Deeper properties of the nonlinear Phi-four and Gross-Pitaevskii equations arising mathematical physics

In this paper, the rational sine–cosine and rational sinh–cosh methods are applied in extracting some properties of nonlinear Phi-four and Gross–Pitaevskii equations. The singular periodic wave solutions, dark soliton solutions and hyperbolic function solutions are reported. The solitary waves are observed from the traveling waves under the values of the parameters. Modulation instability analysis is also observed in various simulations. We also plot to observe the wave distributions of parameters of stability in 2D and 3D visuals via package program.

Optical solutions and other solutions for Radhakrishnan-Kundu-Laksmannan equation by using improved modified extended tanh-function method

This paper studies Radhakrishnan-Kundu-Laksmannan equation which is used to describe the pulse propagation in optical fiber communications. By using improved modified extended tanh-function method various types of solutions are extracted such as bright solitons, singular solitons, singular periodic wave solutions, Jacobi elliptic solutions, periodic wave solutions and Weierstrass elliptic doubly periodic solutions. Moreover, some of the obtained solutions are represented graphically.

Lax pair, Darboux transformation and rogue-periodic waves of a nonlinear Schrödinger–Hirota equation with the spatio-temporal dispersion and Kerr law nonlinearity in nonlinear optics

For a nonlinear Schrödinger–Hirota equation with the spatio-temporal dispersion and Kerr law nonlinearity in nonlinear optics, we derive a Lax pair, a Darboux transformation and two families of the periodic-wave solutions via the Jacobian elliptic functions dn and cn. We construct the linearly-independent and non-periodic solutions of that Lax pair, and substitute those solutions into the Darboux transformation to get the rogue-periodic-wave solutions. When the third-order dispersion or group velocity dispersion (GVD) or inter-modal dispersion (IMD) increases, the maximum amplitude of the rogue-periodic wave remains unchanged. From the rogue-dn-periodic-wave solutions, when the GVD decreases, the minimum amplitude of the rogue-dn-periodic wave decreases. When the third-order dispersion decreases, the minimum amplitude of the rogue-dn-periodic wave rises. Decrease of the IMD causes the period of the rogue-dn-periodic wave to decrease. From the rogue-cn-periodic-wave solutions, when the GVD increases, the minimum amplitude of the rogue-cn-periodic wave decreases. Increase of the third-order dispersion or IMD leads to the decrease of the period.