Quintic-Septic Nonlinear Schrödinger Equation with a Third-Order Dispersion Term

In the present study, the quintic-septic nonlinear modulation of a longitudinal wave propagating to contribute the dispersive and higher-order nonlinear effects in a generalized cubically nonlinear elastic medium is considered. In recent work, for the modulation of a longitudinal wave, a cubic nonlinear Schrödinger equation with a third-order dispersive term is obtained by using a multi-scale expansion of quasi-monochromatic wave solutions. The third- quintic-septic longitudinal wave, by choosing specific values of material constants and wave number for which some coefficients of nonlinear terms are disappeared. In this case, a new perturbation expansion is needed to balance nonlinear effects with dispersive effects. As a result, a quintic-septic nonlinear Schrödinger equation with a third-order dispersion term is obtained as a new model that balances quintic-septic nonlinearity with a third-order dispersion term.