scholarly journals Some Smooth and Nonsmooth Traveling Wave Solutions for KP-MEW(2, 2) Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Liping He ◽  
Yuanhua Lin ◽  
Hongying Zhu
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Integral Constant ◽  
Wave Solutions ◽  
Planar Dynamical Systems

In this paper, we consider the KP-MEW(2, 2) equation by the theory of bifurcations of planar dynamical systems when integral constant is considered. The periodic peakon solution and peakon and smooth periodic solutions are given.

Bifurcations and Exact Traveling Wave Solutions of a Modified Nonlinear Schrödinger Equation

2016 ◽  
Vol 26 (06) ◽  
pp. 1650106 ◽  
Author(s):  
KitIan Kou ◽  
Jibin Li
Keyword(s):  
Dynamical Systems ◽  
Traveling Wave ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
One Dimensional ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Planar Dynamical Systems

In this paper, we consider two singular nonlinear planar dynamical systems created from the studies of one-dimensional bright and dark spatial solitons for one-dimensional beams in a nonlocal Kerr-like media. On the basis of the investigation of the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we obtain all possible explicit exact parametric representations of solutions (including solitary wave solutions, periodic wave solutions, peakon and periodic peakons, compacton solutions, etc.) under different parameter conditions.

Bifurcations and Exact Traveling Wave Solutions of Degenerate Coupled Multi-KdV Equations

2016 ◽  
Vol 26 (03) ◽  
pp. 1650045 ◽  
Author(s):  
Jibin Li ◽  
Fengjuan Chen
Keyword(s):  
Dynamical Systems ◽  
Traveling Wave ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Kdv Equations ◽  
Wave Solutions ◽  
Kink Wave ◽  
Multiplicity Formula ◽  
Planar Dynamical Systems

In this paper, we consider the degenerate coupled multi-KdV equations. Depending on the coupled multiplicity [Formula: see text], the study of the traveling wave solutions for this model derives a series of planar dynamical systems. We consider the cases of [Formula: see text] On the basis of the investigation on the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we obtain all possible explicit exact parametric representations of solutions (including solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions) under different parameter conditions.

scholarly journals Periodic and Solitary-Wave Solutions for a Variant of theK(3,2)Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Jiangbo Zhou ◽  
Lixin Tian
Keyword(s):  
Solitary Wave ◽  
Periodic Solutions ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Polynomial Differential Systems ◽  
Wave Solutions ◽  
Paper Supplement ◽  
Planar Dynamical Systems

We employ the bifurcation method of planar dynamical systems and qualitative theory of polynomial differential systems to derive new bounded traveling-wave solutions for a variant of theK(3,2)equation. For the focusing branch, we obtain hump-shaped and valley-shaped solitary-wave solutions and some periodic solutions. For the defocusing branch, the nonexistence of solitary traveling wave solutions is shown. Meanwhile, some periodic solutions are also obtained. The results presented in this paper supplement the previous results.

scholarly journals New groups of solutions to the Whitham-Broer-Kaup equation

2020 ◽  
Vol 41 (11) ◽  
pp. 1735-1746
Author(s):  
Yaji Wang ◽  
Hang Xu ◽  
Q. Sun
Keyword(s):  
Dynamical Systems ◽  
Riccati Equation ◽  
Bifurcation Analysis ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Phase Portraits ◽  
Tsunami Waves ◽  
Wave Solutions ◽  
Planar Dynamical Systems ◽  
Projective Riccati Equation Method

Abstract The Whitham-Broer-Kaup model is widely used to study the tsunami waves. The classical Whitham-Broer-Kaup equations are re-investigated in detail by the generalized projective Riccati-equation method. 20 sets of solutions are obtained of which, to the best of the authors’ knowledge, some have not been reported in literature. Bifurcation analysis of the planar dynamical systems is then used to show different phase portraits of the traveling wave solutions under various parametric conditions.

scholarly journals A Class of Approximate Damped Oscillatory Solutions to Compound KdV-Burgers-Type Equation with Nonlinear Terms of Any Order: Preliminary Results

2014 ◽  
Vol 2014 ◽  
pp. 1-19
Author(s):  
Yan Zhao ◽  
Weiguo Zhang
Keyword(s):  
Dynamical Systems ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Type Equation ◽  
Approximate Solutions ◽  
Exponential Form ◽  
Oscillatory Solutions ◽  
Wave Solutions ◽  
Existence Conditions ◽  
Planar Dynamical Systems

This paper is focused on studying approximate damped oscillatory solutions of the compound KdV-Burgers-type equation with nonlinear terms of any order. By the theory and method of planar dynamical systems, existence conditions and number of bounded traveling wave solutions including damped oscillatory solutions are obtained. Utilizing the undetermined coefficients method, the approximate solutions of damped oscillatory solutions traveling to the left are presented. Error estimates of these approximate solutions are given by the thought of homogeneous principle. The results indicate that errors between implicit exact damped oscillatory solutions and approximate damped oscillatory solutions are infinitesimal decreasing in the exponential form.

