A Study on Rational Solutions to a KP-like Equation

2015 ◽  
Vol 70 (4) ◽  
pp. 263-268 ◽  
Author(s):  
Yufeng Zhang ◽  
Wen-Xiu Ma
Keyword(s):  
Differential Equation ◽  
Prime Number ◽  
Nonlinear Differential Equation ◽  
Symbolic Computation ◽  
Bilinear Form ◽  
Differential Operators ◽  
Rational Solutions ◽  
Polynomial Solutions ◽  
Bilinear Equation

AbstractA KP-like nonlinear differential equation is introduced through a generalised bilinear equation which possesses the same bilinear form as the standard KP bilinear equation. By symbolic computation, nine classes of rational solutions to the resulting KP-like equation are generated from a search for polynomial solutions to the corresponding generalised bilinear equation. Three generalised bilinear differential operators adopted are associated with the prime number p=3.

MULTI-SOLITON SOLUTIONS AND THEIR INTERACTIONS FOR THE (2+1)-DIMENSIONAL SAWADA-KOTERA MODEL WITH TRUNCATED PAINLEVÉ EXPANSION, HIROTA BILINEAR METHOD AND SYMBOLIC COMPUTATION

2009 ◽  
Vol 23 (25) ◽  
pp. 5003-5015 ◽  
Author(s):  
XING LÜ ◽  
TAO GENG ◽  
CHENG ZHANG ◽  
HONG-WU ZHU ◽  
XIANG-HUA MENG ◽  
...  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Symbolic Computation ◽  
Bilinear Form ◽  
Soliton Solutions ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Truncated Painlevé Expansion ◽  
Hirota Bilinear ◽  
Bilinear Equations

In this paper, the (2+1)-dimensional Sawada-Kotera equation is studied by the truncated Painlevé expansion and Hirota bilinear method. Firstly, based on the truncation of the Painlevé series we obtain two distinct transformations which can transform the (2+1)-dimensional Sawada-Kotera equation into two bilinear equations of different forms (which are shown to be equivalent). Then employing Hirota bilinear method, we derive the analytic one-, two- and three-soliton solutions for the bilinear equations via symbolic computation. A formula which denotes the N-soliton solution is given simultaneously. At last, the evolutions and interactions of the multi-soliton solutions are graphically discussed as well. It is worthy to be noted that the truncated Painlevé expansion provides a useful dependent variable transformation which transforms a partial differential equation into its bilinear form and by means of the bilinear form, further study of the original partial differential equation can be conducted.

Construction of rational solutions for the (2+1)-dimensional Broer–Kaup system

2019 ◽  
Vol 33 (30) ◽  
pp. 1950377 ◽  
Author(s):  
Yun-Hu Wang
Keyword(s):  
Linear Equation ◽  
Symbolic Computation ◽  
Linear Form ◽  
Rational Solutions ◽  
Dynamical Features ◽  
Truncated Painlevé Expansion ◽  
Bilinear Equation

Based on the quartic–linear form of the (2[Formula: see text]+[Formula: see text]1)-dimensional Broer–Kaup system which is derived from its truncated Painlevé expansion, three kinds of rational solutions are obtained through ansatz and symbolic computation with Maple. In general, these kinds of solutions obtained from quartic–linear equation are different from the ones which are generated via bilinear equation. Figures are presented to show the dynamical features of these solutions.

scholarly journals Rational solution to a shallow water wave-like equation

2016 ◽  
Vol 20 (3) ◽  
pp. 875-880
Author(s):  
Hong-Cai Ma ◽  
Ke Ni ◽  
Guo-Ding Ruan ◽  
Ai-Ping Deng
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Shallow Water ◽  
Prime Number ◽  
Water Wave ◽  
Rational Solution ◽  
Rational Solutions ◽  
Shallow Water Wave ◽  
Linear Differential ◽  
New Equation

Two classes of rational solutions to a shallow water wave-like non-linear differential equation are constructed. The basic object is a generalized bilinear differential equation based on a prime number, p = 3. Through this new transformation and with the help of symbolic computation with MAPLE, both the new equation and its rational solutions are obtained.

On Linear Ordinary Differential Equation With Orthogonal Polynomial Solutions

2018 ◽  
Vol 1 (20) ◽  
pp. 559-572
Author(s):  
ولدان وليد محمود
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Fourth Order ◽  
Differential Operators ◽  
Linear Ordinary Differential Equation ◽  
Large Piece ◽  
Polynomial Solutions ◽  
The Subject

: The subject of orthogonal polynomials cuts across a large piece of mathematics and its applications. In this paper we give a  survey of the orthogonal polynomial solutions of second-order and fourth-order linear ordinary differential equation, on the generated self-adjoint differential operators .

scholarly journals Lump Solutions to a (2+1)-Dimensional Fifth-Order KdV-Like Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Sumayah Batwa ◽  
Wen-Xiu Ma
Keyword(s):  
Prime Number ◽  
Bilinear Form ◽  
Kdv Equation ◽  
Quadratic Function ◽  
Lump Solutions ◽  
Free Parameters ◽  
New Equation ◽  
Fifth Order ◽  
Generalized Bilinear Equation ◽  
Bilinear Equation

A (2+1)-dimensional fifth-order KdV-like equation is introduced through a generalized bilinear equation with the prime number p=5. The new equation possesses the same bilinear form as the standard (2+1)-dimensional fifth-order KdV equation. By Maple symbolic computation, classes of lump solutions are constructed from a search for quadratic function solutions to the corresponding generalized bilinear equation. We get a set of free parameters in the resulting lump solutions, of which we can get a nonzero determinant condition ensuring analyticity and rational localization of the solutions. Particular classes of lump solutions with special choices of the free parameters are generated and plotted as illustrative examples.

scholarly journals Oscillatory behavior of a second order nonlinear advanced differential equation with mixed neutral terms

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hongwei Shi ◽  
Yuzhen Bai
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Oscillation Criteria ◽  
Order Nonlinear Differential Equation

AbstractIn this paper, we present several new oscillation criteria for a second order nonlinear differential equation with mixed neutral terms of the form $$ \bigl(r(t) \bigl(z'(t)\bigr)^{\alpha }\bigr)'+q(t)x^{\beta } \bigl(\sigma (t)\bigr)=0,\quad t\geq t_{0}, $$(r(t)(z′(t))α)′+q(t)xβ(σ(t))=0,t≥t0, where $z(t)=x(t)+p_{1}(t)x(\tau (t))+p_{2}(t)x(\lambda (t))$z(t)=x(t)+p1(t)x(τ(t))+p2(t)x(λ(t)) and α, β are ratios of two positive odd integers. Our results improve and complement some well-known results which were published recently in the literature. Two examples are given to illustrate the efficiency of our results.

Sign in / Sign up

Export Citation Format

Share Document