Global analytic solutions and traveling wave solutions of the Cauchy problem for the Novikov equation

2017 ◽  
Vol 146 (4) ◽  
pp. 1537-1550
Author(s):  
Xinglong Wu
Keyword(s):  
Cauchy Problem ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Analytic Solutions ◽  
Wave Solutions ◽  
The Cauchy Problem ◽  
Novikov Equation

scholarly journals Remarks on solitary waves and Cauchy problem for Half-wave-Schrödinger equations

2020 ◽  
pp. 2050058
Author(s):  
Yakine Bahri ◽  
Slim Ibrahim ◽  
Hiroaki Kikuchi
Keyword(s):  
Cauchy Problem ◽  
Traveling Wave ◽  
Critical Case ◽  
Orbital Stability ◽  
Traveling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Half Wave ◽  
The Cauchy Problem ◽  
Interesting Open Problem

In this paper, we study solitary wave solutions of the Cauchy problem for Half-wave-Schrödinger equation in the plane. First, we show the existence and the orbital stability of the ground states. Second, we prove that given any speed [Formula: see text], traveling wave solutions exist and converge to the zero wave as the velocity tends to [Formula: see text]. Finally, we solve the Cauchy problem for initial data in [Formula: see text], with [Formula: see text]. The critical case [Formula: see text] still stands as an interesting open problem.

scholarly journals Single peaked traveling wave solutions to a generalized μ-Novikov Equation

2020 ◽  
Vol 10 (1) ◽  
pp. 66-75
Author(s):  
Byungsoo Moon
Keyword(s):  
Linear Combination ◽  
Traveling Wave ◽  
Wave Solution ◽  
Traveling Wave Solutions ◽  
Traveling Wave Solution ◽  
Wave Solutions ◽  
Quadratic Nonlinearities ◽  
Novikov Equation

Abstract In this paper, we study the existence of peaked traveling wave solution of the generalized μ-Novikov equation with nonlocal cubic and quadratic nonlinearities. The equation is a μ-version of a linear combination of the Novikov equation and Camassa-Hom equation. It is found that the equation admits single peaked traveling wave solutions.

THE EXACT TRAVELING WAVE SOLUTIONS AND THEIR BIFURCATIONS IN THE GARDNER AND GARDNER–KP EQUATIONS

2012 ◽  
Vol 22 (05) ◽  
pp. 1250126 ◽  
Author(s):  
FANG YAN ◽  
CUNCAI HUA ◽  
HAIHONG LIU ◽  
ZENGRONG LIU
Keyword(s):  
Traveling Wave ◽  
Function Method ◽  
Traveling Wave Solutions ◽  
Analytic Solutions ◽  
Auxiliary Function ◽  
Kp Equation ◽  
Gardner Equation ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Kp Equations

By using the method of dynamical systems, this paper studies the exact traveling wave solutions and their bifurcations in the Gardner equation. Exact parametric representations of all wave solutions as well as the explicit analytic solutions are given. Moreover, several series of exact traveling wave solutions of the Gardner–KP equation are obtained via an auxiliary function method.

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.

scholarly journals Exploration on traveling wave solutions to the 3rd-order klein–fock-gordon equation (KFGE) in mathematical physics

2020 ◽  
Vol 8 (1) ◽  
pp. 14 ◽  
Author(s):  
Nur Hasan Mahmud Shahen ◽  
Foyjonnesa . ◽  
Md. Habibul Bashar
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Dynamic Models ◽  
Gordon Equation ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Analytic Solutions ◽  
Hyperbolic Function ◽  
Wave Solutions

In this paper, the -expansion method has been applied to find the new exact traveling wave solutions of the nonlinear evaluation equations (NLEEs) by utilizing 3rd-order Klein–Gordon Equation (KFGE). With the collaboration of symbolic commercial software maple, the competence of this method for inventing these exact solutions has been more exhibited. As an upshot, some new exact solutions are obtained and signified by hyperbolic function solutions, different combinations of trigonometric function solutions, and exponential function solutions. Moreover, the -expansion method is a more efficient method for exploring essential nonlinear waves that enrich a variety of dynamic models that arises in nonlinear fields. All sketching is given out to show the properties of the innovative explicit analytic solutions. Our proposed method is directed, succinct, and reasonably good for the various nonlinear evaluation equations (NLEEs) related treatment and mathematical physics also. 

