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§32.13 Reductions of Partial Differential Equations

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

The modified Korteweg–de Vries (mKdV) equation

32.13.1v_{t}-6v^{2}v_{x}+v_{{xxx}}=0,
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Symbols:
v(x,t): solution
Permalink:
http://dlmf.nist.gov/32.13.E1
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has the scaling reduction

32.13.2
z=x(3t)^{{-1/3}},
v(x,t)=(3t)^{{-1/3}}w(z),
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Symbols:
z: real and v(x,t): solution
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where w(z) satisfies \mbox{P}_{{\mbox{\scriptsize II}}} with \alpha a constant of integration.

The Korteweg–de Vries (KdV) equation

32.13.3u_{t}+6uu_{x}+u_{{xxx}}=0,
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Symbols:
u(x,t): solution
Referenced by:
§32.13(i)
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http://dlmf.nist.gov/32.13.E3
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has the scaling reduction

32.13.4
z=x(3t)^{{-1/3}},
u(x,t)=-(3t)^{{-2/3}}(w^{{\prime}}+w^{2}),
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Symbols:
z: real and u(x,t): solution
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where w(z) satisfies \mbox{P}_{{\mbox{\scriptsize II}}}.

Equation (32.13.3) also has the similarity reduction

32.13.5
z=x+3\lambda t^{2},
u(x,t)=W(z)-\lambda t,
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Symbols:
z: real and u(x,t): solution
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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

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Notes:
See Ablowitz and Segur (1977).
Keywords:
Painlevé transcendents, sine-Gordon equation
Referenced by:
Other Changes
Permalink:
http://dlmf.nist.gov/32.13.ii
Amendment (effective with 1.0.1):
Two references were added at the end of this subsection.

Reported 2011-03-02

The sine-Gordon equation

32.13.6u_{{xt}}=\mathop{\sin\/}\nolimits u,
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Symbols:
\mathop{\sin\/}\nolimits z: sine function and u(x,t): solution
Permalink:
http://dlmf.nist.gov/32.13.E6
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has the scaling reduction

32.13.7
z=xt,
u(x,t)=v(z),
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Symbols:
z: real, u(x,t): solution and v(z): solution
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http://dlmf.nist.gov/32.13.E7
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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.

See also Wong and Zhang (2009a, b).

§32.13(iii) Boussinesq Equation

The Boussinesq equation

32.13.8u_{{tt}}=u_{{xx}}-6(u^{2})_{{xx}}+u_{{xxxx}},
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Symbols:
u(x,t): solution
Permalink:
http://dlmf.nist.gov/32.13.E8
Encodings:
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has the traveling wave solution

32.13.9
z=x-ct,
u(x,t)=v(z),

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

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.

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