Transformations for a generalized variable-coefficient Korteweg-de Vries model from blood vessels, Bose-Einstein condensates, rods and positons with symbolic computation

The variable-coefficient Korteweg-de Vries (KdV)-typed models, although often hard to be studied, are of current interest in describing various real situations. Under investigation hereby is a large class of the generalized variable-coefficient KdV models with external-force and perturbed/dissipative terms. Recent examples of this class include those in blood vessels and circulatory system, arterial dynamics, trapped Bose-Einstein condensates related to matter waves and nonlinear atom optics, Bose gas of impenetrable bosons with longitudinal confinement, rods of compressible hyperelastic material and semiconductor heterostructures with positonic phenomena. In this Letter, based on symbolic computation, four transformations are proposed from this class either to the cylindrical or standard KdV equation when the respective constraint holds. The constraints have nothing to do with the external-force term. Under those transformations, such analytic solutions as those with the Airy, Hermit and Jacobian elliptic functions can be obtained, including the solitonic profiles. The roles for the perturbed and external-force terms to play are observed and discussed. Investigations on this class can be performed through the properties of solutions of cylindrical and standard KdV equations.