In this paper, we combine the unified and the explicit exponential finite difference methods to obtain both analytical and numerical solutions for the Newell‐Whitehead‐Segel–type equations which are very important in mathematical biology. The unified method is utilized to obtain various solitary wave solutions for these equations. Numerical solutions of the specific case studies are investigated by using the explicit exponential finite difference method ensures the accuracy and reliability of the proposed scheme. After obtaining the approximate solutions, convergence analysis and error estimation (the error norms and absolute errors) are presented by comparing these results with the analytical obtained solutions and other methods in the literature through tables and graphs. The obtained analytical and numerical results are in good agreement.

This research paper studies the optical soliton wave solutions of the model of sub-10-fs-pulse propagation by the implementation of the modified Khater method. This model describes the dynamics of light pulses that represent a higher-order nonlinear Schrödinger equation with the non-Kerr nonlinear term. The validity of this model depends on one primary hypothesis, which is the carrier wavelength of the soliton is much shorter than the spatial width. This means that the amplitude of the soliton frequency must be less than the carrier frequency. The shorter femtosecond pulses ([Formula: see text]100 fs) are desired to increase the bit rate of pulse propagation. The losing of distribution in such short-wavelength pulses through waveguides is a negligible loss. Our solitary analytical wave solutions are approved with the waveguide made of highly nonlinear optical materials.

Abstract. In this article, modi ed Korteweg-de Vries (mKdV) equation is solved numerically by using lumped Petrov-Galerkin approach, where weight functions are quadratic and the element shape functions are cubic B-splines. The proposed numerical scheme is tested by applying four test problems including single solitary wave, interaction of two and three solitary waves, and evolution of solitons with the Gaussian initial condition. In order to show the performance of the algorithm, the error norms, L2, L1, and a couple of conserved quantities are computed. For the linear stability analysis of numerical algorithm, Fourier method is also investigated. As a result, the computed results show that the presented numerical scheme is a successful numerical technique for solving the mKdV equation. Therefore, the presented method is preferable to some recent numerical methods.

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A septic B-spline collocation method is implemented to find the numerical solution of the modified regularized long wave (MRLW) equation. Three test problems including the single soliton and interaction of two and three solitons are studied to validate the proposed method by calculating the error norms 2 and ∞ and the invariants 1 , 2 , and 3 . Also, we have studied the Maxwellian initial condition pulse. The numerical results obtained by the method show that the present method is accurate and efficient. Results are compared with some earlier results given in the literature. A linear stability analysis of the method is also investigated.