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The Traveling Wave Solutions to
the Korteweg de Vries Equation (KdV)

The Korteweg de Vries Equation (KdV) has various forms.
It may be expressed as

u_{t} + u_{x} + uu_{x} + u_{xxx} = 0

The KdV derives from the analysis of Korteweg and de Vries in 1895 to derive an equation
for water waters that would explain the existence of a smoothly humped wave observed in
nature. When the KdV equation was used in numerical simulations in the 1950's the
investigators found that the wave solutions persisted after interactions. These wave solutions
were called solitons.

Traveling Wave Solutions

Some aspects of the solutions to the KdV equations may be derived from analysis. A
traveling wave solution is of the form u(x-vt-x_{0}). Letting z=x-vt-x_{0}
the KdV equation becomes:

(1 −v)u_{z} + ½(u²)_{z} + u_{zzz} = 0

This may be immediately integrated with respect to z to give:

(1 −v)u + ½u² + u_{zz} = c_{0}

where c_{0} is an arbitrary constant.

The above equation may be multiplied by u_{z} to give

(1 −v)uu_{z} + ½u²u_{z} + u_{zz}u_{z} =
c_{0}u_{z} which is the same as
(1−v)½(u²)_{z} + 1/6(u³)_{z}
+ (½u_{z}²)_{z} =
c_{0}u_{z}

This obviously can be integrated with respect to z to give

(1−v)½u² + 1/6u³ + ½u_{z}² =
c_{0}u + c_{1}

In principle this equation could be solved for u_{z} and the result integrated.
As a practical matter this would be too cumbersome. Instead let us check to see if
U(z) = a·sech²(bz) is a solution to

Substituting these expressions into the KdV equation and dividing by -2ab·sech²(bz)tanh(bz)
gives

(1-v)+4vb² + (a − 12vb²)sech²(bz) = 0

Thus for the KdV equation to be satisfied for all z it must hold that

(1-v) + 4vb² = 0
and
a − 12b² = 0

These conditions imply that

b = ½[a/12]^{1/2}
and
v = 1/(1-4b²) = 1/(1 - a/3)

The parameter a represents the amplitude of the wave. Parameter b represents the inverse of
the width of the wave. The parameter v is the speed of the wave. A positive value of v
indicates movement to the right and a negative value movement to the left. Once any one of the
three parameters is specified the other two are determined.