Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640018 ◽  
Author(s):  
Wen-Xiu Ma ◽  
Yuan Zhou ◽  
Rachael Dougherty
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Symbolic Computations ◽  
Polynomial Solutions ◽  
Quartic Polynomial ◽  
Dependent Variables ◽  
Bilinear Equations

Lump-type solutions, rationally localized in many directions in the space, are analyzed for nonlinear differential equations derived from generalized bilinear differential equations. By symbolic computations with Maple, positive quadratic and quartic polynomial solutions to two classes of generalized bilinear differential equations on [Formula: see text] are computed, and thus, lump-type solutions are presented to the corresponding nonlinear differential equations on [Formula: see text], generated from taking a transformation of dependent variables [Formula: see text].

A class of nonlinear heat-type equations and their multi-wave solutions by a generalized bilinear technique

2016 ◽  
Vol 30 (06) ◽  
pp. 1650067
Author(s):  
Jun-Rong Liu
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
New Technique ◽  
Bilinear Operators ◽  
Wave Solutions ◽  
Dependent Variables ◽  
A New Technique ◽  
Bilinear Equations

Multi-wave resonant solutions of a class of nonlinear heat-type equations are investigated by their corresponding generalized bilinear equations. We develop a new technique for searching for resonant solutions to those generalized bilinear equations, using the idea of weights of dependent variables. The results show that generalized bilinear operators and generalized bilinear equations are powerful and irreplaceable tools for dealing with nonlinear differential equations.

New Liapunov Function for Nonlinear Time-Varying Systems

1968 ◽  
Vol 90 (2) ◽  
pp. 208-212
Author(s):  
A. K. Newman
Keyword(s):  
Differential Equations ◽  
Constant Coefficient ◽  
Quadratic Forms ◽  
Nonlinear Differential Equations ◽  
Liapunov Function ◽  
Time Varying ◽  
Liapunov Functions ◽  
Dependent Variables ◽  
Time Varying Systems ◽  
Derivatives Of

A new method of generating Liapunov functions is described that is useful for time-varying nonlinear differential equations. Constant coefficient quadratic forms are often used as Liapunov functions for linear constant coefficient differential equations. However, if the differential equation coefficients are time-varying and nonlinear, better results are usually obtained by using variable quadratic forms as Liapunov functions. This variation in V often introduces undesirable terms in V˙, which are cancelled by modifying V by subtracting integrals of certain partial derivatives of V with respect to the dependent variables. With few restrictions, V is proved to remain positive definite after the modification. The method often directly extends a Liapunov function useful for constant coefficient differential equations to cover the case when the coefficients are time-varying and nonlinear. Two examples are presented, including the incremental circuit for a time-varying nonlinear transmission line with hysteresis and the equations for an N-body collision avoidance problem.

Application of the exp(-φ)-expansion method to the Pochhammer-Chree equation

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3347-3354 ◽  
Author(s):  
Nematollah Kadkhoda ◽  
Michal Feckan ◽  
Yasser Khalili
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Present Article ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Rational Functions ◽  
Expansion Method ◽  
Exponential Functions

In the present article, a direct approach, namely exp(-?)-expansion method, is used for obtaining analytical solutions of the Pochhammer-Chree equations which have a many of models. These solutions are expressed in exponential functions expressed by hyperbolic, trigonometric and rational functions with some parameters. Recently, many methods were attempted to find exact solutions of nonlinear partial differential equations, but it seems that the exp(-?)-expansion method appears to be efficient for finding exact solutions of many nonlinear differential equations.

scholarly journals Solving nonlinear differential equations with differentiable quantum circuits

2021 ◽  
Vol 103 (5) ◽  
Author(s):  
Oleksandr Kyriienko ◽  
Annie E. Paine ◽  
Vincent E. Elfving
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Quantum Circuits

scholarly journals Nonlinear Differential Equations Modeling the Antarctic Circumpolar Current

2021 ◽  
Vol 23 (4) ◽  
Author(s):  
Jifeng Chu ◽  
Kateryna Marynets
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Topological Degree ◽  
Antarctic Circumpolar Current ◽  
Fixed Point Theorems ◽  
Nonlinear Differential Equations ◽  
Existence Results ◽  
Circumpolar Current ◽  
The Antarctic

AbstractThe aim of this paper is to study one class of nonlinear differential equations, which model the Antarctic circumpolar current. We prove the existence results for such equations related to the geophysical relevant boundary conditions. First, based on the weighted eigenvalues and the theory of topological degree, we study the semilinear case. Secondly, the existence results for the sublinear and superlinear cases are proved by fixed point theorems.

Asymptotics for third-order nonlinear differential equations: Non-oscillatory and oscillatory cases

2021 ◽  
pp. 1-19
Author(s):  
Calogero Vetro ◽  
Dariusz Wardowski
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Oscillatory Behavior ◽  
Third Order ◽  
First Order ◽  
Third Order Differential Equation ◽  
Comparison Technique ◽  
First Order Differential Equations

We discuss a third-order differential equation, involving a general form of nonlinearity. We obtain results describing how suitable coefficient functions determine the asymptotic and (non-)oscillatory behavior of solutions. We use comparison technique with first-order differential equations together with the Kusano–Naito’s and Philos’ approaches.

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