Exploration of the algebraic traveling wave solutions of a higher order model

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Jian-Gen Liu ◽  
Yi-Ying Feng ◽  
Hong-Yi Zhang
Keyword(s):  
Dynamical Systems ◽  
Shallow Water ◽  
Traveling Wave ◽  
Water Wave ◽  
Traveling Wave Solutions ◽  
Content Type ◽  
Shallow Water Wave ◽  
Wave Solutions ◽  
Wave Phenomena ◽  
Planar Dynamical Systems

Purpose The purpose of this paper is to construct the algebraic traveling wave solutions of the (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsve (KdV-Z-K) equation, which can be usually used to express shallow water wave phenomena. Design/methodology/approach The authors apply the planar dynamical systems and invariant algebraic cure approach to find the algebraic traveling wave solutions and rational solutions of the (3 + 1)-dimensional modified KdV-Z-K equation. Also, the planar dynamical systems and invariant algebraic cure approach is applied to considered equation for finding algebraic traveling wave solutions. Findings As a result, the authors can find that the integral constant is zero and non-zero, the algebraic traveling wave solutions have different evolutionary processes. These results help to better reveal the evolutionary mechanism of shallow water wave phenomena and find internal connections. Research limitations/implications The paper presents that the implemented methods as a powerful mathematical tool deal with (3 + 1)-dimensional modified KdV-Z-K equation by using the planar dynamical systems and invariant algebraic cure. Practical implications By considering important characteristics of algebraic traveling wave solutions, one can understand the evolutionary mechanism of shallow water wave phenomena and find internal connections. Originality/value To the best of the authors’ knowledge, the algebraic traveling wave solutions have not been reported in other places. Finally, the algebraic traveling wave solutions nonlinear dynamics behavior was shown.

Exact Cuspon and Compactons of the Novikov Equation

2014 ◽  
Vol 24 (03) ◽  
pp. 1450037 ◽  
Author(s):  
Jibin Li
Keyword(s):  
Dynamical Systems ◽  
Qualitative Analysis ◽  
Traveling Wave ◽  
Wave Solution ◽  
Traveling Wave Solutions ◽  
Phase Portraits ◽  
Breaking Wave ◽  
Wave Solutions ◽  
Novikov Equation

In this paper, we apply the method of dynamical systems to the traveling wave solutions of the Novikov equation. Through qualitative analysis, we obtain bifurcations of phase portraits of the traveling system and exact cuspon wave solution, as well as a family of breaking wave solutions (compactons). We find that the corresponding traveling system of Novikov equation has no one-peakon solution.

On the existence of traveling wave solutions and upper and lower bounds for some Fisher–Kolmogorov type equations

2014 ◽  
Vol 07 (05) ◽  
pp. 1450050 ◽  
Author(s):  
Juan Belmonte-Beitia
Keyword(s):  
Mathematical Biology ◽  
Dynamical Systems ◽  
Lower Bounds ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Systems Approach ◽  
Traveling Wave Solutions ◽  
Upper And Lower Bounds ◽  
Kolmogorov Type ◽  
Wave Solutions

In this paper, we use a dynamical systems approach to prove the existence of traveling waves solutions for the Fisher–Kolmogorov density-dependent equation. Moreover, we prove the existence of upper and lower bounds for these traveling wave solutions found previously. Finally, we present a particular example which has several applications in the mathematical biology field.

scholarly journals Approximate Damped Oscillatory Solutions for Generalized KdV-Burgers Equation and Their Error Estimates

2011 ◽  
Vol 2011 ◽  
pp. 1-26 ◽  
Author(s):  
Weiguo Zhang ◽  
Xiang Li
Keyword(s):  
Dynamical Systems ◽  
Error Estimates ◽  
Traveling Wave ◽  
Wave Solution ◽  
Burgers Equation ◽  
Traveling Wave Solutions ◽  
Approximate Solutions ◽  
Traveling Wave Solution ◽  
Oscillatory Solutions ◽  
Wave Solutions

We focus on studying approximate solutions of damped oscillatory solutions of generalized KdV-Burgers equation and their error estimates. The theory of planar dynamical systems is employed to make qualitative analysis to the dynamical systems which traveling wave solutions of this equation correspond to. We investigate the relations between the behaviors of bounded traveling wave solutions and dissipation coefficient, and give two critical valuesλ1andλ2which can characterize the scale of dissipation effect, for right and left-traveling wave solution, respectively. We obtain that for the right-traveling wave solution if dissipation coefficientα≥λ1, it appears as a monotone kink profile solitary wave solution; that if0<α<λ1, it appears as a damped oscillatory solution. This is similar for the left-traveling wave solution. According to the evolution relations of orbits in the global phase portraits which the damped oscillatory solutions correspond to, we obtain their approximate damped oscillatory solutions by undetermined coefficients method. By the idea of homogenization principle, we give the error estimates for these approximate solutions by establishing the integral equations reflecting the relations between exact and approximate solutions. The errors are infinitesimal decreasing in the exponential form.

scholarly journals Periodic Peakon and Smooth Periodic Solutions for KP-MEW(3,2) Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Junning Cai ◽  
Minzhi Wei ◽  
Liping He
Keyword(s):  
Dynamical System ◽  
Periodic Solution ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Phase Portrait ◽  
Traveling Wave ◽  
Bifurcation Theory ◽  
Wave System ◽  
Integral Constant ◽  
Straight Line

In this paper, we consider the KP-MEW(3,2) equation by the bifurcation theory of dynamical systems when integral constant is considered. The corresponding traveling wave system is a singular planar dynamical system with one singular straight line. The phase portrait for c < 0 , 0 < c < 1 , and c > 1 is drawn. Exact parametric representations of periodic peakon solutions and smooth periodic solution are presented.

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