scholarly journals Construction of Different Types Analytic Solutions for the Zhiber-Shabat Equation

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 908 ◽  
Author(s):  
Asıf Yokus ◽  
Hülya Durur ◽  
Hijaz Ahmad ◽  
Shao-Wen Yao
Keyword(s):  
Traveling Wave ◽  
Evolution Equations ◽  
Solution Process ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Analytic Solutions ◽  
Nonlinear Evolution Equations ◽  
Important Place ◽  
Advantages And Disadvantages ◽  
Wave Solutions

In this paper, a new solution process of ( 1 / G ′ ) -expansion and ( G ′ / G , 1 / G ) -expansion methods has been proposed for the analytic solution of the Zhiber-Shabat (Z-S) equation. Rather than the classical ( G ′ / G , 1 / G ) -expansion method, a solution function in different formats has been produced with the help of the proposed process. New complex rational, hyperbolic, rational and trigonometric types solutions of the Z-S equation have been constructed. By giving arbitrary values to the constants in the obtained solutions, it can help to add physical meaning to the traveling wave solutions, whereas traveling wave has an important place in applied sciences and illuminates many physical phenomena. 3D, 2D and contour graphs are displayed to show the stationary wave or the state of the wave at any moment with the values given to these constants. Conditions that guarantee the existence of traveling wave solutions are given. Comparison of ( G ′ / G , 1 / G ) -expansion method and ( 1 / G ′ ) -expansion method, which are important instruments in the analytical solution, has been made. In addition, the advantages and disadvantages of these two methods have been discussed. These methods are reliable and efficient methods to obtain analytic solutions of nonlinear evolution equations (NLEEs).

Different types analytic solutions of the (1+1)-dimensional resonant nonlinear Schrödinger’s equation using (G′/G)-expansion method

2019 ◽  
Vol 34 (03) ◽  
pp. 2050036 ◽  
Author(s):  
Hülya Durur
Keyword(s):  
Applied Sciences ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Analytic Solutions ◽  
Important Place ◽  
Wave Solutions ◽  
Different Types ◽  
Hyperbolic Complex ◽  
Rational Type

In this paper, an alternative method has been studied for traveling wave solutions of mathematical models which have an important place in applied sciences and balance term is not integer. With this method, the trigonometric, hyperbolic, complex and rational type traveling wave solutions of the (1[Formula: see text]+[Formula: see text]1)-dimensional resonant nonlinear Schrödinger’s (RNLS) equation with the parabolic law have constructed. This method can be applied reliably and effectively in many differential equations.

Analysis of a heterogeneous model for riot dynamics: the effect of censorship of information

2015 ◽  
Vol 27 (3) ◽  
pp. 554-582 ◽  
Author(s):  
H. BERESTYKI ◽  
N. RODRIGUEZ
Keyword(s):  
Numerical Simulations ◽  
Traveling Wave ◽  
Heterogeneous Media ◽  
Traveling Wave Solutions ◽  
Global Solutions ◽  
Open Problems ◽  
Heterogeneous Model ◽  
Wave Solutions ◽  
The Cauchy Problem ◽  
Barrier Problem

This paper is concerned with modelling the dynamics of social outbursts of activity, such as protests or riots. In this sequel to our work in Berestycki et al. (Networks and Heterogeneous Media, vol. 10, no. 3, 1–34), written in collaboration with J-P. Nadal, we model the effect of restriction of information and explore its impact on the existence of upheaval waves. The system involves the coupling of an explicit variable representing the intensity of rioting activity and an underlying (implicit) field of social tension. We prove the existence of global solutions to the Cauchy problem in ${\mathbb R}^d$ as well as the existence of traveling wave solutions in certain parameter regimes. We furthermore explore the effects of heterogeneities in the environment with the help of numerical simulations, which lead to pulsating waves in certain cases. We analyse the effects of periodic domains as well as the barrier problem with the help of numerical simulations. The barrier problem refers to the potential blockage of a wavefront due to a spatial heterogeneity in the system which leads to an area of low excitability (referred to as the barrier). We conclude with a variety of open problems.

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