Lump and new interaction solutions to the (3+1)-dimensional nonlinear evolution equation
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Asma Issasfa,1,∗, Ji Lin,2,∗1College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua 321004, China 2Department of Physics, Zhejiang Normal University, Jinhua 321004, China
First author contact: Authors to whom any correspondence should be addressed. Received:2020-06-29Revised:2020-08-17Accepted:2020-08-27Online:2020-11-12
Abstract In this paper, a new (3+1)-dimensional nonlinear evolution equation is introduced, through the generalized bilinear operators based on prime number p=3. By Maple symbolic calculation, one-, two-lump, and breather-type periodic soliton solutions are obtained, where the condition of positiveness and analyticity of the lump solution are considered. The interaction solutions between the lump and multi-kink soliton, and the interaction between the lump and breather-type periodic soliton are derived, by combining multi-exponential function or trigonometric sine and cosine functions with a quadratic one. In addition, new interaction solutions between a lump, periodic-solitary waves, and one-, two- or even three-kink solitons are constructed by using the ansatz technique. Finally, the characteristics of these various solutions are exhibited and illustrated graphically. Keywords：generalized (3+1)-dimensional nonlinear evolution equation;lump solution;breather-type periodic soliton;interaction solution;generalized bilinear form
PDF (896KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Asma Issasfa, Ji Lin. Lump and new interaction solutions to the (3+1)-dimensional nonlinear evolution equation. Communications in Theoretical Physics, 2020, 72(12): 125003- doi:10.1088/1572-9494/abb7d3
1. Introduction
Recently, finding the exact solution of nonlinear partial differential equations (NPDEs) has become an interesting topic of intensive investigations in the field of sciences, especially in mathematics and physics. The exact solutions, such as rational and solitary wave solutions, play an essential role in studying the natural phenomena that appeared in fluid engineering and nonlinear optics [1-6], which helps us to understand their physical mechanisms.
Lump solution is a type of rational function, analytic and localized in all space directions; it appears in many physical phenomena such as optic media [7], oceanographic engineering [8], plasma physics [9], etc. It has been studied for many (N+1)-dimensional NPDEs, including the Kadomtsev-Petviashvili (KP) equation [10], the Boussinesq equation [11], the (2 + 1)-dimensional bidirectional Sawada-Kotera (bSK) equation [12], the (3 + 1)-dimensional Shallow Water Wave equation [13], and the B-Kadomtsev-Petviashviii (BKP) equation [14], which are constructed via the Bäcklund transformation and the Hirota bilinear method [15].
Recently, for a better understanding of the complex physical phenomena, many authors have applied the direct Hirota bilinear method to derive the interaction solution among lump-kink [16-19] and lump-soliton solutions [20-24] using the auxiliary function as a special ansatz by collecting a quadratic function with the exponential function. Later, a class of mixture solutions consisting of lump, multi-soliton and periodic wave solutions [24, 25] were obtained by extending the ansatz to the combination of quadratic, multi-exponential and periodic functions. Therefore, from the bilinear form of the NPDEs, the direct ansatz technique becomes a more and more effective method to construct diverse kinds of interaction solutions. Besides, this method is still valid for the NPDEs constructed via the generalized bilinear derivatives [26, 27], which involve different prime numbers, p=3, p=5, p=7... It is interesting to obtain lumps and more interaction solutions to these kinds of NPDEs.
