New exact solutions of two-dimensional integrable equations using the $\bar \partial $ -dressing method

2011 ◽  
Vol 167 (3) ◽  
pp. 725-739 ◽  
Author(s):  
V. G. Dubrovsky ◽  
A. V. Topovsky ◽  
M. Yu. Basalaev
Keyword(s):  
Exact Solutions ◽  
Two Dimensional ◽  
Dressing Method ◽  
Integrable Equations ◽  
New Exact Solutions

scholarly journals New Classes of Solutions of Dynamical Problems of Plasticity

2020 ◽  
pp. 792-796
Author(s):  
Sergei I. Senashov ◽  
Olga V. Gomonova ◽  
Irina L. Savostyanova ◽  
Olga N. Cherepanova
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Plastic Layer ◽  
Two Dimensional ◽  
Science And Engineering ◽  
Spatial Variables ◽  
One Dimensional ◽  
New Exact Solutions

Dynamical problems of the theory of plasticity have not been adequately studied. Dynamical problems arise in various fields of science and engineering but the complexity of original differential equations does not allow one to construct new exact solutions and to solve boundary value problems correctly. One-dimensional dynamical problems are studied rather well but two-dimensional problems cause major difficulties associated with nonlinearity of the main equations. Application of symmetries to the equations of plasticity allow one to construct some exact solutions. The best known exact solution is the solution obtained by B.D. Annin. It describes non-steady compression of a plastic layer by two rigid plates. This solution is a linear one in spatial variables but includes various functions of time. Symmetries are also considered in this paper. These symmetries allow transforming exact solutions of steady equations into solutions of non-steady equations. The obtained solution contains 5 arbitrary functions

Some new exact solutions for the two-dimensional Navier–Stokes equations

2007 ◽  
Vol 371 (5-6) ◽  
pp. 438-452 ◽  
Author(s):  
Chiping Wu ◽  
Zhongzhen Ji ◽  
Yongxing Zhang ◽  
Jianzhong Hao ◽  
Xuan Jin
Keyword(s):  
Exact Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
New Exact Solutions

scholarly journals Exact Solutions of the Two-Dimensional Cattaneo Model Using Lie Symmetry Transformations

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Khudija Bibi ◽  
Khalil Ahmad
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Lie Symmetry ◽  
Two Dimensional ◽  
Linear Combinations ◽  
Similarity Transformations ◽  
New Exact Solutions ◽  
Symmetry Transformations

In this article, symmetry technique is utilized to obtain new exact solutions of the Cattaneo equation. The infinitesimal symmetries, linear combinations of these symmetries, and corresponding similarity variables are determined, which lead to many exact solutions of the considered equation. By applying similarity transformations, the mentioned partial differential equation is reduced to some ordinary differential equations of second order. Solutions of these ordinary differential equations have yielded many exact solutions of the Cattaneo equation.

scholarly journals New Soliton Solutions of The Nonlinear Radhakrishnan-Kundu-Lakshmanan Equation With the Beta-Derivative

2021 ◽  
Author(s):  
Yusuf Pandir ◽  
Yusuf Gurefe ◽  
Tolga Akturk
Keyword(s):  
Exact Solutions ◽  
Rational Function ◽  
Function Method ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Dark Soliton ◽  
Two Dimensional ◽  
New Exact Solutions ◽  
Definition Of ◽  
Exponential Function Method

Abstract In this article, the modified exponential function method is applied to find the exact solutions of the Radhakrishnan-Kundu-Lakshmanan equation with Atangana’s conformable beta-derivative. The definition of the conformable beta derivative and its properties proposed by Atangana are given. With the proposed method, exact solutions of the nonlinear Radhakrishnan-Kundu-Lakshmanan equation which can be stated with the conformable beta-derivative of Atangana are obtained. The exact solutions found as a result of the application of the method seem to be 1-soliton solutions, dark soliton solutions, periodic soliton solutions and rational function solutions. According to the obtained results, we can say that the Radhakrishnan-Kundu-Lakshmanan equation with Atangana’s conformable beta-derivative have different soliton solutions. Also, three-dimensional contour and density graphs and two- dimensional graphs drawn with different parameters are given of these new exact solutions.

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