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.
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:
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