Reductions and New Exact Solutions of ZK, Gardner KP, and Modified KP Equations via Generalized Double Reduction Theorem

Table 1

Multipliers and conserved vectors for (14).

Multipliers	Conserved vector
Λ1=𝑥−𝑎𝑡−𝑏𝑡𝑢	𝑇^𝑡1𝑢=𝑥𝑢−𝑢𝑎𝑡−^2𝑏𝑡2
	𝑇^𝑥1𝑢=−^3𝑏^2𝑡3+𝑢^2𝑏𝑥2−𝑎𝑏𝑡𝑢^2−𝑏𝑐𝑡𝑢𝑢𝑥𝑥+𝑏𝑐𝑡𝑢^2𝑥2
	−𝑏𝑑𝑡𝑢𝑢𝑦𝑦2+𝑎𝑢𝑥−𝑎^2𝑡𝑢+𝑐𝑥𝑢𝑥𝑥−𝑎𝑐𝑡𝑢𝑥𝑥−𝑐𝑢𝑥+𝑑𝑥𝑢𝑦𝑦−𝑎𝑑𝑡𝑢𝑦𝑦
	𝑇^𝑦1=−𝑏𝑑𝑡𝑢𝑢𝑥𝑦2+𝑏𝑑𝑡𝑢𝑦𝑢𝑥2−𝑑𝑢𝑦
Λ2=𝑢	𝑇^𝑡2=𝑢^22
	𝑇^𝑥2=𝑢^3𝑏3+𝑢^2𝑎2+𝑐𝑢𝑢𝑥𝑥−𝑐𝑢^2𝑥2+𝑑𝑢𝑢𝑦𝑦2
	𝑇^𝑦2=𝑑𝑢𝑢𝑥𝑦2−𝑑𝑢𝑥𝑢𝑦2
Λ3=𝑓(𝑦)	𝑇^𝑡3=𝑓(𝑦)𝑢
	𝑇^𝑥3=𝑓𝑦𝑏𝑢^22+𝑎𝑢+𝑐𝑢𝑥𝑥
	𝑇^𝑦3=𝑓(𝑦)𝑑𝑢𝑥𝑦