Table 2 Reduction of the DSWS

(1) \(X_{1}\)

ξ = x, u(x,t)=F(ξ), v(x,t)=H(ξ).

\(H' = 0, 3kFH' + 3bF'H - aH''' = 0\).

(2) \(X_{2}\)

ξ = t, u(x,t)=F(ξ), v(x,t)=H(ξ).

\(F' = 0\), \(H' = 0\).

(3) \(X_{3}\)

\(\xi = x{t^{-{{1 \over 3}}}}\), \(u ( {x,t} ) = {t^{ - { {2 \over 3}}}}F ( \xi )\), \(v ( {x,t} ) = {t^{ - { {2 \over 3}}}}H ( \xi ) \).

\(\xi F' - 6HH' + 2F = 0\), \(- 9kFH' - 9bF'H + \xi H' + 3aH''' + 2H = 0\). (A)

(4) \(X_{2}-X_{1}\)

ξ = x + t, u(x,t)=F(ξ), v(x,t)=H(ξ).

\(2HH' + F' = 0,3bF'H + 3kFH' - aH''' + H' = 0\).

(5) \(X_{2}+X_{1}\)

ξ = x − t, u(x,t)=F(ξ), v(x,t)=H(ξ).

\(2HH' - F' = 0\), \(3bF'H + 3kFH' - aH''' - H' = 0\). (B)