We study solitary wave interactions in the Euler–Poisson equations modeling ion acoustic plasmas and their approximation by KdV n-solitons. Numerical experiments are performed and solutions compared to appropriately scaled KdV n-solitons. While largely correct qualitatively the soliton solutions did not accurately capture the scattering shifts experienced by the solitary waves. We propose corre...

We obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation ut + 6uux + ǫ uxxx = 0, u(x, t = 0, ǫ) = u0(x), for ǫ small, near the point of gradient catastrophe (xc, tc) for the solution of the dispersionless equation ut + 6uux = 0. The sub-leading term in this expansion is described by the smooth solution of a fourth order ODE, which is a hi...

The new generalized Harry Dym equation of Z. Popowicz is transformed into the Hirota–Satsuma system of coupled KdV equations.

We propose a scheme for nonlinearizing linear equations to generate integrable systems of both the AKNS and the KN classes, based on the simple idea of dimensional analysis and detecting the building blocks of the Lax pair. Along with the well known equations we discover a novel integrable hierarchy of higher order nonholonomic deformations for the AKNS family, e.g. the KdV, the mKdV, the NLS a...

We reduce the vectorial binary Darboux transformation for the Manin-Radul supersymmetric KdV system in such a way that it preserves the Manin-Radul-Mathieu supersymmetric KdV equation reduction. Expressions in terms of bosonic Pfaffians are provided for transformed solutions and wave functions. We also consider the implications of these results for the supersymmetric sine-Gordon equation. On le...

The KdV equation has many applications in mechanics and wave dynamics. Therefore, researchers are carrying out work to develop analyze modified generalized forms of the standard equation. In this paper, we inspect KdV-mKdV equation, which is a form ordinary We use fractional operator Caputo sense examine some theoretical results concerned with solution’s existence, uniqueness, stability. employ...

A general class of KdV-type wave equations regularized with a convolution-type nonlocality in space is considered. The differs from the nonlinear nonlocal unidirectional previously studied by addition linear convolution term involving third-order derivative. To solve Cauchy problem we propose semi-discrete numerical method based on uniform spatial discretization, that an extension published wor...

We propose a scheme for nonlinearizing linear equations to generate integrable nonlinear systems of both the AKNS and the KN classes, based on the simple idea of dimensional analysis and detecting the building blocks of the Lax pair. Along with the well known equations we discover a novel integrable hierarchy of higher order nonholonomic deformations for the AKNS family, e.g. for the KdV, the m...

In this paper, the application of the method of lines (MOL) to the KdV equation was presented. The MOL approach of the KdV equation led to a system of ordinary differential equations. Solution of the system was obtained by applying Euler’s method. In order to show the accuracy of the presented method, two test problem whose exact solutions are known were used. The numerical solutions obtained w...