# §32.13 Reductions of Partial Differential Equations

## §32.13(i) Korteweg–de Vries and Modified Korteweg–de Vries Equations

The modified Korteweg–de Vries (mKdV) equation

32.13.1

has the scaling reduction

32.13.2

where satisfies  with a constant of integration.

The Korteweg–de Vries (KdV) equation

32.13.3

has the scaling reduction

32.13.4

where satisfies .

Equation (32.13.3) also has the similarity reduction

32.13.5

where is an arbitrary constant and is expressible in terms of solutions of . See Fokas and Ablowitz (1982) and P. J. Olver (1993b, p. 194).

## §32.13(ii) Sine-Gordon Equation

The sine-Gordon equation

32.13.6

has the scaling reduction

32.13.7

where satisfies (32.2.10) with and . In consequence if , then satisfies  with and .

## §32.13(iii) Boussinesq Equation

The Boussinesq equation

32.13.8

has the traveling wave solution

32.13.9

where is an arbitrary constant and satisfies

with and constants of integration. Depending whether or , is expressible in terms of the Weierstrass elliptic function (§23.2) or solutions of , respectively.