DGH Equation Research Articles

New Soliton and Periodic Wave Solutions to the Fractional DGH Equation Describing Water Waves in a Shallow Regime

New Soliton and Periodic Wave Solutions to the Fractional DGH Equation Describing Water Waves in a Shallow Regime

Dispersive solitary wave structures with MI Analysis to the unidirectional DGH equation via the unified method

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.

Invariant Subspaces and Exact Solutions to the Generalized Strongly Dispersive DGH Equation

In this paper, the invariant subspaces of the generalized strongly dispersive DGH equation are given, and the exact solutions of the strongly dispersive DGH equation are obtained. Firstly, transform nonlinear partial differential Equation (PDE) into ordinary differential Equation (ODE) systems by using the invariant subspace method. Secondly, combining with the dynamical system method, we use the invariant subspaces which have been obtained to construct the exact solutions of the equation. In the end, the figures of the exact solutions are given.

Well-posedness and blow-up solution for the stochastic Dullin-Gottwald-Holm equation

In this paper, to describe the uncertainty from external environment, we propose a stochastic Dullin-Gottwald-Holm (SDGH) equation by adding an additive noise to the well-known DGH equation. Using regularization, the local existence and uniqueness of the solution of SDGH are proved. By means of a priori estimates, we find that the local solution may blow up in a finite time if the initial value satisfies certain conditions.

Symmetry analysis, conserved quantities and applications to a dissipative DGH equation

In this paper, we study a weakly dissipative Dullin–Gottwald–Holm equation from the viewpoint of Lie symmetry analysis. We first perform symmetry analysis and the nonlinear self-adjointness of this equation. Due to a mixed derivatives term in the equation, we need to rewrite the corresponding form Lagrangian in symmetric form to construct conservation laws. From the viewpoint, we present a general procedure of how these conserved quantities come about. Based on these conserved quantities, blow-up analysis and global existence of strong solutions are presented. Finally, we show that this equation admits a weak peakon-type solution.

Blow-up Phenomena and Persistence Properties of Solutions to the Two-Component DGH Equation

This paper is concerned with blow-up phenomena and persistence properties for an integrable two-component Dullin-Gottwald-Holm shallow water system. We give sufficient conditions on the initial data which guarantee blow-up phenomena of solutions in finite time for both periodic and nonperiodic cases, respectively. Furthermore, the persistence properties of solutions to the system are investigated.

A New Blow‐Up Criterion for the DGH Equation

We investigate the DGH equation. Analogous to the Camassa‐Holm equation, this equation possesses the blow‐up phenomenon. We establish a new blow up criterion on the initial data to guarantee the formulation of singularities in finite time.

Stumpons and fractal-like wave solutions to the Dullin–Gottwald–Holm equation

The traveling wave solutions to the Dullin–Gottwald–Holm equation (called DGH equation) are classified by an improved qualitative analysis method. Meanwhile, the influence of the parameters on the traveling wave forms is specifically considered. The equation is shown to admit more traveling wave forms solutions, especially new solutions such as stumpons and fractal-like waves are first given. We also point out that the smooth solutions can converge to non-smooth ones under certain conditions. Furthermore, the new explicit forms of peakons with period are obtained.

Application of Exp-function method to Dullin–Gottwald–Holm equation

In this paper, we adopt the Exp-function method and the traveling-wave transformation to study the so-called DGH equation, as a result a number of exact solutions of this equation have been found. The family of solution including some exact solutions such as solitary wave pattern, periodic traveling-wave solution, kink-wave solution and new bounded-wave solutions. And explained some of the solutions physical meaning.

New exotic solitary waves in one type of nonlinear dispersive equations

New exotic solitary wave and the painlevé integrability of one type of the nonlinear dispersive generalized DGH equation are studied. By Painlevé analysis, we discover that the nonlinear dispersive generalized DGH equation with m＝2 is integrable, which is a new integrable equation. By the new variable transformation and the auto-Backlund transformation，we obtain abundant exotic solitary wave solutions, such as compactons, peakons, new double solitary waves with peak points, and double solitary waves with blow-up points.

Blow up and propagation speed of solutions to the DGH equation

A wave-breaking mechanism for solutionswith certain initial profiles and propagation speed are discussed inthis paper. Firstly, we apply the best constant to give sufficientcondition via an appropriate integral form of the initial data,which guarantees finite time singularity formation for thecorresponding solution, then we establish blow up criteria via theconserved quantities. Finally, persistence properties of the strongsolutions are presented and infinite propagation speed is alsoinvestigated in the sense that the corresponding solution $u(x,t)$does not have compact spatial support for $t>0$ though $u_0 \inC_0^{\infty}(\mathbb{R})$.

Optimal control of the viscous Dullin–Gottwalld–Holm equation

This paper studies the problem for optimal control of the viscous DGH equation. The existence and uniqueness of weak solution to the equation are proved in a short interval. The optimal control of the viscous DGH equation under boundary condition is given and the existence of optimal solution to the equation is proved.

Blow-up of solutions to the DGH equation

In this paper, firstly we find best constants for two convolution problems on the unit circle via a variational method. Then we apply the best constants on a nonlinear integrable shallow water equation (the DGH equation) to give sufficient conditions on the initial data, which guarantee finite time singularity formation for the corresponding solutions. Finally, we discuss the blow-up phenomena for the nonperiodic case.

On the Well-Posedness Problem and the Scattering Problem for the Dullin-Gottwald-Holm Equation

In this paper, we study the well-posedness of the Cauchy problem and the scattering problem for a new nonlinear dispersive shallow water wave equation (the so-called DGH equation) which was derived by Dullin, Gottwald and Holm. The issue of passing to the limit as the dispersive parameter tends to zero for the solution of the DGH equation is investigated, and the convergence of solutions to the DGH equation as α2→0 is studied, and the scattering data of the scattering problem for the equation can be explicitly expressed; the new exact peaked solitary wave solutions are obtained in the DGH equation. After giving the condition of existing peakon in the DGH equation, it turns out to be nonlinearly stable for the peakon in the DGH equation.