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Investigation of solitons and mixed lump wave solutions with (3+1)-dimensional potential-YTSF equation

https://doi.org/10.1016/j.cnsns.2020.105544Get rights and content

Highlights

  • We study new exact soliton solutions, lump wave and mixed lump wave solutions to the YTSF equation.

  • Different types of interaction solutions are investigated using the extended three soliton test approach.

  • The graphical behaviors have also been depicted with different values of parameters.

Abstract

The article investigates the exact and mixed lump wave solitons to the (3+1)-dimensional potential YTSF equation, which is an extension of the Bogoyavlenskii-Schif equation. Different types of interaction solutions in terms of a new merge of positive quadratic functions, trigonometric functions and hyperbolic functions are obtained, which are investigated using the extended three soliton test approach. The dynamical behavior of solutions has also depicted in different 3D representations. The representations show that the soliton solutions are obtained in the form of train and cusp. Some other solutions like lamb wave with high peak, and lump solutions with kink background are also observed in these representations.

Introduction

The study of exact wave solutions has become an important and interesting research area in the field of nonlinear sciences. The nonlinear models are studied in different fields, such as nonlinear optics, fluid dynamics and biological nerve propagation. That’s why the exact solutions are constructed using different transformations and techniques [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12].

In recent times, the researchers are more interested in the study of exact solitons and lump wave solutions, see also [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]. The lump waves are localized in all directions in space, appear from nowhere and disappear without a trace [29], [30], [31], [32]. We employ the extended three soliton test function to obtained the exact and lump wave solutions of (3+1)-dimensional potential YTSF equation [33], [34] and is read as,4ux3z+3ux2uxz42uxt+32ux2uz+32uy2=0.Setting u=34wx, putting it into Eq. (1) and integrate with respect to x yields.4wx3z+32wx22wxz42wxt+32wy2+c=0.By considering the integration’s constant c=0 and then the Hirota Bilinear form, of this equation takes the form(Dx3Dz4DxDt+3Dy2)F.F=0,which is connected with the following bilinear operator [35], [36].DxmDtn(a.b)=(xx)m(tt)n×a(x,t).b(x,t)|x=x,t=t,and with the transformation w=2lnF which is equal tou=34(w)x=32(lnF)x,the Eq. (3) becomes2FxxxzF2FxxxFz6FxxzFx+6FxxFxz4FxtF+4FxFt+3FyyF3Fy2=0.The topic is of current interest and the study will enrich the existing literature on related topics. In recent times, a (2+1) dimensional YTSF equation is also considered in [37]. There are various recent studies on a new kind of lump solutions [38] and their interaction solutions [39], even for linear PDEs [40] and abundant studies on solitons for nonlocal equation [41] and soliton solutions for three-component models [42], [43]. Therefore, in this article, the lump and exact solutions are extracted.

In the following section, the exact solutions are constructed.

Section snippets

Exact solutions

A test function of the following type is used to study the interaction of solitons. For details see also [32]. Thus, we haveF=a1cosζ1+a2coshζ2+a3exp(ζ3)+exp(ζ3),where ζi=kix+hiy+liz+wit, and  i=1,2,3. The real parameters for the amplitude of the ith soliton are ki, hi, li, where as ai(i=1,2,3) are real parameters and wi represents the wave speed. Using Eq. (6) into Eq. (5) and balancing power of sin ζ1exp ( ± ζ3), cos ζ1exp ( ± ζ3), sinh ζ2exp ( ± ζ3),

cosh ζ2exp ( ± ζ3), exp ( ± ζ3), sin ζ1

Lump wave solutions

We chose the special function F in the formF=h0+ζ12+ζ22+ζ32,where ζi=kix+hiy+liz+wit, and,  i=1,2,3. Also the constants ki, hi, li and wi are to be calculated after putting the Eq. (7) into Eq. (5)and comparing the coefficients of ζ12,ζ22,ζ32,ζ1ζ2,ζ1ζ3,ζ2ζ3 and ζ10ζ20ζ30 equating to zero, we obtained the particular solutions about h0, h1, h3, w1, w2 and w3 as follows:h0=4(k1l1+k2l2+k3l3)(k12+k22+k32)(6k12h26k22+k1k2h2)+h2+h22+(6k32h26k22+k2k3h2)+(6k12h26k22+k1k2h2)2+(6k32h26k22+k2k3h2)2,h1=6k

Graphical description

The dynamical behaviour of the solution u2(x, y, z, t) has depicted in Fig. 1. The Fig. 1(a) and (b) show the 3D graphical representation and contourplot of train of the mixed lump wave with kink background. Also, the dynamical feather wave (as velocity and amplitude) has retained the same value along xaxis. Whereas the values of yaxis and zaxis are chosen z=y=2. These representations are considered for the values of a1=1.2, a3=1.5, k1=0.8, k3=0.3, h1=0.3, and h3=2.

The dynamical behaviour of

Conclusion

The article studies new exact soliton solutions, lump wave and mixed lump wave solutions to the YTSF equation. The dynamics of these solutions are discussed in different cases. Furthermore, the graphical behaviors have also been depicted with different values of parameters. The dynamics show that these solutions are in the form of mixed lump wave with kink back ground, interaction cusp of soliton and anti-soliton, sharp wave with exceptional amplitude.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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