The Painleve Test and Reducibility to the Canonical Forms for Higher-Dimensional Soliton Equations with Variable-Coefficients

The general KdV equation (gKdV) derived by T. Chou is one of the famous (1+1) dimensional soliton equations with variable coefficients. It is well-known that the gKdV equation is integrable. In this paper a higher-dimensional gKdV equation, which is integrable in the sense of the Painleve test, is presented. A transformation that links this equation to the canonical form of the Calogero-Bogoyavlenskii-Schiff equation is found. Furthermore, the form and similar transformation for the higher-dimensional modified gKdV equation are also obtained.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Similar oscillations on both sides of a shock. Part I. Even-odd alternative dispersions and general assignments of Fourier dispersions towards a unfication of dispersion models

We consider assigning different dispersions for different dynamical modes, particularly with the distinguishment and alternation of opposite signs for alternative Fourier components. The Korteweg-de Vries (KdV) equation with periodic boundary condition and longest-wave sinusoidal initial field, as used by N. Zabusky and M. D. Kruskal, is chosen for our case study with such alternating-dispersion of the Fourier modes of (normalized) even and odd wavenumbers. Numerical results verify the capability of our new model to produce two-sided (around the shock) oscillations, as appear on both sides of some ion-acoustic and quantum shocks, not admitted by models such as the KdV(-Burgers) equation, but also indicate even more, including singular zero-dispersion limit or non-convergence to the classical shock (described by the entropy solution), non-thermalization (of the Galerkin-truncated models) and applicability to other models (showcased by the modified KdV equation with cubic nonlinearity). A unification of various dispersive models, keeping the essential mathematical elegance (such as the variational principle and Hamiltonian formulation) of each, for phenomena with complicated dispersion relation is thus suggested with a further explicit example of two even-order dispersions (from the Hilbert transforms) extending the Benjamin-Ono model. The most general situation can be simply formulated by the introduction of the dispersive derivative, the indicator function and the Fourier transform, resulting in an integro-differential dispersion equation. Other issues such as the real-number order dispersion model and the transition from non-thermalization to thermalization and, correspondingly, from regularization to non-regularization for untruncated models are also briefly remarked

New exact solutionsand numerical approximations of the generalized kdv equation

This paper is devoted to create new exact and numerical solutions of the generalized Korteweg-de Vries (GKdV) equation with ansatz method and Galerkin finite element method based on cubic B-splines over finite elements. Propagation of single solitary wave is investigated to show the efficiency and applicability of the proposed methods. The performance of the numerical algorithm is proved by computing L2 and L∞ error norms. Also, three invariants I1, I2, and I3 have been calculated to determine the conservation properties of the presented algorithm. The obtained numerical solutions are compared with some earlier studies for similar parameters. This comparison clearly shows that the obtained results are better than some earlier results and they are found to be in good agreement with exact solutions. Additionally, a linear stability analysis based on Von Neumann’s theory is surveyed and indicated that our method is unconditionally stable

The Painlevé Test and Reducibility to the Canonical Forms for Higher-Dimensional Soliton Equations with Variable-Coefficients

The general KdV equation (gKdV) derived by T. Chou is one of the famous (1 + 1) dimensional soliton equations with variable coefficients. It is well-known that the gKdV equation is integrable. In this paper a higher-dimensional gKdV equation, which is integrable in the sense of the Painlevé test, is presented. A transformation that links this equation to the canonical form of the Calogero-Bogoyavlenskii-Schiff equation is found. Furthermore, the form and similar transformation for the higher-dimensional modified gKdV equation are also obtained

On Integrability of Nonautonomous Nonlinear Schroedinger Equations

We show, in general, how to transform the nonautonomous nonlinear Schroedinger equation with quadratic Hamiltonians into the standard autonomous form that is completely integrable by the familiar inverse scattering method in nonlinear science. Derivation of the corresponding equivalent nonisospectral Lax pair is outlined. A few simple integrable systems are discussed.Comment: 15 pages, no figure