Two (2+1)-dimensional hierarchies of evolution equations and their Hamiltonian structures

Advances in Mathematical Physics

2021

Self Cite

It is well known that the celebrated Kadomtsev-Petviashvili (KP) equation has many important applications. The aim of this article is to use fractional KP equation to not only simulate shallow ocean waves but also construct novel spatial structures. Firstly, the definitions of the conformable fractional partial derivatives and integrals together with a physical interpretation are introduced and then a fractional integrable KP equation consisting of fractional KPI and KPII equations is derived. Secondly, a formula for the fractional n -soliton solutions of the derived fractional KP equation is obtained and fractional line one-solitons with bend, wavelet peaks, and peakon are constructed. Thirdly, fractional X-, Y- and 3-in-2-out-type interactions in the fractional line two- and three-soliton solutions of the fractional KPII equation are simulated for shallow ocean waves. Besides, a falling and spreading process of a columnar structure in the fractional line two-soliton solution is also simulated. Finally, a fractional rational solution of the fractional KP equation is obtained including the lump solution as a special case. With the development of time, the nonlinear dynamic evolution of the fractional lump solution of the fractional KPI equation can change from ring and conical structures to lump structure.

Section: Discussionmentioning

confidence: 99%

Line Soliton Interactions for Shallow Ocean Waves and Novel Solutions with Peakon, Ring, Conical, Columnar, and Lump Structures Based on Fractional KP Equation

Advances in Mathematical Physics

2021

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“…At the same time, extending HBM to the tfnisAKNS hierarchy ( 3) is worth exploring. Integrable couplings [28][29][30][31][32] include the original integrable systems as sub-systems. Extending HBM and IST to the integrable couplings and their fractional order generalizations is worthy of study.…”

Section: Discussionmentioning

confidence: 99%

Fractional isospectral and non-isospectral AKNS hierarchies and their analytic methods for N-fractal solutions with Mittag-Leffler functions

2021

Adv Differ Equ

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Ablowitz–Kaup–Newell–Segur (AKNS) linear spectral problem gives birth to many important nonlinear mathematical physics equations including nonlocal ones. This paper derives two fractional order AKNS hierarchies which have not been reported in the literature by equipping the AKNS spectral problem and its adjoint equations with local fractional order partial derivative for the first time. One is the space-time fractional order isospectral AKNS (stfisAKNS) hierarchy, three reductions of which generate the fractional order local and nonlocal nonlinear Schrödinger (flnNLS) and modified Kortweg–de Vries (fmKdV) hierarchies as well as reverse-t NLS (frtNLS) hierarchy, and the other is the time-fractional order non-isospectral AKNS (tfnisAKNS) hierarchy. By transforming the stfisAKNS hierarchy into two fractional bilinear forms and reconstructing the potentials from fractional scattering data corresponding to the tfnisAKNS hierarchy, three pairs of uniform formulas of novel N-fractal solutions with Mittag-Leffler functions are obtained through the Hirota bilinear method (HBM) and the inverse scattering transform (IST). Restricted to the Cantor set, some obtained continuous everywhere but nondifferentiable one- and two-fractal solutions are shown by figures directly. More meaningfully, the problems worth exploring of constructing N-fractal solutions of soliton equation hierarchies by HBM and IST are solved, taking stfisAKNS and tfnisAKNS hierarchies as examples, from the point of view of local fractional order derivatives. Furthermore, this paper shows that HBM and IST can be used to construct some N-fractal solutions of other soliton equation hierarchies.

“…Tu Guizhang put forward a scheme for generating (2 + 1)-dimensional hierarchies by using a residue operator with an associative algebra [ ] which includes all pseudodifferential operators ∑ =−∞ , where the operator is defined by = + ( ), ∈ . Recently, Zhang gave some (2 + 1)-dimensional integrable hierarchies [15,16], but it is difficult to solve these equations by using the usual ways which are mentioned in [15,16]. Meanwhile, the applications of these equations were not mentioned.…”

Section: Introductionmentioning

confidence: 99%

(2+1)-Dimensional mKdV Hierarchy and Chirp Effect of Rossby Solitary Waves

Wang

Advances in Mathematical Physics

Yin

et al. 2015

By constructing a kind of generalized Lie algebra, based on generalized Tu scheme, a new (2+1)-dimensional mKdV hierarchy is derived which popularizes the results of (1+1)-dimensional integrable system. Furthermore, the (2+1)-dimensional mKdV equation can be applied to describe the propagation of the Rossby solitary waves in the plane of ocean and atmosphere, which is different from the (1+1)-dimensional mKdV equation. By virtue of Riccati equation, some solutions of (2+1)-dimensional mKdV equation are obtained. With the help of solitary wave solutions, similar to the fiber soliton communication, the chirp effect of Rossby solitary waves is discussed and some conclusions are given.