相关论文: High accuracy power series method for solving scal…
Let $L$ be the $n$-th order linear differential operator $Ly = \phi_0y^{(n)} + \phi_1y^{(n-1)} + \cdots + \phi_ny$ with variable coefficients. A representation is given for $n$ linearly independent solutions of $Ly=\lambda r y$ as power…
We consider an initial- and boundary- value problem for the nonlinear Schr\"odinger equation with homogeneous Dirichlet boundary conditions in the one space dimension case. We discretize the problem in space by a central finite difference…
A method for finding exact solutions of nonlinear differential equations is presented. Our method is based on the application of the Newton polygons corresponding to nonlinear differential equations. It allows one to express exact solutions…
We assess the applicability and efficiency of time-adaptive high-order splitting methods applied for the numerical solution of (systems of) nonlinear parabolic problems under periodic boundary conditions. We discuss in particular several…
Inverse problems are in many cases solved with optimization techniques. When the underlying model is linear, first-order gradient methods are usually sufficient. With nonlinear models, due to nonconvexity, one must often resort to…
Quantum theory has been remarkably successful in providing an understanding of physical systems at foundational scales. Solving the Schr\"odinger equation provides full knowledge of all dynamical quantities of the physical system. However…
The algorithm for generation of exact solutions of the nonlinear equation in partial derivatives of a divergent type which is included in the formulation of magnetostatics, hydro-and aerodynamics, quantum mechanics (stationary Schr\"odinger…
One of old methods for finding exact solutions of nonlinear differential equations is considered. Modifications of the method are discussed. Application of the method is illustrated for finding exact solutions of the Fisher equation and…
When a system of first order linear ordinary differential equations has eigenvalues of large magnitude, its solutions exhibit complicated behaviour, such as high-frequency oscillations, rapid growth or rapid decay. The cost of representing…
By solving the Schr\"odinger equation one obtains the whole energy spectrum, both the bound and the continuum states. If the Hamiltonian depends on a set of parameters, these could be tuned to a transition from bound to continuum states.…
We revisit the following nonlinear Schr\"odinger system \begin{align*}\begin{cases} -\epsilon^{2}\Delta u +P(x) u= \mu_1 u^3 +\beta uv^2, &~\text{in}\;\mathbb {R}^3,\\ -\epsilon^{2}\Delta v+Q(x) v= \mu_2 v^3 +\beta u^2v,…
Solving partial differential equations is crucial to analysing and predicting complex, large-scale physical systems but pushes conventional high-performance computers to their limits. Application specific photonic processors are an exciting…
This article demonstrates how variation of parameters can be successfully implemented in combination with other classical techniques, such as the method of characteristics, to derive novel classes of solutions to nonlinear partial…
In this study, we consider two nonlinear Schr\"{o}dinger-type models that are derived by R L. Frank, F. M\'{e}hats, C. Sparber [arXiv:1611.01574] to study 3D nonlinear Schr\"{o}dinger equations under strong magnetic fields. One model is…
Treating the nonlinear term of the Gross-Pitaevskii nonlinear Schrodinger equation as a perturbation of an isolated discrete eigenvalue of the linear problem one obtains a Rayleigh-Schrodinger power series. This power series is proved to be…
We address the open problem of determining which classes of time-dependent linear Schr\"odinger equations and focusing and defocusing cubic and quintic non-linear Schr\"odinger equations (NLS) on unbounded domains that can be computed by an…
The Schr\"odingerization method converts linear partial and ordinary differential equations with non-unitary dynamics into systems of Schr\"odinger-type equations with unitary evolution. It does so via the so-called warped phase…
The error behavior of exponential operator splitting methods for nonlinear Schr{\"o}dinger equations in the semiclassical regime is studied. For the Lie and Strang splitting methods, the exact form of the local error is determined and the…
In this paper, we introduce an algorithm that provides approximate solutions to semi-linear ordinary differential equations with highly oscillatory solutions, which, after an appropriate change of variables, can be rewritten as…
The power flow equations are at the core of most of the computations for designing and operating electric power systems. The power flow equations are a system of multivariate nonlinear equations which relate the power injections and…