§32.13 Reductions of Partial Differential Equations

The modified Korteweg–de Vries (mKdV) equation

has the scaling reduction

where $w(z)$ satisfies ${\text{P}}_{\text{II}}$ with $\alpha$ a constant of integration.

The Korteweg–de Vries (KdV) equation

has the scaling reduction

where $w(z)$ satisfies ${\text{P}}_{\text{II}}$.

Equation (32.13.3) also has the similarity reduction

where $\lambda$ is an arbitrary constant and $W(z)$ is expressible in terms of solutions of ${\text{P}}_{\text{I}}$. See Fokas and Ablowitz (1982) and P. J. Olver (1993b, p. 194).

The sine-Gordon equation

has the scaling reduction

where $v(z)$ satisfies (32.2.10) with $\alpha =\frac{1}{2}$ and $\gamma =0$. In consequence if $w=\exp \left(-\mathrm{i}v\right)$, then $w(z)$ satisfies ${\text{P}}_{\text{III}}$ with $\alpha =-\beta =\frac{1}{2}$ and $\gamma =\delta =0$.

See also Wong and Zhang (2009b).

The Boussinesq equation

has the traveling wave solution

where $c$ is an arbitrary constant and $v(z)$ satisfies

with $A$ and $B$ constants of integration. Depending whether $A=0$ or $A\ne 0$, $v(z)$ is expressible in terms of the Weierstrass elliptic function (§23.2) or solutions of ${\text{P}}_{\text{I}}$, respectively.