High-accuracy power series solutions with arbitrarily large radius of convergence for the fractional nonlinear Schrödinger-type equations

2018 ◽  
Vol 51 (23) ◽  
pp. 235201 ◽  
Author(s):  
U Al Khawaja ◽  
M Al-Refai ◽  
Gavriil Shchedrin ◽  
Lincoln D Carr
Keyword(s):  
Power Series ◽  
High Accuracy ◽  
Radius Of Convergence ◽  
Large Radius ◽  
Series Solutions ◽  
Nonlinear Schrödinger ◽  
Power Series Solutions

Optical solitons, complexitons and power series solutions of a (2+1)-dimensional nonlinear Schrödinger equation

2018 ◽  
Vol 32 (28) ◽  
pp. 1850336 ◽  
Author(s):  
Wei-Qi Peng ◽  
Shou-Fu Tian ◽  
Tian-Tian Zhang
Keyword(s):  
Power Series ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Dynamical Behavior ◽  
Nonlinear Schrodinger Equation ◽  
Series Solutions ◽  
Ansatz Method ◽  
Nonlinear Schrödinger ◽  
Power Series Solutions

In this paper, a (2+1)-dimensional generalized nonlinear Schrödinger equation is investigated, which is an important model in the field of optical fiber propagation. By employing the ansatz method, we obtain the bright soliton, dark soliton and complexiton of the equation. Some constraint conditions are also derived to ensure the existence of the solitons. Moreover, its power series solutions with the convergence analysis are also provided. Some graphical analyses of those solutions are presented in order to better understand their dynamical behavior.

A symmetry-preserving difference scheme and analytical solutions of a generalized higher-order beam equation

2021 ◽  
Vol 477 (2255) ◽  
Author(s):  
Shou-Fu Tian ◽  
Mei-Juan Xu ◽  
Tian-Tian Zhang
Keyword(s):  
Power Series ◽  
Vector Fields ◽  
Higher Order ◽  
Beam Equation ◽  
Series Solutions ◽  
Local Conservation ◽  
Power Series Method ◽  
Potential System ◽  
Power Series Solutions ◽  
The Difference

Under investigation in this work is a generalized higher-order beam equation, which is an important physical model and describes the vibrations of a rod. By considering Lie symmetry analysis, and using the power series method, we derive the geometric vector fields, symmetry reductions, group invariant solutions and power series solutions of the equation, respectively. The convergence analysis of the power series solutions are also provided with detailed proof. Furthermore, by virtue of the multipliers, the local conservation laws of the equation are derived as well. Furthermore, an effective and direct approach is proposed to study the symmetry-preserving discretization for the equation via its potential system. Finally, the invariant difference models of the generalized beam equation are successfully constructed. Our results show that it is very useful to construct the difference models of the potential system instead of the original equation.

The semi-analytic theory of standing waves

1987 ◽  
Vol 411 (1840) ◽  
pp. 123-137 ◽  
Keyword(s):  
Numerical Calculation ◽  
Power Series ◽  
Deep Water ◽  
Standing Waves ◽  
Affirmative Answer ◽  
Rigorous Proof ◽  
Series Solutions ◽  
Wave Problem ◽  
Power Series Solutions ◽  
Formal Power

The algorithm proposed by Schwartz & Whitney ( J. Fluid Mech . 107, 147–171 (1981)) for the numerical calculation of formal power series solutions of the classical standing-wave problem is vindicated by a rigorous proof that resonances do not occur in the calculations. A detailed account of a successful algorithm is given. The analytical question of the convergence of the power series whose coefficients have been calculated remains open. An affirmative answer would be a first demonstration of the existence of standing waves on deep water.

Sign in / Sign up

Export Citation Format

Share Document