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Numerical Computation of Travelling Wave Solutions for the Nonlinear Ito System |
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PP: 75-83 |
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Author(s) |
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Ali H Bhrawy,
Robert A. Van Gorder,
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Abstract |
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| The Ito equation (a coupled nonlinear wave equation which generalizes the KdV equation) has previously been shown to
admit a reduction to a single nonlinear Casimir equation governing the wave solutions. Some analytical properties of the solutions
to this equation in certain parameter regimes have been studied recently. However, for general parameter regimes where the analytical
approach is not so useful, a numerical method would be desirable. Therefore, in this paper, we proceed to show that the Casimir equation
for the Ito system can be solved numerically by use of the shifted Jacobi-Gauss collocation (SJC) spectral method. First, we present the
general solution method, which is follows by implementation of the method for specific parameter values. The presented results in this
article demonstrate the accuracy and efficiency of the method. In particular, we demonstrate that relatively few notes permit very low
residual errors in the approximate numerical solutions. We are also able to show that the coefficients of the higher order terms in the
shifted Jacobi polynomials decrease exponentially, meaning that accurate solutions can be obtained after relatively few terms are used.
With this, we have a numerical method which can accurately and efficiently capture the behavior of nonlinear waves in the Ito equation. |
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