The (3+1)-dimensional NPDE reads$\begin{eqnarray}3{u}_{{xz}}-{(2{u}_{t}+{u}_{{xxx}}-\eta {{uu}}_{x}-\kappa {{uu}}_{y})}_{y}+\lambda {({u}_{x}{\partial }_{x}^{-1}{u}_{y})}_{x}=0,\end{eqnarray}$where u=u(x, y, z, t), while η, κ and λ are arbitrary constants, and $\eta +\kappa +\lambda \ne 0$. The inverse operator ${\partial }_{x}^{-1}$ is introduced as$\begin{eqnarray}({{\rm{\partial }}}_{x}^{-1}g)(x)={\int }_{-\infty }^{x}g(t){\rm{d}}t,\end{eqnarray}$under condition of decomposition at the infinity, with ${\partial }_{x}^{-1}{\partial }_{x}={\partial }_{x}{\partial }_{x}^{-1}=1$. The integrability and multi-soliton solutions of equation (1) were proposed by Wazwaz using the Hirota bilinear method [28]. For η=2, κ=0, and λ=2, equation (1) is reduced to a model equation, which was first introduced in the study of the algebraic geometrical solutions [29]. In 2013, Zhaqilao obtained a rogue wave and rational solution of this equation by using a simple symbolic computation approach [30]. Shi et al applied the KP-hierarchy reduction method to derive the rogue wave solutions [31]. Chen et al obtained a lump solution to the reduced (3+1)-dimensional equation [32]. After that, M-lump solutions to this equation were constructed by taking a ‘long wave limit' of N-soliton solutions [33].
In this article, we construct a new (3+1)-dimensional nonlinear evolution equation from the generalized bilinear derivatives of NPDE type involving a prime number, p=3, and research for new interaction solutions among lump, periodic-solitary waves, and multi-kink soliton solutions using the ansatz technique, which have not been reported in other studies.
By employing the following dependent variable bilinear transformation$\begin{eqnarray}u=\displaystyle \frac{12}{\eta +\kappa +\lambda }{(\mathrm{ln}f)}_{{xx}}.\end{eqnarray}$Equation (1) can be converted to the following bilinear form$\begin{eqnarray}\begin{array}{l}(3{{\rm{D}}}_{x}{{\rm{D}}}_{z}-2{{\rm{D}}}_{y}{{\rm{D}}}_{t}-{{\rm{D}}}_{x}^{3}{{\rm{D}}}_{y})f\cdot f\\ \quad =\,2(3{f}_{{xz}}f-3{f}_{x}{f}_{z}-2{f}_{{yt}}f+2{f}_{y}{f}_{t}-{{ff}}_{3{xy}}\\ \qquad +\,{f}_{y}{f}_{3x}-3{f}_{{xy}}{f}_{2x}+3{f}_{x}{f}_{2{xy}})=0,\end{array}\end{eqnarray}$where ${{\rm{D}}}_{i}\ (i=x,y,z,t)$ represent the Hirota's bilinear differential operator [34], defined by$\begin{eqnarray*}\begin{array}{l}P({{\rm{D}}}_{{\rm{x}}},{{\rm{D}}}_{y},{{\rm{D}}}_{z},{{\rm{D}}}_{t})f(x,y,z,t,\ldots ).g(x,y,z,t...)\\ \quad =\,P({\partial }_{x}-{\partial }_{x^{\prime} },{\partial }_{y}-{\partial }_{y^{\prime} },{\partial }_{z}-{\partial }_{z^{\prime} },{\partial }_{t}-{\partial }_{t^{\prime} },\ldots )\\ \qquad \times \,f(x,y,z,t...)\cdot g(x,y,z,t...){| }_{x^{\prime} =x,y^{\prime} =y,z^{\prime} =z,t^{\prime} =t},\end{array}\end{eqnarray*}$where P is a polynomial of Dx, Dy, Dz, Dt,....
