POSITON, NEGATON, SOLITON AND COMPLEXITON SOLUTIONS TO A FOUR-DIMENSIONAL NONLINEAR EVOLUTION EQUATION

2009 ◽  
Vol 23 (25) ◽  
pp. 2971-2991 ◽  
Author(s):  
ZHAQILAO ◽  
ZHI-BIN LI
Keyword(s):  
Evolution Equation ◽  
Exact Solutions ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Direct Method ◽  
Sufficient Conditions ◽  
Comprehensive Approach ◽  
Wronskian Determinant ◽  
Interaction Solutions ◽  
Complexiton Solutions

A generalized Wronskian formulation is presented for a four-dimensional nonlinear evolution equation. The representative systems are explicitly solved by selecting a broad set of sufficient conditions which make the Wronskian determinant a solution to the bilinearized four-dimensional nonlinear evolution equation. The obtained solution formulas provide us with a comprehensive approach to construct explicit exact solutions to the four-dimensional nonlinear evolution equation, by which positons, negatons, solitons and complexitons are computed for the four-dimensional nonlinear evolution equation. Applying the Hirota's direct method, multi-soliton, non-singular complexiton, and their interaction solutions of the four-dimensional nonlinear evolution equation are also obtained.

A NONLINEAR EVOLUTION EQUATION IN AN ORDERED SPACE, ARISING FROM KINETIC THEORY

2007 ◽  
Vol 09 (02) ◽  
pp. 217-251
Author(s):  
CECIL P. GRÜNFELD
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Sufficient Conditions ◽  
Existence Theory ◽  
The Cauchy Problem ◽  
Ordered Space ◽  
Coagulation Equation ◽  
Boltzmann Model ◽  
Classical Boltzmann Equation

We investigate the Cauchy problem for a nonlinear evolution equation, formulated in an abstract Lebesgue space, as a generalization of various Boltzmann kinetic models. Our main result provides sufficient conditions for the existence, uniqueness, and positivity of global in time solutions. The analysis extends nontrivially monotonicity methods, originally developed in the context of the existence theory for the classical Boltzmann equation in L1. Our application examples are Smoluchowski's coagulation equation, a Povzner-like equation with dissipative collisions, and a Boltzmann model with chemical reactions, for which we obtain a unitary existence theory, with improved results, compared to the literature.

scholarly journals Residual Symmetry Reduction and Consistent Riccati Expansion to a Nonlinear Evolution Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Lamine Thiam ◽  
Xi-zhong Liu
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Symmetry Reduction ◽  
Lie Symmetry ◽  
Residual Symmetry ◽  
Interaction Solutions ◽  
Lie Symmetry Method ◽  
Consistent Riccati Expansion ◽  
Symmetry Method

The residual symmetry of a (1 + 1)-dimensional nonlinear evolution equation (NLEE) ut+uxxx−6u2ux+6λux=0 is obtained through Painlevé expansion. By introducing a new dependent variable, the residual symmetry is localized into Lie point symmetry in an enlarged system, and the related symmetry reduction solutions are obtained using the standard Lie symmetry method. Furthermore, the (1 + 1)-dimensional NLEE equation is proved to be integrable in the sense of having a consistent Riccati expansion (CRE), and some new Bäcklund transformations (BTs) are given. In addition, some explicitly expressed solutions including interaction solutions between soliton and cnoidal waves are derived from these BTs.

scholarly journals Bäcklund Transformation and Exact Solutions to a Generalized (3 + 1)-Dimensional Nonlinear Evolution Equation

2022 ◽  
Vol 2022 ◽  
pp. 1-12
Author(s):  
Yali Shen ◽  
Ying Yang
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Bäcklund Transformation ◽  
Dynamic Properties ◽  
Traveling Wave Solutions ◽  
Mixed Solution ◽  
Backlund Transformation ◽  
Bilinear Method ◽  
Interaction Solutions

In this article, a generalized (3 + 1)-dimensional nonlinear evolution equation (NLEE), which can be obtained by a multivariate polynomial, is investigated. Based on the Hirota bilinear method, the N-soliton solution and bilinear Bäcklund transformation (BBT) with explicit formulas are successfully constructed. By using BBT, two traveling wave solutions and a mixed solution of the generalized (3 + 1)-dimensional NLEE are obtained. Furthermore, the lump and the interaction solutions for the equation are constructed. Finally, the dynamic properties of the lump and the interaction solutions are described graphically.

scholarly journals Lump Solutions and Interaction Solutions for the Dimensionally Reduced Nonlinear Evolution Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Baoyong Guo ◽  
Huanhe Dong ◽  
Yong Fang
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Quadratic Functions ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Dynamical Characteristics ◽  
Interaction Solutions ◽  
Physical Quantities ◽  
Hirota Bilinear

In this paper, by means of the Hirota bilinear method, a dimensionally reduced nonlinear evolution equation is investigated. Through its bilinear form, lump solutions are obtained. We construct interaction solutions between lump solutions and one soliton solution by choosing quadratic functions and exponential function. Interaction solutions with the combinations of exponential functions and sine function are also given. Meanwhile, the figures of these solutions are plotted. The dynamical characteristics and properties of obtained solutions are discussed, respectively. The results show that the corresponding physical quantities and properties of nonlinear waves are associated with the values of the parameters.

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