We study the Cauchy problem for the weakly dissipative Dullin–Gottwald–Holm equation describing unidirectional propagation of surface waves in a shallow water ...regime:ut−α2uxxt+c0ux+3uux+γuxxx+λ(u−α2uxx)=α2(2uxuxx+uuxxx). In the present paper we demonstrate the simple conditions on the initial data that lead to blow-up of the solution in finite time.
In this paper we classify all bounded travelling wave solutions for the integrable Dullin–Gottwald–Holm equation. It is shown that it decomposes in two known cases: the Camassa–Holm and the ...Korteweg–de Vries equation. For the former, the classification is similar to the one presented in (2005) 12, while for the latter it is only possible to obtain smooth solutions.
In this paper, a numerical scheme to solve the time‐fractional Dullin–Gottwald–Holm equation has been proposed. The proposed scheme is based on approximating the solution using multiquadric radial ...basis function. The fractional derivatives are described in the Caputo sense. The obtained numerical results are compared with exact solution to confirm the accuracy and effectiveness of the presented scheme.
Considered herein is the well-posedness problem of the periodic two-component Dullin–Gottwald–Holm (DGH) system on the circle, which can be derived from Eulerʼs equation with constant vorticity in ...shallow water waves moving over a linear shear flow. The result of blow-up solutions for certain initial profiles in a manner which corresponds to wave-breaking is established.
Considered herein is the Cauchy problem of a modified two-component Dullin–Gottwald–Holm system. We derive a new sufficient condition on the initial data to guarantee that the corresponding solution ...to this system blows up in finite time.
This article delves into new solitary wave solutions for the unidirectional Dullin–Gottwald–Holm (DGH) model. Dynamical characterizations of wave prorogation in shallow water are investigated using ...two modern analytical approaches (generalized Tanh-function (GTF) methods and the improved Fan-expansion (IFE) method). In addition, the solutions achieved by using He’s variational iteration (He’s VI) approach are examined by demonstrating agreement between the built analytical and numerical solutions, providing more evidence for the correctness of the acquired results. Numerous sorts of solutions are produced, including solitary, shock, kink, periodic, cone, anti-kink, etc. The numerical simulations of these solutions are shown as 3D, 2D, and contour graphs. Some stream graphs illustrate the local direction of the vector field at each point, and a relatively uniform density across the property, indicating no background scalar field, further elucidates the interaction between solutions. In light of prior literature, the authors of this research provide an explanation for the originality of their findings. The comparison reveals up-to-date analytical and numerical solutions to the model under study.
•Analytical handling of the unidirectional Dullin–Gottwald–Holm (DGH) model.•Studying the stability property of the model and obtained solitary wave solutions.•Explaining the obtained solutions through some distinct types of sketches.
This article discusses the dynamical structures of unique traveling wave solutions to the unidirectional time-fractional Dullin–Gottwald–Holm equation, as well as the modulation instability analysis ...of solitary wave prorogations in shallow water mechanics. Different sorts of explicit solutions are obtained which are expressed as the kink, singular kink, singular soliton, multiple solitons and other forms of soliton with specified parameters by utilizing the unified method. Three-dimensional plots, density plots, and their two-dimensional combined line plots of the obtained novel solutions satisfying the corresponding equation of interest are given to comprehend the underlying mechanisms of the identified family.
The obtained novel wave solutions can motivate applied scientists to refine their theories to the best of their abilities and can be utilized to verify the outcomes of numerical simulations for wave propagation in shallow water and other nonlinear cases. The implemented methods are shown to be straightforward and effective for approximating the considered equation, and it may be utilized to resolve numerous classes of nonlinear partial differential equations that arise in engineering and mathematical physics.
In this paper, the asymptotic stability of the solutions near the explicit singular waves of Dullin–Gottwald–Holm equation is studied based on the commutator estimate, the semi-group theory of linear ...operator and the Banach fixed point theorem.
In this paper we construct the conservation laws for the Camassa–Holm equation, the Dullin–Gottwald–Holm equation (DGH) and the generalized Dullin–Gottwald–Holm equation (generalized DGH). The ...variational derivative approach is used to derive the conservation laws. Only first order multipliers are considered. Two multipliers are obtained for the Camassa–Holm equation. For the DGH and generalized DGH equations the variational derivative approach yields two multipliers; thus two conserved vectors are obtained.
PDF
