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Abstract
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Solving partial differential equations has always been one of the significant mathematical tools
for modeling applied phenomena in various fields. In this paper, our main motivation is to find
the traveling wave solutions of unstable nonlinear Schr¨odinger equation (UNLSE) which describes
the two layer baroclinic instability, and two lossless symmetric stream plasma instability, and also
disturbance of time period in both stable and unstable media. The methods which are used to
endure these solutions are extended rational sine-cosine/sinh-cosh. New solutions are constructed
in the different forms such as hyperbolic and trigonometric. The achieved results present that
proposed techniques are consequential for exploring several types of nonlinear partial differential
equations (NLPDEs) in applied sciences and solutions are presented in the form of novel periodic,
dark, bright, and periodically bright solitons. These methods are highly effective, robust, and
provide an alternative approach for establishing new soliton solutions for various types of partial
differential equations (PDEs) used in mathematical physics.
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