Abstract
In this paper, a trial function method is employed to find exact solutions to the nonlinear Schrödinger equations with high-order time-dependent coefficients. This system might be used to describe the propagation of ultrashort optical pulses in nonlinear optical fibers, with self-steepening and self-frequency shift effects. The new general solutions are found for the general case a 0 ≠ 0 including the Jacobi elliptic function solutions, solitary wave solutions, and rational function solutions which are presented in comparison with the previous ones obtained by Triki and Wazwaz, who only studied the special case a 0 = 0 .
Highlights
It is well known that many physical phenomena can be described by a nonlinear Schrödinger equation (NLSE), which is found in many diverse fields such as plasma physics [1], fluid dynamics [2], nonlinear optics [3], quantum mechanics [4], hydrodynamics [5], and biology
Finding the exact solutions to the NLSE has an important theoretical and practical significance in understanding the physical phenomena described by the NLSE
In “Exact Solutions,” we first apply the trial function method to obtain its exact solutions by using a suitable transformation, and we illustrate the shapes of the wave amplitude of different solutions by taking appropriate parameters for those varying coefficients
Summary
It is well known that many physical phenomena can be described by a nonlinear Schrödinger equation (NLSE), which is found in many diverse fields such as plasma physics [1], fluid dynamics [2], nonlinear optics [3], quantum mechanics [4], hydrodynamics [5], and biology. It should be recognized that most of the methods mentioned above are related to constant coefficient models It becomes more difficult than those constant coefficient counterparts when we study the NLSE with time-dependent coefficients. We are going to apply Liu’s method to find the exact solutions of the following cubic-quintic NLSE with time-dependent coefficients: iqt. One of the present authors has obtained analytical traveling-wave solutions to a generalized Gross-Pitaevskii (GP) equation [22] with some new time- and space-varying coefficients and external fields [24] because of their possible applications to the BECs [25–. In “Exact Solutions,” we first apply the trial function method to obtain its exact solutions by using a suitable transformation, and we illustrate the shapes of the wave amplitude of different solutions by taking appropriate parameters for those varying coefficients. In “Concluding Remarks,” we summarize the results found in this work
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