This paper obtains the exact 1-soliton solution of the perturbed Korteweg-de Vries equation with power law nonlinearity. The topological soliton solutions are obtained. The solitary wave ansatz is used to carry out this integration. The domain restrictions are identified in the process and the parameter constraints are also obtained. It has been proved that topological solitons exist only when ...

A. general theory for determining Hamiltonian model equations from noncanonical perturbation expansions of Hamiltonian systems is applied to the Boussinesq expan sion fcr long, small amplitude waves in shallow water, leading to the Korteweg-deVries equation. New Hamiltonian model equations, including a natural "Hamiltonian ver-cn» of the KdV equation, are proposed. The method also provides a di...

We apply the method of operator splitting on the generalized Korteweg{de Vries (KdV) equation ut +f(u)x+"uxxx = 0, by solving the nonlinear conservation law ut +f(u)x = 0 and the linear dispersive equation ut + "uxxx = 0 sequentially. We prove that if the approximation obtained by operator splitting converges, then the limit function is a weak solution of the generalized KdV equation. Convergen...

A new relaxation analysis and two acceleration schemes are proposed for the ve-point Red-Black Gauss-Seidel smoothing in multigrid for solving two dimensional Poisson equation. For a multigrid V cycle, we discovered that under-relaxation is applicable to restriction half cycle and overrelaxation is applicable to interpolation half cycle. Numerical experiments using modi ed multigrid V cycle alg...

The quantum Liouville equation in the Wigner representation is solved numerically by using Monte Carlo methods. For incremental time steps, the propagation is implemented as a classical evolution in phase space modi ed by a quantum correction. The correction, which is a momentum jump function, is simulated in the quasi-classical approximation via a stochastic process. In this paper the techniqu...

In this paper, a least mean square (LMS) t ype algorithm is devised for unbiased system identi cation in the presence of white input and output noise, assuming that the ratio of the noise powers is known. The proposed approach aims to minimi ze the mean square value of the equation-error function under a constantnorm constrain t, and is equivalent to minimizin g a modi ed mean square error func...

This paper describes an adaptive algorithm for optimal control of time-dependent partial di erentialalgebraic equation (PDAE) systems. A direct method based on a modi ed multiple shooting type technique and sequential quadratic programming (SQP) is used for solving the optimal control problem, while an adaptive mesh re nement (AMR) algorithm is employed to dynamically adapt the spatial integrat...

In this present work, the Kudryashov method and the functional variable method are used to construct exact solutions of the complex KdV equation. The Kudryashov method and the functional variable method are powerful methods for obtaining exact solutions of nonlinear evolution equations.

We construct compactly supported self-similar solutions of the Modi ed PorousMedium Equation (MPME) ut = um for um > 0 (1 + ") um for um < 0: They have the form u(x; t) = t U(x t ); where the similarity exponents and depend on ", m and the dimension N . This corresponds to what is known in the literature as anomalous exponents or self-similarity of the second kind, a not completely understood p...

By using solutions of an ordinary differential equation, an auxiliary equationmethod is described to seek exact solutions of variablecoefficient KdV-MKdV equation. As a result, more new exact nontravelling solutions, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions, and combined Jacobi elliptic function solutions, for...