Optical wave propagation to a nonlinear phenomenon with pulses in optical fiber

Lump waves, bright-dark solitons and some novel interaction solutions in (3+1)-dimensional shallow water wave equation

“…Nonlinear phenomena pervade various fields of science and engineering, making the research on nonlinear evolution equations (NLEEs) highly valuable for practical applications. Since the 1970s, after extensive efforts and exploration over an extended period, researchers in this field have compiled numerous methods for solving NLEEs, such as, Darboux transformation [1,2], Bäcklund transformation [3,4], inverse scattering transform [5,6], Hirota bilinear method [7,8], sine-Gordon equation expansion method [9,10], Riemann-hilbert method [11,12], rational function method [13,14]. The Hirota bilinear method transforms the equations into bilinear form by transforming the variables, then the D-operator is used to derive the exact solution to the equation, thereby avoiding solving complex integral equations or inverse scattering problems.…”

Analytical wave solutions of an electronically and biologically important model via two efficient schemes

“…Different types of phenomena occurring chemically, biologically, economically and others come from nonlinear partial differential equations (NLPDEs). Many different methods have been developed to gain analytical wave solutions of these NLPDEs, i.e., optical soliton solutions of coupled nonlinear Schrödinger equations have been gained with use of Kudryashov R-function technique, [1] some new kinds of optical soliton solutions of time-fractional perturbed nonlinear Schrödinger equations have been achieved by using the generalized Kudryashov scheme, [2] by applying the modified auxiliary equation technique, optical wave solutions of timefractional resonant nonlinear Schrödinger equations have been obtained, [3] new optical wave solutions for the time-fractional perturbed nonlinear Schrödinger equations have been achieved by utilizing the improved tan[φ (ζ /2)]-expansion scheme, [4] different kinds of optical wave solitons of time-fractional paraxial wave equations have been gained by using the Sardar sub-equation method, [5] various optical wave solutions of three-component coupled nonlinear Schrödinger equations have been attained with the help of generalized exponential rational function scheme, [6] dark, bright, singular and periodic solitary wave solutions of generalized fractional Davey-Stewartson equations have been obtained by applying the generalized projective Riccati equation technique, [7] some exact wave solutions of the Lax equation have been achieved by applying the extended sinh-Gordon expansion technique, [8] kink solitons of the Sharma-Tasso-Olver-Burgers equation have been attained by using Kudryashov and exponential techniques, [9] traveling wave solutions of perturbed Biswas-Milovic equations have been gained with the use of improved F-expansion technique, [10] some new optical wave solutions of complex Korteweg-de Vries equations have been obtained by applying the unified scheme. [11] Similarly, Hirota bilinear method, [12] modified extended tanh expansion method, [13] modified simplest equation technique, [14] extended Jacobi elliptic function scheme, [15] sech and tanh function solutions are obtained by using the sine-Gordon expansion scheme, [16] sinh, cosh, sin and cos involving solutions are gained by utilizing the rational sine-Gordon expansion technique [17] and many other techniques.…”

The explicit power series solution formation and computationof Lie point infinitesimals generators: Lie symmetry approach

“…The Lie symmetry analysis has been discussed and the optimal system introduced for partial differential equations by Malik et al [9,10]. Numerous types of soliton and their interaction have been studied for many partial differential equations [11][12][13][14][15][16][17][18][19].…”