Lie Group Analysis of a Forced KdV Equation

Table 1

Classification results.

No.	𝐹	Condition on consts.	Extension of principal Lie algebra
1.	𝐹0(𝑛+𝑡)^−5/3+𝑝5𝛼	𝛼, 𝐹0, 𝑝≠0	𝑋3=6𝛼(𝑛+𝑡)𝜕𝑡+𝛼𝑝𝑡^2+2𝑥𝜕𝑥+2𝑝𝑡−2(𝑐+𝛼𝑢)𝜕𝑢
2.	𝐹0+𝑞𝑡3𝛼	𝛼, 𝑞≠0	𝑋3=6𝛼𝜕𝑡+𝛼𝑞𝑡^2𝜕𝑥+2𝑞𝑡𝜕𝑢
3.	𝐹0(𝑛+𝑡)^−5/3	𝐹0≠0	𝑋3=3𝛼(𝑛+𝑡)𝜕𝑡+𝛼𝑥𝜕𝑥−2(𝑐+𝛼𝑢)𝜕𝑢
4.	𝐹0	𝐹0≠0	𝑋3=𝜕𝑡,
			𝑋4=6𝛼𝜕𝑡+𝛼5𝛼𝐹0𝑡^2+2𝑥𝜕𝑥+25𝛼𝐹0𝑡−2(𝑐+𝛼𝑢)𝜕𝑢

Here 𝐹₀, 𝑛, 𝑝 and 𝑞 are arbitrary constants.