:

Solving yields:

Giving:

And therefore:

So the general solution is:

Which we can rewrite as:

Where ε and μ are defined as:

Then by the little aside, the solution to the Nonlinear non-dispersive wave equation is:

Where *ε²>0, μ²>0* and