scholarly journals Unconditional well-posedness for the Kawahara equation

2021 ◽  
Vol 502 (2) ◽  
pp. 125282
Author(s):  
Dan-Andrei Geba ◽  
Bai Lin
Keyword(s):  
Well Posedness ◽  
Kawahara Equation

On the classical solutions for a Rosenau–Korteweg-deVries–Kawahara type equation

2021 ◽  
pp. 1-23
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Cauchy Problem ◽  
Small Amplitude ◽  
Type Equation ◽  
Discrete Systems ◽  
Finite Depth ◽  
Capillary Waves ◽  
Classical Solutions ◽  
Well Posedness ◽  
Kawahara Equation ◽  
The Cauchy Problem

The Rosenau–Korteweg-deVries–Kawahara equation describes the dynamics of dense discrete systems or small-amplitude gravity capillary waves on water of a finite depth. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem.

scholarly journals Well posedness for the Kawahara equation on the half-line

2020 ◽  
Author(s):  
Boling Guo ◽  
fengxia liu
Keyword(s):  
Initial Data ◽  
Local Existence ◽  
Half Line ◽  
Well Posedness ◽  
Kawahara Equation ◽  
Low Regularity ◽  
Regularity Properties

We study the low-regularity properties of the Kawahara equation on the half line. We obtain the local existence, uniqueness, and continuity of the solution. Moreover, We obtain that the nonlinear terms of the solution are smoother than the initial data.

scholarly journals Well-posedness of stochastic modified Kawahara equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
P. Agarwal ◽  
Abd-Allah Hyder ◽  
M. Zakarya
Keyword(s):  
Cauchy Problem ◽  
Fixed Point ◽  
Water Wave ◽  
Well Posedness ◽  
Kawahara Equation ◽  
Fourier Restriction ◽  
The Cauchy Problem ◽  
Restriction Method ◽  
Fifth Order ◽  
Point Argument

AbstractIn this paper we consider the Cauchy problem for the stochastic modified Kawahara equation, which is a fifth-order shallow water wave equation. We prove local well-posedness for data in $H^{s}(\mathbb{R})$Hs(R), $s\geq -1/4$s≥−1/4. Moreover, we get the global existence for $L^{2}( \mathbb{R})$L2(R) solutions. Due to the non-zero singularity of the phase function, a fixed point argument and the Fourier restriction method are proposed.

scholarly journals On Solitary Wave Solutions for the Camassa-Holm and the Rosenau-RLW-Kawahara Equations with the Dual-Power Law Nonlinearities

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Nattakorn Sukantamala ◽  
Supawan Nanta
Keyword(s):  
Solitary Wave ◽  
Nonlinear Wave ◽  
Traveling Wave ◽  
Analytic Solution ◽  
Traveling Wave Solutions ◽  
Well Posedness ◽  
Solitary Wave Solutions ◽  
Kawahara Equation ◽  
Ansatz Method ◽  
Wave Solutions

The nonlinear wave equation is a significant concern to describe wave behavior and structures. Various mathematical models related to the wave phenomenon have been introduced and extensively being studied due to the complexity of wave behaviors. In the present work, a mathematical model to obtain the solution of the nonlinear wave by coupling the classical Camassa-Holm equation and the Rosenau-RLW-Kawahara equation with the dual term of nonlinearities is proposed. The solution properties are analytically derived. The new model still satisfies the fundamental energy conservative property as the original models. We then apply the energy method to prove the well-posedness of the model under the solitary wave hypothesis. Some categories of exact solitary wave solutions of the model are described by using the Ansatz method. In addition, we found that the dual term of nonlinearity is essential to obtain the class of analytic solution. Besides, we provide some graphical representations to illustrate the behavior of the traveling wave solutions.

scholarly journals Well-posedness and controllability of Kawahara equation in weighted Sobolev spaces

2021 ◽  
Vol 207 ◽  
pp. 112267
Author(s):  
Roberto de A. Capistrano-Filho ◽  
Milena Monique de S. Gomes
Keyword(s):  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Well Posedness ◽  
Kawahara Equation

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

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