Based on the generalized of the Hirota bilinear operators, a kind of generalized bilinear operators on prime number p is defined as [35]$\begin{eqnarray}\begin{array}{l}{{\rm{D}}}_{p,x}^{\beta }{{\rm{D}}}_{p,y}^{\gamma }{{\rm{D}}}_{p,z}^{\delta }{{\rm{D}}}_{p,t}^{\theta }f\cdot g\\ =\,{\left(\displaystyle \frac{\partial }{\partial x}+{\alpha }_{p}\displaystyle \frac{\partial }{\partial x^{\prime} }\right)}^{\beta }{\left(\displaystyle \frac{\partial }{\partial y}+{\alpha }_{p}\displaystyle \frac{\partial }{\partial y^{\prime} }\right)}^{\gamma }{\left(\displaystyle \frac{\partial }{\partial z}+{\alpha }_{p}\displaystyle \frac{\partial }{\partial z^{\prime} }\right)}^{\delta }\\ {\left(\displaystyle \frac{\partial }{\partial t}+{\alpha }_{p}\displaystyle \frac{\partial }{\partial t^{\prime} }\right)}^{\theta }f(x,y,z,t)\cdot f(x^{\prime} ,y^{\prime} ,z^{\prime} ,t^{\prime} ){| }_{x=x^{\prime} ,y=y^{\prime} ,z=z^{\prime} ,t=t^{\prime} }\\ =\,\displaystyle \sum _{i=0}^{\beta }\displaystyle \sum _{j=0}^{\gamma }\displaystyle \sum _{k=0}^{\delta }\displaystyle \sum _{l=0}^{\theta }\left(\begin{array}{c}\beta \\ i\end{array}\right)\left(\begin{array}{c}\gamma \\ j\end{array}\right)\left(\begin{array}{c}\delta \\ k\end{array}\right)\\ \ \ \times \,\left(\begin{array}{c}\theta \\ l\end{array}\right){\alpha }_{p}^{i}{\alpha }_{p}^{j}{\alpha }_{p}^{k}{\alpha }_{p}^{l}\displaystyle \frac{{\partial }^{\beta -i}}{\partial {x}^{\beta -i}}\displaystyle \frac{{\partial }^{i}}{\partial {x}^{{\prime} i}}\displaystyle \frac{{\partial }^{\gamma -j}}{\partial {y}^{\gamma -j}}\\ \displaystyle \frac{{\partial }^{j}}{\partial {y}^{{\prime} j}}\displaystyle \frac{{\partial }^{\delta -k}}{\partial {z}^{\delta -k}}\displaystyle \frac{{\partial }^{k}}{\partial {z}^{{\prime} k}}\displaystyle \frac{{\partial }^{\theta -l}}{\partial {t}^{\theta -l}}\displaystyle \frac{{\partial }^{l}}{\partial {t}^{{\prime} l}}f(x,y,z,t)\\ \cdot \,f(x^{\prime} ,y^{\prime} ,z^{\prime} ,t^{\prime} ){| }_{x=x^{\prime} ,y=y^{\prime} ,z=z^{\prime} ,t=t^{\prime} },\end{array}\end{eqnarray}$where β, γ, δ, θ≥0, ${\alpha }_{p}^{s}={\left(-1\right)}^{{r}_{p}(s)}$, if $s\equiv {r}_{p}(s)$ mod p.
By taking p=3, we can generalize (4) into$\begin{eqnarray}\begin{array}{l}(3{{\rm{D}}}_{3,x}{{\rm{D}}}_{3,z}-2{{\rm{D}}}_{3,y}{{\rm{D}}}_{3,t}-{{\rm{D}}}_{3,x}^{3}{{\rm{D}}}_{3,y})f\cdot f\\ =\,2(3{f}_{{xz}}f-3{f}_{x}{f}_{z}-2{f}_{{yt}}f+2{f}_{y}{f}_{t}-3{f}_{{xy}}{f}_{2x})=0.\end{array}\end{eqnarray}$According to the Bell polynomial and linear superposition of integral equations [36], under the following transformation$\begin{eqnarray}w=2{(\mathrm{ln}f)}_{x},\end{eqnarray}$the generalized bilinear form (6) can be transformed to the following (3+1)-dimensional nonlinear evolution equation$\begin{eqnarray}\begin{array}{l}3{w}_{z}-2{\partial }_{x}^{-1}{w}_{{yt}}-\displaystyle \frac{3}{4}{w}^{2}{w}_{y}-\displaystyle \frac{3}{8}{w}^{3}{\partial }_{x}^{-1}{w}_{y}\\ \quad -\,\displaystyle \frac{3}{4}{{ww}}_{x}{\partial }_{x}^{-1}{w}_{y}-\displaystyle \frac{3}{2}{w}_{x}{w}_{y}=0.\end{array}\end{eqnarray}$Note that equation (7) is a special transformation by the link between the function f and w, which can transform generalized bilinear form to nonlinear equation. It is a distinctive feature of founding Bell polynomial theories to integral equations.
