Bounded Explicit Traveling Wave Solutions of Non-Linear Non-Dispersive Wave Equations

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

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.


Traveling Wave General Form



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

Evaluating the KdV Equation

Next, we integrate with respect to z:

Integrating w.r.t. z

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

Setting Integration Constant to zero

Which we can rearrange into:

Rearranged form

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

General Second Order ODE

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:
Multiplying through

Solving for φ'(z) yields:
Solving for the derivative

Multiplying by 2*m*[phi(z)]^(2m-1) yields:
After Multiplying through

Some substitutions

After subsituting in

Simplifying yields:
After some simplification

Another substitution

Solving for Psi

Solving yields:
Solving ODE

Solving the ODE

And therefore:
General Solution

So the general solution is:
Solution for phi

Which we can rewrite as:
Solution for phi

Where ε and μ are defined as:
Substitution for the above solution

Then by the little aside, the solution to the Nonlinear non-dispersive wave equation is:
Solving using the aside lemma
Where ε²>0, μ²>0 and Nonlinear f(u)