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

Table 3

Multipliers and conserved vectors for (78).

Multipliers	Conserved vector
Λ1=16𝑐−𝑓𝑡𝑦^3+6𝑐𝑓(𝑡)𝑥𝑦	𝑇^𝑡1=16𝑐−𝑓𝑡𝑦^3+6𝑐𝑓(𝑡)𝑥𝑦𝑢𝑥
	𝑇^𝑥11=−(𝑡6𝑐𝑦𝑓)−6𝑑𝑐𝑥𝑢^2𝑢𝑥+2𝑑𝑐𝑢^3−6𝑎𝑐𝑥𝑢𝑢𝑥+3𝑐𝑎𝑢^2+6𝑏𝑐𝑢𝑥𝑥+6𝑐^2𝑢−6𝑐^2𝑥𝑢𝑥−6𝑏𝑐𝑥𝑢𝑥𝑥𝑥
	+1𝑓6𝑐^𝑡𝑡𝑦^31𝑢−6𝑐𝑦𝑓𝑡𝑑𝑦^2𝑢^2𝑢𝑥+𝑎𝑦^2𝑢𝑢𝑥+6𝑐𝑥𝑢+𝑐𝑦^2𝑢𝑥+𝑏𝑦^2𝑢𝑥𝑥𝑥
	𝑇^𝑦1=𝑓(𝑡)−𝑐𝑥𝑢+𝑐𝑥𝑦𝑢𝑦+𝑓𝑡12𝑦^21𝑢−6𝑦^3𝑢𝑦
Λ2=12𝑐−𝑓𝑡𝑦^2+2𝑐𝑥𝑓(𝑡)	𝑇^𝑡21=−𝑢2𝑐^𝑥𝑓𝑡𝑦^2−2𝑓(𝑡)𝑐𝑥
	𝑇^𝑥21=−𝑓(𝑡6𝑐)−6𝑑𝑐𝑥𝑢^2𝑢𝑥+2𝑑𝑐𝑢^3−6𝑎𝑐𝑥𝑢𝑢𝑥+3𝑐𝑎𝑢^2+6𝑏𝑐𝑢𝑥𝑥+6𝑐^2𝑢−6𝑐^2𝑥𝑢𝑥−6𝑏𝑐𝑥𝑢𝑥𝑥𝑥
	+1𝑓2𝑐^𝑡𝑡𝑦^21𝑢−𝑓6𝑐^𝑡6𝑐𝑥𝑢+3𝑐𝑦^2𝑢𝑥+3𝑏𝑦^2𝑢𝑥𝑥𝑥+3𝑑𝑦^2𝑢^2𝑢𝑥+3𝑎𝑦^2𝑢𝑢𝑥
	𝑇^𝑦2=𝑓𝑡1𝑦𝑢−2𝑦^2𝑢𝑦+𝑓(𝑡)𝑐𝑥𝑢𝑦
Λ3=𝑦𝑓(𝑡)	𝑇^𝑡3=𝑓(𝑡)𝑦𝑢𝑥
	𝑇^𝑥3=𝑦𝑓(𝑡)𝑑𝑢^2𝑢𝑥+𝑎𝑢𝑢𝑥+𝑐𝑢𝑥+𝑏𝑢𝑥𝑥𝑥−𝑦𝑓𝑡𝑢
	𝑇^𝑦3=−𝑓(𝑡)𝑐𝑢+𝑓(𝑡)𝑐𝑦𝑢𝑦
Λ4=𝑓(𝑡)	𝑇^𝑡4=𝑓(𝑡)𝑢𝑥
	𝑇^𝑥4=𝑓(𝑡)𝑑𝑢^2𝑢𝑥+𝑎𝑢𝑢𝑥+𝑐𝑢𝑥+𝑏𝑢𝑥𝑥𝑥−𝑓𝑡𝑢
	𝑇^𝑦4=𝑓(𝑡)𝑐𝑢𝑦