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32 Painlevé TranscendentsApplications

§32.13 Reductions of Partial Differential Equations

Contents
  1. §32.13(i) Korteweg–de Vries and Modified Korteweg–de Vries Equations
  2. §32.13(ii) Sine-Gordon Equation
  3. §32.13(iii) Boussinesq Equation

§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:
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See also:
Annotations for §32.13(i), §32.13 and Ch.32

has the scaling reduction

32.13.2
z=x(3t)^{-1/3},
v(x,t)=(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), §32.13 and Ch.32

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:
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See also:
Annotations for §32.13(i), §32.13 and Ch.32

has the scaling reduction

32.13.4
z=x(3t)^{-1/3},
u(x,t)=-(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), §32.13 and Ch.32

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,
ⓘ
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), §32.13 and Ch.32

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

ⓘ
Keywords:
Painlevé transcendents, applications, sine-Gordon equation
Notes:
See Ablowitz and Segur (1977).
Referenced by:
§32.11(v), Erratum (V1.0.1) for References, Erratum (V1.0.7) for Other
Permalink:
http://dlmf.nist.gov/32.13.ii
Correction (effective with 1.0.7):
The reference Wong and Zhang (2009a), formerly incorrectly placed in this subsection, has been moved to the end of §32.11(v).
Amendment (effective with 1.0.1):
Two references were added at the end of this subsection.
See also:
Annotations for §32.13 and Ch.32

The sine-Gordon equation

32.13.6 u_{xt}=\sin u,
ⓘ
Symbols:
\sin\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), §32.13 and Ch.32

has the scaling reduction

32.13.7
z=xt,
u(x,t)=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), §32.13 and Ch.32

where v(z) satisfies (32.2.10) with \alpha=\tfrac{1}{2} and \gamma=0. In consequence if w=\exp\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 (2009b).

§32.13(iii) Boussinesq Equation

ⓘ
Keywords:
Boussinesq equation, Painlevé transcendents, applications, partial differential equations
Notes:
See Ablowitz and Segur (1977).
Permalink:
http://dlmf.nist.gov/32.13.iii
See also:
Annotations for §32.13 and Ch.32

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), §32.13 and Ch.32

has the traveling wave solution

32.13.9
z=x-ct,
u(x,t)=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), §32.13 and Ch.32

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

32.13.10 v^{\prime\prime}=6v^{2}+(c^{2}-1)v+Az+B,

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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