A Nonlinearly Dispersive Fifth Order Integrable Equation and its Hierarchy

Das, Ashok; Popowicz, Ziemowit

doi:10.2991/jnmp.2005.12.1.9

Cited by 8 publications

(11 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This part of the work is grouped into two subsections. The first subsection deals with the construction of the analytical hybrid solitary wave solutions of Equation (1). The second sub-section, for its part, is working on an intense numerical simulation in order to reassure itself of the stability of the obtained solutions with a view to a probable future application and for a possible confirmation of the hybrid characters (planned when choosing of the ansatz given by Equation ( 18) below) of these obtained solutions.…”

Section: Resultsmentioning

confidence: 99%

Higher Order Solitary Wave Solutions of the Standard KdV Equations

Tchaho¹,

Omanda²,

Mbourou³

et al. 2021

OJAppS

View full text Add to dashboard Cite

Considered under their standard form, the fifth-order KdV equations are a sort of reading table on which new prototypes of higher order solitary waves residing there, have been uncovered and revealed to broad daylight. The mathematical tool that made it possible to explore and analyze this equation is the Bogning-Djeumen Tchaho-Kofané method extended to the new implicit Bogning' functions. The analytical form of the solutions chosen in this manuscript is particular in the sense that it contains within its bosom, a package of solitary waves made up of three solitons, especially, the bright type soliton, the hybrid soliton and the dark type soliton which we estimate capable in their interactions of generating new hybrid or multi-form solitons. Existence conditions of the obtained solitons have been determined. It emerges that, these existence conditions of the chosen ansatz could open the way to other new varieties of fifth-order KdV equations including to which it will be one of the solutions. Some of the obtained solitons are exact solutions. Intense numerical simulations highlighted numerical stability and confirmed the hybrid character of the obtained solutions. These results will help to model new nonlinear wave phenomena, in plasma media and in fluid dynamics, especially, on the shallow water surface.

show abstract

Section: Resultsmentioning

confidence: 99%

Higher Order Solitary Wave Solutions of the Standard KdV Equations

Tchaho¹,

Omanda²,

Mbourou³

et al. 2021

OJAppS

View full text Add to dashboard Cite

show abstract

“…in the case when , the equation (2A) is reduced to (3) with the change of variable (4) the equation (3) takes the form (5) The solutions (0.11) and (2.A) were derived using the following Maple code restart:with(inttrans): eq:=diff(P(x,t),t)=eta*diff(P(x,t),x,x,x); eq1:=P(x,0)=Dirac(x-X); eq2:=fourier(eq,x,k); fourier(P(x,t),x,k)=1/sqrt(2*Pi)*Int(P(x,t)*exp(-I*k*x),x=-infinity..infinity); eq3:=fourier(eq1,x,k); fourier(P(x,t),x,k)=P(t); eq4:=subs(fourier(P(x,t),x,k)=P(t),eq2); P(0)=rhs(eq3); eq5:=dsolve({eq4,P(0)=rhs(eq3)}); P(x,t)=1/sqrt(2*Pi)*Int(rhs(eq5)*exp(I*k*x),k=-infinity..infinity); eq6:=P(x,t)=invfourier(rhs(eq5),k,x) assuming X>0 and eta>0 and t>0; eq7:=P(x,t)=simplify(convert(rhs(eq6),AiryAi),power,symbolic); eq8:=P(x,y,t)=subs(eta=eta [1],rhs(eq7))*subs(x=y,X=Y,eta=eta [2],rhs(eq7));…”

Section: Methodsmentioning

confidence: 99%

“…The AiryKaup-Kupershmidt filter was designed applying the Kaup-Kupershmidt operator [2] on the solution of the Airy diffusion equation [3,4]. In [1] was showed that the Airy-Kaup-Kupershmidt filter is a powerful edge detector [5,6] and is also a powerful enhancement tool in image processing.…”

Section: Introductionmentioning

confidence: 99%

“…x,y,sigma)=Int(Int(BA(x-eta,y-xi)*Img(eta,xi),eta=-infinity..infinity),xi=-infinity..infinity); > KK:=(i,j,alpha)->evalf(subs({x=i,X=6,y=j,Y=6,sigma=alpha},rhs(eq7BC))); > > KKA:=(alpha)->Matrix((i,j)->evalf(KK(i,j,alpha)),11); > > geomKK:=(alpha)->Convolution(geom,KKA(alpha),weight=1); > edgeKK :=(alpha)-> Array (abs (geomKK(alpha)), datatype=float[8]);min_v:= (alpha)-> rtable_scanblock (geomKK(alpha), [rtable_dims (geomKK(alpha))], 'Minimum'); max_v := (alpha)-> rtable_scanblock (geomKK(alpha), [rtable_dims (geomKK(alpha))],'Maximum'); delta_v :=(alpha)-> max_v(alpha) -min_v(alpha); img_edgeKK := (alpha)-> Array ((edgeKK(alpha)-min_v(alpha))/delta_v(alpha), order=C_order, datatype=float[([Scale(geom,0.5),Scale(lasg,0.5),Scale(loc,0.5)]); > loc[2,4];…”

mentioning

confidence: 99%

See 1 more Smart Citation

Vessel extraction using the Buckmaster-Airy filter

Sánchez

2016

SPIE Proceedings

View full text Add to dashboard Cite

A new and powerful technique for vessel extraction from biomedical images using the so called BuckmasterAiry Filter is designed, prototyped and tested. The design, the prototyping and the testing were performed using computer algebra software, specifically the Maple package ImageTools. Some preliminary experiments were performed ant the results were excellent. Our new technique is based on partial differential equations.. Specifically two dimensional Airy diffusion equation and the two dimensional Buckmaster equation were used for designing the new Buckmaster-Airy Filter. Such new filter is able to enhance the quality of an image, producing simultaneously noise elimination, but without altering the edges of the image. The new Bukmaster-Airy filter is applied to the target image via discrete convolution. The results of some experiments of vessel extraction will be presented; and some lines for future research such as the possible implementation of the Buckmaster-Airy Filter as a new plugging for the program ImageJ, will be proposed.

show abstract

“…Tam and Hu 16 found three Soliton solutions of KK-type equations using Hirota method and Mathematica. Das and Popowicz 17 studied the nonlinear dispersive fifth-order integrable equation and its hierarchy. Inc 18 determine the Soliton solution of the KK equation numerically and convergence analysis of the decomposition method.…”

Section: Introductionmentioning

confidence: 99%