MIXED FOURIER-LEGENDRE SPECTRAL METHODS FOR THE MULTIPLE SOLUTIONS OF THE SCHRODINGER EQUATION ON THE UNIT DISK

In this paper, we first compute the multiple non-trivial solutions of the Schrodinger equation on a unit disk, by using the Liapunov-Schmidt reduction and symmetry-breaking bifurcation theory, combined with the mixed Fourier-Legendre spectral and pseudospectral methods. After that, we propose the extended systems, which can detect the symmetry-breaking bifurcation points on the branch of the O(2) symmetric positive solutions. We also compute the multiple positive solutions with various symmetries of the Schrodinger equation by the branch switching method based on the Liapunov-Schmidt reduction. Finally, the bifurcation diagrams are constructed, showing the symmetry/peak breaking phenomena of the Schr¨odinger equation. Numerical results demonstrate the effectiveness of these approaches.

Solving Operator Equation Based on Expansion Approach

To date, researchers usually use spectral and pseudospectral methods for only numerical approximation of ordinary and partial differential equations and also based on polynomial basis. But the principal importance of this paper is to develop the expansion approach based on general basis functions (in particular case polynomial basis) for solving general operator equations, wherein the particular cases of our development are integral equations, ordinary differential equations, difference equations, partial differential equations, and fractional differential equations. In other words, this paper presents the expansion approach for solving general operator equations in the form Lu+Nu=g(x),x∈Γ, with respect to boundary condition Bu=λ, where L, N and B are linear, nonlinear, and boundary operators, respectively, related to a suitable Hilbert space, Γ is the domain of approximation, λ is an arbitrary constant, and g(x)∈L2(Γ) is an arbitrary function. Also the other importance of this paper is to introduce the general version of pseudospectral method based on general interpolation problem. Finally some experiments show the accuracy of our development and the error analysis is presented in L2(Γ) norm.

Spectral methods for internal waves: indistinguishable density profiles and double-humped solitary waves

Internal solitary waves are widely observed in both the oceans and large lakes. They can be described by a variety of mathematical theories, covering the full spectrum from first order asymptotic theory (i.e. Korteweg-de Vries, or KdV, theory), through higher order extensions of weakly nonlinear-weakly nonhydrostatic theory, to fully nonlinear-weakly nonhydrostatic theories and finally exact theory based on the Dubreil-Jacotin-Long (DJL) equation that is formally equivalent to the full set of Euler equations. We discuss how spectral and pseudospectral methods allow for the computation of novel phenomena in both approximate and exact theories. In particular we construct markedly different density profiles for which the coefficients in the KdV theory are very nearly identical. These two density profiles yield qualitatively different behaviour for both exact, or fully nonlinear, waves computed using the DJL equation and in dynamic simulations of the time dependent Euler equations. For exact, DJL, theory we compute exact solitary waves with two-scales, or so-called double-humped waves.