Riccati-type Bäcklund transformations of nonisospectral and generalized variable-coefficient KdV equations

We extend the method of constructing Bäcklund transformations for integrable equations through Riccati equations to the nonisospectral and the variable-coefficient equations. By taking nonisospectral and generalized variable-coefficient Korteweg–de Vries (KdV) equations as examples, their Bäcklund transformations are obtained under a more generalized constrain condition. In addition, the Lax pairs and infinite numbers of conservation laws of these equations are given. Especially, some classical equations such as the cylindrical KdV equation are just the special cases of the constrain condition.