Generalized Wronskian relations, one-dimensional Schrödinger equation and nonlinear partial differential equations solvable by the inverse-scattering method

1976 ◽  
Vol 31 (2) ◽  
pp. 229-249 ◽  
Author(s):  
F. Calogero
Keyword(s):  
Partial Differential Equations ◽  
Schrödinger Equation ◽  
Differential Equations ◽  
Inverse Scattering ◽  
Schrodinger Equation ◽  
Nonlinear Partial Differential Equations ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Partial Differential ◽  
One Dimensional

scholarly journals Compatible flat metrics

2002 ◽  
Vol 2 (7) ◽  
pp. 337-370 ◽  
Author(s):  
Oleg I. Mokhov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Inverse Scattering ◽  
Nonlinear Partial Differential Equations ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Dressing Method ◽  
Partial Differential ◽  
Differential Reduction ◽  
Component Case

We solve the problem of description of nonsingular pairs of compatible flat metrics for the generalN-component case. The integrable nonlinear partial differential equations describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) are found and integrated. The integrating of these equations is based on reducing to a special nonlinear differential reduction of the Lamé equations and using the Zakharov method of differential reductions in the dressing method (a version of the inverse scattering method).

scholarly journals Solusi Persamaan Schrodinger dengan Menggunakan Metode Transformasi Diferensial

2021 ◽  
Vol 4 (1) ◽  
pp. 47
Author(s):  
Muhammad Abdy ◽  
Hisyam Ihsan ◽  
Dhea Ayu Rossyana Dewi
Keyword(s):  
Partial Differential Equations ◽  
Schrödinger Equation ◽  
Differential Equations ◽  
Schrodinger Equation ◽  
Nonlinear Partial Differential Equations ◽  
Transformation Method ◽  
Differential Transformation Method ◽  
Partial Differential ◽  
Differential Transformation ◽  
Initial Values

Abstrak. Penelitian ini membahas tentang solusi persamaan diferensial parsial linier yaitu persamaan Schrodinger. Solusi persamaan ini dilakukan dengan menggunakan metode transformasi diferensial yang merupakan metode semi-numerik-analitik yang dapat digunakan untuk menyelesaikan persamaan diferensial biasa ataupun persamaan diferensial parsial linier dan nonlinier. Metode transformasi diferensial merupakan metode yang menggunakan teori ekspansi deret pangkat pada bentuk transformasinya untuk menentukan solusi. Pada penelitian ini digunakan dua nilai awal pada persamaan Schrodinger yang diberikan. Solusi dengan kedua nilai awal yang diberikan diperoleh dengan menggunakan ekspansi deret Maclaurin. Kemudian solusi tersebut disimulasikan menggunakan software Maple18. Akibatnya, metode transformasi diferensial pada penelitian ini merupakan salah satu metode yang mampu menghasilkan solusi untuk persamaan Schrodinger..Kata Kunci: Persamaan Schrodinger, Metode Transformasi DiferensialAbstract. This study discusses the solution of linear partial differential equations, namely Schrodinger equation. The solution of the equation is done by using the differential transformation method which is a semi-numerical-analytical method, it can be used to solve both ordinary differential equations and linear or nonlinear partial differential equations. Differential transformation method is a method uses the theory of rank expansion in the form of transformation to determine solutions. In this study, two initial values in the given Schrodinger equation were used. Solutions with both initial values given are obtained using the Maclaurin series expansion. Then, the solution is simulated using Maple18 software. As a result, the differential transformation method in this study is one method that is able to solve a solution to the Schrodinger equation.Keywords: Schrodinger Equation, Differential Transformation Method

scholarly journals Wave Structures for Nonlinear Schrodinger Types Fractional Partial Differential Equations Arise in Physical Sciences

2021 ◽  
Vol 9 (2) ◽  
pp. 101-110
Author(s):  
Mst. Nasrin Nahar ◽  
Md. Tarikul Islam ◽  
Diganta Broto Kar
Keyword(s):  
Partial Differential Equations ◽  
Schrödinger Equation ◽  
Differential Equations ◽  
Schrodinger Equation ◽  
Nonlinear Partial Differential Equations ◽  
Expansion Method ◽  
Hyperbolic Function ◽  
Nonlinear Phenomena ◽  
Partial Differential ◽  
Nonlinear Schrödinger

Nonlinear partial differential equations are mostly renowned for depicting the underlying behavior of nonlinear phenomena relating to the nature of the real world. In this paper, we discuss analytic solutions of fractional-order nonlinear Schrodinger types equations such as the space-time fractional nonlinear Schrodinger equation and the (2+1)-dimensional time-fractional Schrodinger equation. The considered equations are converted into ordinary differential equations with the help of wave variable transformation and then the recently established rational ( )-expansion method is employed to construct the exact solutions. The obtained solutions have appeared in the forms of a trigonometric function, hyperbolic function, and rational function which are compared with those of literature and claimed to be different. The graphical representations of the solutions are finally brought out for their physical appearances. The applied method is seemed to be efficient, concise, and productive which might be used for further research. Mathematics Subject Classifications: 35C08, 35R11

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