A third-order nonlinear Schrödinger equation: the exact solutions, group-invariant solutions and conservation laws

Abstract

In this study, we consider the third order nonlinear Schrödinger equation (TONSE) that models the wave pulse transmission in a time period less than one-trillionth of a second. With the help of the extended modified method, we obtain numerous exact travelling wave solutions containing sets of generalized hyperbolic, trigonometric and rational solutions that are more general than classical ones. Secondly, we construct the transformation groups which left the equations invariant and vector fields with the Lie symmetry groups approach. With the help of these vector fields, we obtain the symmetry reductions and exact solutions of the equation. The obtained groupinvariant solutions are Jacobi elliptic function and exponential type. We discuss the dynamic behaviour and structure of the exact solutions for distinct solutions of arbitrary constants. Lastly, we obtain conservation laws of the considered equation by construing the complex equation as a system of two real partial differential equations (PDEs).

Traveling, periodic and localized solitary waves solutions of the (4+1)-dimensional nonlinear Fokas equation

“…In Fig. 1c, we show the 2D profile of the periodic solution (32) with parameter values as = 1.5 , = 1.8, a 0 = 1, c 2 = −2.0 and c 0 = 0.5 within range of −∞ to +∞ . In Fig.…”

New analytical formalisms used in finite element analysis of robots with elastic elements

“…In the Gibbs-Appell case, the energy of accelerations is used, a notion most engineers are less familiar with. In the future, it is presumed that the equivalent forms developed by the analytical mechanics and which apparently have no practical applicability will be reevaluated and new fields of applicability will be found in the engineering studies in which they will prove their usefulness [27][28][29][30][31][32][33][34][35][36][37].…”

On bounds for topological descriptors of φ-sum graphs

“…Moreover, for α = 2 and α = 3, FGZI becomes first Zagreb index and forgotten topological index. Similarly, for β = 1 and β = − 1 2 , GRI becomes second Zagreb index and classical Randić index respectively, see [26][27][28][29][30][31][32].…”