Approximate Symmetries of the Harry Dym Equation

Table 3

Approximately differential invariants for the perturbed Harry Dym equation.

Operator	Approximate differential invariants
𝐯1	𝑡,𝑢
𝐯2	𝑥,𝑢
𝐯3	𝑢𝑡,𝑥
𝐯4	𝑥,𝑢𝑡^1/3
𝐯5	𝑢𝑡,𝑥^2
𝐯7+𝑎𝐯8	−ln𝑥𝑎𝑢+𝑡,𝑥
𝐯6+𝐯8	𝑢𝑡,𝑥+1
𝐯6−𝐯7+𝐯8	𝑢ln(𝑥+1)+𝑡,𝑥+1
𝐯6+𝐯7+𝐯8	𝑢−ln(𝑥+1)+𝑡,𝑥+1
𝐯2+𝑎𝐯8	−ln𝑥𝑢𝑎𝜀+𝑡,𝑥
𝐯2−𝐯6+𝑎𝐯8	−ln(𝑎𝑥−1)𝑢𝑎𝜀+𝑡,𝑎𝑥−1
𝐯2+𝐯6+𝑎𝐯8	−ln(𝑎𝑥+1)𝑢𝑎𝜀+𝑡,𝑎𝑥+1
𝐯1+𝑎𝐯2+𝑏𝐯7	−𝑏𝜀𝑥−𝑎𝜀+𝑡,𝑢
𝑎𝐯1+𝑏𝐯2+𝐯5; +𝑐𝐯6+𝑑𝐯7	−𝑑𝜀−𝑏√𝑥𝑐𝜀+𝑎arctan√𝑢𝑐𝜀+𝑎+𝑡,𝑥^2+𝑐𝜀+𝑎