Abstract
In this paper, the classification of single traveling wave solutions to a special kind of space-time fractional Schrödinger type equation with high-order nonlinearities is obtained by the complete discrimination system for polynomial method, where the fractional derivative is defined by Atangana’s conformable definition. The descriptions of the method and the fractional derivative are given, and the concrete procedure of the method is also presented in the paper. Especially, some new solutions such as hyperelliptic functions solution could be found and all the existing results about traveling wave solutions to the equation could also be seen. Moreover, concrete examples indicate that all the solutions obtained in the paper can be realized.
Highlights
In order to portray anomalous phenomena in science and engineering, nonlinear equation with fractional order derivative is proposed and has been widely applied in the fields such as anomalous diffusion [1, 2], and finding exact solutions to such a special kind of equation is an important and significant task
Atangana et al proposed a special kind of definition called Atangana’s fractional derivative [7], which is easy to apply due to the reason that it obeys almost all the properties satisfied by the conventional Newtonian concept of derivative such as chain rules
Equation (3) into an ordinary differential equation by a special kind of fractional complex traveling wave transformation and use the complete discrimination system for polynomial method of 5th order to get the classification of the traveling wave solutions to it
Summary
In order to portray anomalous phenomena in science and engineering, nonlinear equation with fractional order derivative is proposed and has been widely applied in the fields such as anomalous diffusion [1, 2], and finding exact solutions to such a special kind of equation is an important and significant task. Equation (3) into an ordinary differential equation by a special kind of fractional complex traveling wave transformation and use the complete discrimination system for polynomial method of 5th order to get the classification of the traveling wave solutions to it Some new solutions such as hyperelliptic function solutions are given, and concrete examples with concrete parameters are presented to ensure the existence of each solution obtained in the paper. This indicates that the complete discrimination system for polynomial method is efficient and powerful in handing the nonlinear problems arising in mathematics and physics.
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