Weierstrass elliptic function solutions and degenerate solutions of a variable coefficient higher-order Schrödinger equation

Abstract

In this paper, the auxiliary equation method is used to study the Weierstrass elliptic function solutions and degenerate solutions of the variable coefficient higher order Schrödinger equation, including Jacobian elliptic function solutions, trigonometric function solutions and hyperbolic function solutions. The types of solutions of the variable coefficient higher-order Schrödinger equation are enriched, and the method of seeking precise and accurate solutions is extended. It is concluded that the types of degenerate solutions are related to the coefficients of the equation itself when the degenerate solutions are obtained from the solutions of the Weierstrass elliptic functions. In addition, the solutions form of the equation is extended from the power series expansion form to the Laurent series expansion form, and the corresponding solutions are obtained. After the conversion formula between the Weierstrass elliptic function solutions and the Jacobian elliptic function solutions is constructed, the Jacobian elliptic function solutions of the higher order Schrödinger equation with variable coefficients are also obtained. These have not been previously studied.

Explicit and exact travelling wave solutions for Hirota equation and computerized mechanization

“…For certain significant PDEs in the fields of Mathematics and Physics, if they can be converted into the form of elliptic Eq (23), then their solutions are readily obtained. In addition, to the best of our knowledge, the solutions q 1 , q 4 − q 10 and q 15 of Eq (51) are new, which can not be found in [14][15][16]. In addition, since Eqs (33)-(35) are similar to Eq (32), here we only list the solutions obtained by substituting Eqs (32) and (36) into Eq (23) along with Eqs (9)-( 22).…”

“…One can see that in one limit of α = 0, the equation reduces to the nonlinear Schro ¨dinger equation which describes plane self-focusing and one-dimensional self-modulation of waves in nonlinear dispersive media. Eq (51) has been studied in some literatures [14][15][16].…”