Abstract
This article is concerned with the numerical solution of nonlinear hyperbolic Schro¨dinger equations (NHSEs) via an efficient Haar wavelet collocation method (HWCM). The time derivative is approximated in the governing equations by the central difference scheme, while the space derivatives are replaced by finite Haar series, which transform it to full algebraic form. The experimental rate of convergence follows the theoretical statements of convergence and the conservation laws of energy and mass are also presented, which strengthens the proposed method to be convergent and conservative. The Haar wavelets based on numerical results for solitary wave shape of |φ| are discussed in detail. The proposed approach provides a fast convergent approximation to the NHSEs. The reliability and efficiency of the method are illustrated by computing the maximum error norm and the experimental rate of convergence for different problems. Comparisons are performed with various existing methods in recent literature and better performance of the proposed method is shown in various tables and figures.
Highlights
The nonlinear hyperbolic Schrödinger equation govern most scientific and physical processes, and they play an essential role in nonlinear optics, Biomolecular dynamics, plasma physics, and water waves
The real and imaginary part of the numerical solution are compared with the exact solution in Figure 1, Figure 2 and with the Galerkin method in Table 4, where the same order of accuracy has been obtained for a lesser number of collocation points than the Galerkin method
We have proposed the Haar wavelet collocation method (HWCM) for the numerical solution of second-order
Summary
The nonlinear hyperbolic Schrödinger equation govern most scientific and physical processes, and they play an essential role in nonlinear optics, Biomolecular dynamics, plasma physics, and water waves. Because finding the exact solution to these types of NHSEs is difficult due to the nonlinear term, numerical methods are an alternative method of determining their solution. We considered the following NHSEs [1] ∂2 φ ∂φ − + μ − iμ2 − iμ.
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