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32 Painlevé TranscendentsApplications

§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 vt-6v2vx+vxxx=0,

has the scaling reduction

32.13.2 z =x(3t)-1/3,
v(x,t) =(3t)-1/3w(z),

where w(z) satisfies PII with α a constant of integration.

The Korteweg–de Vries (KdV) equation

32.13.3 ut+6uux+uxxx=0,

has the scaling reduction

32.13.4 z =x(3t)-1/3,
u(x,t) =-(3t)-2/3(w+w2),

where w(z) satisfies PII.

Equation (32.13.3) also has the similarity reduction

32.13.5 z =x+3λt2,
u(x,t) =W(z)-λt,

where λ is an arbitrary constant and W(z) is expressible in terms of solutions of PI. 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 uxt=sinu,

has the scaling reduction

32.13.7 z =xt,
u(x,t) =v(z),

where v(z) satisfies (32.2.10) with α=12 and γ=0. In consequence if w=exp(-iv), then w(z) satisfies PIII with α=-β=12 and γ=δ=0.

See also Wong and Zhang (2009b).

§32.13(iii) Boussinesq Equation

The Boussinesq equation

32.13.8 utt=uxx-6(u2)xx+uxxxx,

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

32.13.10 v′′=6v2+(c2-1)v+Az+B,

with A and B constants of integration. Depending whether A=0 or A0, v(z) is expressible in terms of the Weierstrass elliptic function (§23.2) or solutions of PI, respectively.