This work studies a generalized Camassa–Holm equation with higher order nonlinearities (g-kbCH). The Camassa–Holm, the Degasperis–Procesi and the Novikov equations are integrable members of this ...family of equations. g-kbCH is well-posed in Sobolev spaces Hs, s>3/2, on both the line and the circle and its solution map is continuous but not uniformly continuous. In this work it is shown that the solution map is Hölder continuous in Hs equipped with the Hr-topology for 0⩽r<s, and the Hölder exponent is expressed in terms of s and r.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
This article presents a new approach for solving the fuzzy fractional Degasperis–Procesi (FFDP) and Camassa–Holm equations using the iterative transform method (ITM). The fractional ...Degasperis–Procesi (DP) and Camassa–Holm equations are extended from the classical DP and Camassa–Holm equations by incorporating fuzzy sets and fractional derivatives. The ITM is a powerful technique widely used for solving nonlinear differential equations. This approach transforms the fuzzy fractional differential equations into a series of ordinary differential equations, which are then solved iteratively using a recursive algorithm. Numerical simulations demonstrate the proposed approach’s accuracy and effectiveness. The results show that the ITM provides an efficient and accurate method for solving the FFDP and Camassa–Holm equations. The proposed method can be extended to solve other fuzzy fractional differential equations.
The deterministic Degasperis-Procesi equation admits weak multi-shockpeakon solutions of the form
<disp-formula> <tex-math id="FE1"> \begin{document}$ u(x, t) = \sum\limits_{i = ...1}^nm_i(t)e^{-|x-x_i(t)|}-\sum\limits_{i = 1}^ns_i(t){\rm sgn}(x-x_i(t))e^{-|x-x_i(t)|}, $\end{document} </tex-math></disp-formula>
where $ {\rm sgn}(x) $ denotes the signum function with $ {\rm sgn}(0) = 0 $, if and only if the time-dependent parameters $ x_i(t) $ (positions), $ m_i(t) $ (momenta) and $ s_i(t) $ (shock strengths) satisfy a system of $ 3n $ ordinary differential equations. We prove that a stochastic perturbation of the Degasperis-Procesi equation also has weak multi-shockpeakon solutions if and only if the positions, momenta and shock strengths obey a system of $ 3n $ stochastic differential equations.
•General Degasperis-Procesi model.
We consider the general Degasperis-Procesi model of shallow water out-flows. This six parametric family of conservation laws contains, in particular, KdV, ...Benjamin-Bona-Mahony, Camassa-Holm, and Degasperis-Procesi equations. The main result consists of criterions which guarantee the existence of solitary wave solutions: solitons and peakons (“peaked solitons”).
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
In the present study, we consider the q-homotopy analysis transform method to find the solution for modified Camassa-Holm and Degasperis-Procesi equations using the Caputo fractional operator. Both ...the considered equations are nonlinear and exemplify shallow water behaviour. We present the solution procedure for the fractional operator and the projected solution procedure gives a rapidly convergent series solution. The solution behaviour is demonstrated as compared with the exact solution and the response is plotted in 2D plots for a diverse fractional-order achieved by the Caputo derivative to show the importance of incorporating the generalised concept. The accuracy of the considered method is illustrated with available results in the numerical simulation. The convergence providence of the achieved solution is established in -curves for a distinct arbitrary order. Moreover, some simulations and the important nature of the considered model, with the help of obtained results, shows the efficiency of the considered fractional operator and algorithm, while examining the nonlinear equations describing real-world problems.
Focusing on the local geometric properties of the shockpeakon for the Degasperis–Procesi equation, a multi-symplectic method for the quasi-Degasperis–Procesi equation is proposed to reveal the jump ...discontinuity of the shockpeakon for the Degasperis–Procesi equation numerically in this paper. The main contribution of this paper lies in the following: (1) the uniform multi-symplectic structure of the b-family equation is constructed; (2) the stable jump discontinuity of the shockpeakon for the Degasperis–Procesi equation is reproduced by simulating the peakon–antipeakon collision process of the quasi-Degasperis–Procesi equation. First, the multi-symplectic structure and several local conservation laws are presented for the b-family equation with two exceptions (b=3 and b=4). And then, the Preissman Box multi-symplectic scheme for the multi-symplectic structure is constructed and the mathematical proofs for the discrete local conservation laws of the multi-symplectic structure are given. Finally, the numerical experiments on the peakon–antipeakon collision of the quasi-Degasperis–Procesi equation are reported to investigate the jump discontinuity of shockpeakon of the Degasperis–Procesi equation. From the numerical results, it can be concluded that the peakon–antipeakon collision of the quasi-Degasperis–Procesi equation can be simulated well by the multi-symplectic method and the simulation results can reveal the jump discontinuity of shockpeakon of the Degasperis–Procesi equation approximately.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK
For s<3/2, it is known (see 24) that the Cauchy problem for the b-family of equations is ill-posed in Sobolev spaces Hs when b>1. The proof of ill-posedness depends naturally on the value of b, and ...is based on the construction of peakon–antipeakon solutions with interesting properties which allows to make conclusion on ill-posedness. In this context the construction of such type of the solution for b<1 is very attractive problem which help shed light on the ill-posedness problem in this case. Thus, in the present paper we consider the ill-posedness of the b-family of equations with additional term for insufficiently investigated case b<1 on the line.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Kako misliti postutopično utopijo oziroma misliti utopijo v njeni očitni nenavzočnosti? Nekoč je utopija šla pred spoznavanjem, celo usmerjala ga je, bila kot nekakšen kažipot, danes je drugače, ...kajti spoznanje gre pred utopijo, pogosto celo brez nje. In tu nekaj manjka, krepko manjka. Preboja naprej brez nečesa utopičnega ni in ga tudi ne more biti. Tudi doba, v kateri živimo, se deklarira za postutopično in soočeni smo z retrotopijo, z vzvratnimi, tj. povratnimi procesi svetovnih razsežnosti.
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