# §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 $v_{t}-6v^{2}v_{x}+v_{xxx}=0,$ Symbols: $v(x,t)$: solution Permalink: http://dlmf.nist.gov/32.13.E1 Encodings: TeX, pMML, png See also: Annotations for 32.13(i)

has the scaling reduction

 32.13.2 $\displaystyle z$ $\displaystyle=x(3t)^{-1/3},$ $\displaystyle v(x,t)$ $\displaystyle=(3t)^{-1/3}w(z),$ Symbols: $z$: real and $v(x,t)$: solution Permalink: http://dlmf.nist.gov/32.13.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 32.13(i)

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

The Korteweg–de Vries (KdV) equation

 32.13.3 $u_{t}+6uu_{x}+u_{xxx}=0,$ Symbols: $u(x,t)$: solution Referenced by: §32.13(i) Permalink: http://dlmf.nist.gov/32.13.E3 Encodings: TeX, pMML, png See also: Annotations for 32.13(i)

has the scaling reduction

 32.13.4 $\displaystyle z$ $\displaystyle=x(3t)^{-1/3},$ $\displaystyle u(x,t)$ $\displaystyle=-(3t)^{-2/3}(w^{\prime}+w^{2}),$ Symbols: $z$: real and $u(x,t)$: solution Permalink: http://dlmf.nist.gov/32.13.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 32.13(i)

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

Equation (32.13.3) also has the similarity reduction

 32.13.5 $\displaystyle z$ $\displaystyle=x+3\lambda t^{2},$ $\displaystyle u(x,t)$ $\displaystyle=W(z)-\lambda t,$ Symbols: $z$: real and $u(x,t)$: solution Permalink: http://dlmf.nist.gov/32.13.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 32.13(i)

where $\lambda$ is an arbitrary constant and $W(z)$ is expressible in terms of solutions of $\mbox{P}_{\mbox{\scriptsize I}}$. 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 $u_{xt}=\mathop{\sin\/}\nolimits u,$ Symbols: $\mathop{\sin\/}\nolimits\NVar{z}$: sine function and $u(x,t)$: solution Permalink: http://dlmf.nist.gov/32.13.E6 Encodings: TeX, pMML, png See also: Annotations for 32.13(ii)

has the scaling reduction

 32.13.7 $\displaystyle z$ $\displaystyle=xt,$ $\displaystyle u(x,t)$ $\displaystyle=v(z),$ Symbols: $z$: real, $u(x,t)$: solution and $v(z)$: solution Permalink: http://dlmf.nist.gov/32.13.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 32.13(ii)

where $v(z)$ satisfies (32.2.10) with $\alpha=\tfrac{1}{2}$ and $\gamma=0$. In consequence if $w=\mathop{\exp\/}\nolimits\!\left(-iv\right)$, then $w(z)$ satisfies $\mbox{P}_{\mbox{\scriptsize III}}$ with $\alpha=-\beta=\tfrac{1}{2}$ and $\gamma=\delta=0$.

## §32.13(iii) Boussinesq Equation

The Boussinesq equation

 32.13.8 $u_{tt}=u_{xx}-6(u^{2})_{xx}+u_{xxxx},$ Symbols: $u(x,t)$: solution Permalink: http://dlmf.nist.gov/32.13.E8 Encodings: TeX, pMML, png See also: Annotations for 32.13(iii)

has the traveling wave solution

 32.13.9 $\displaystyle z$ $\displaystyle=x-ct,$ $\displaystyle u(x,t)$ $\displaystyle=v(z),$ Symbols: $z$: real, $u(x,t)$: solution, $c$: constant and $v(z)$: solution Permalink: http://dlmf.nist.gov/32.13.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 32.13(iii)

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

 32.13.10 $v^{\prime\prime}=6v^{2}+(c^{2}-1)v+Az+B,$ Symbols: $z$: real, $c$: constant, $v(z)$: solution, $A$: integration constant and $B$: integration constant Permalink: http://dlmf.nist.gov/32.13.E10 Encodings: TeX, pMML, png See also: Annotations for 32.13(iii)

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