Abstract
We apply the generalized projective Riccati equations method with the aid of Maple software to construct many new soliton and periodic solutions with parameters for two higher-order nonlinear partial differential equations (PDEs), namely, the nonlinear Schrödinger (NLS) equation with fourth-order dispersion and dual power law nonlinearity and the nonlinear quantum Zakharov-Kuznetsov (QZK) equation. The obtained exact solutions include kink and antikink solitons, bell (bright) and antibell (dark) solitary wave solutions, and periodic solutions. The given nonlinear PDEs have been derived and can be reduced to nonlinear ordinary differential equations (ODEs) using a simple transformation. A comparison of our new results with the well-known results is made. Also, we drew some graphs of the exact solutions using Maple. The given method in this article is straightforward and concise, and it can also be applied to other nonlinear PDEs in mathematical physics.
Highlights
The investigation of exact traveling wave solutions to nonlinear partial differential equations (PDEs) plays an important role in the study of nonlinear physical phenomena
The investigation of exact traveling wave solutions to nonlinear PDEs plays an important role in the study of nonlinear physical phenomena
Conte and Musette [12] presented an indirect method to find solitary wave solutions of some nonlinear PDEs that can be expressed as polynomials in two elementary functions which satisfy a projective Riccati equation [19]
Summary
The investigation of exact traveling wave solutions to nonlinear PDEs plays an important role in the study of nonlinear physical phenomena. Many powerful tools have been established to determine soliton and periodic wave solutions of nonlinear PDEs, such as the (G/G)-expansion method [1,2,3,4,5,6], the extended auxiliary equation method [7, 8], the new mapping method [9,10,11], the generalized projective Riccati equations method [12,13,14,15,16,17], and the (G/G, 1/G)-expansion method [18]. Conte and Musette [12] presented an indirect method to find solitary wave solutions of some nonlinear PDEs that can be expressed as polynomials in two elementary functions which satisfy a projective Riccati equation [19].
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