A number ordinary diffential equations have a solution of the form y(z)=a·sech²(bz). This represents curves such as are shown below:

This is the shape of the bow-wave that John Scot Russell observed in the Union Canal near Edinburgh in 1834 and followed for two miles.

A higher value of b represents a narrower solution as shown below:

A negative value for the parameter a gives a curve such as shown below:

The derivative terms of this function can be evaluated

dy/dz = 2a·sech(bz)(dsech(bz)/dz)b = -2ab·sech²(bz)tanh(bz)
and hence
y(dy/dz) = -2a²b·sech⁴(bz)tanh(bz)
 

The second derivative is given by:

d²y/dz² = 4ab²sech²(bz)tanh²(bz) - 2ab²sech⁴(bz)
but since tanh²(bz)=1−sech²(bz)
d²y/dz² = 4ab²sech²(bz) - 6ab²sech⁴(bz)
 

Therefore

d³y/dz³ = −8ab³sech²(bz)tanh(bz) + 24ab³sech⁴(bz)tanh(bz)
 

One ordinary differential equation which y(z)=a·sech²(bz) satisfies is:

(1 −v)y_z + ½(y²)_z + vy_zzz = 0
 

where v is a parameter and y_z denotes the derivative dy/dz.

In this case there are relationsips between the parameters a, b and v which must be satisfied.

The above ordinary differential equation is derived from the partial differential equation called the Regularized Long Wave Equation (RLWE). The RLWE has various forms but the relevant form here is:

u_t + u_x + uu_x - u_txx = 0
which is equivalent to
u_t + u_x + (½u²)_x - u_txx = 0
 

The ordinary differential equation shown previously arises from the RLWE if solutions of the form z=x−vt+x₀ are sought.

Another partial differential equation that leads to the same ordinary differential equation shown above for the RLWE is the Korteweg de Vries (KdV) equation. The Korteweg-de Vries equation 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 waves that would explain the existence of smoothly humped waves of the sort observed by John Scot Russell in 1834. 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.

The RLW equation was formulated by Peregrine (1966) as an alternative to KdV equation for studying soliton phenomenon. It was proposed because it would not have the same limitations for the size of the time step in numerical solution that the KdV has.

The General Case

Consider the partial derivatives of y(z)=a·sech²(bz) with respect to the parameters a and b; i.e.,

∂y/∂a = sech²(bz)
∂y/∂b = 2a·sech(bz)[sech(bz)tanh(bz)]z = 2a·sech²(bz)tanh(bz)z

(To be continued.)