This article is arranged as follows. In section 2, one-, two-lump, and breather-type periodic soliton solutions of equation (8) are obtained by taking the function in the generalized bilinear form as a quadratic form, and the combination of exponential with the triangular functions. In section 3, by adding the sum of exponential or trigonometric sine and cosine functions to the quadratic one, the interacted lump waves with multi-kink soliton, the interaction among a lump and breather-type periodic soliton are derived. In section 4, the mixed solutions consisting of lump, multi-kink soliton, and periodic-solitary waves solutions of equation (8) are constructed by using the ansatz technique. The last section contains conclusions and discussions.
2. Lumps and breather-type periodic soliton solutions
2.1. Lump soliton
To obtain the lump-type soliton of equation (8), we take the function in the generalized bilinear form as a quadratic form by assuming f as follows$\begin{eqnarray}f={A}^{2}+{B}^{2}+{a}_{11},\end{eqnarray}$with$\begin{eqnarray}\begin{array}{rcl}A(x,y,z,t) & = & {a}_{1}x+{a}_{2}y+{a}_{3}z+{a}_{4}t+{a}_{5},\\ B(x,y,z,t) & = & {a}_{6}x+{a}_{7}y+{a}_{8}z+{a}_{9}t+{a}_{10},\end{array}\end{eqnarray}$where ${a}_{i}(i=1,\ldots ,11)$ are all undetermined real constants. By substituting equation (9) with (10) into equation (6) and vanishing all the coefficients of different powers of x, y, z, t, we get a system of algebraic equations. After symbolic computation with Maple, we obtain the following set of solutions$\begin{eqnarray}\begin{array}{rcl}{a}_{11} & = & -\displaystyle \frac{({a}_{2}^{2}+{a}_{7}^{2})({a}_{1}^{2}+{a}_{6}^{2})({a}_{1}{a}_{2}+{a}_{6}{a}_{7})}{({a}_{2}{a}_{8}-{a}_{3}{a}_{7})({a}_{1}{a}_{7}-{a}_{2}{a}_{6})},\\ {a}_{4} & = & \displaystyle \frac{3({a}_{1}{a}_{2}{a}_{3}+{a}_{1}{a}_{7}{a}_{8}-{a}_{2}{a}_{6}{a}_{8}+{a}_{3}{a}_{6}{a}_{7})}{2({a}_{2}^{2}+{a}_{7}^{2})},\\ {a}_{9} & = & \displaystyle \frac{3({a}_{1}{a}_{2}{a}_{8}-{a}_{1}{a}_{3}{a}_{7}+{a}_{2}{a}_{3}{a}_{6}+{a}_{6}{a}_{7}{a}_{8})}{2({a}_{2}^{2}+{a}_{7}^{2})}.\end{array}\end{eqnarray}$The other unexpressed parameters in the set are real constants with restricting condition$\begin{eqnarray*}{a}_{2}\left|\begin{array}{cc}{a}_{2} & -{a}_{7}\\ {a}_{7} & {a}_{2}\end{array}\right|\ne 0,\end{eqnarray*}$which makes the solutions well defined, and with the conditions $({a}_{1}{a}_{2}+{a}_{6}{a}_{7})\lt 0$ and $({a}_{2}{a}_{8}-{a}_{3}{a}_{7})({a}_{1}{a}_{7}-{a}_{2}{a}_{6})\ne 0$ indicates that function f is real, positive and guarantees analyticity and rational localization of solutions for equation (8), and the rational function solution $w\to 0$ when ${A}^{2}+{B}^{2}\to \infty $ at any (z, t)∈R2. We take the following choices for the parameters$\begin{eqnarray}\begin{array}{rcl}{a}_{1} & = & 1,\ \ \ {a}_{2}=9,\ \ \ {a}_{3}=2,\ \ \ {a}_{5}=1,\\ {a}_{6} & = & 5,\ \ \ {a}_{7}=2,\ \ \ {a}_{8}=3,\ \ \ {a}_{10}=3.\end{array}\end{eqnarray}$The lump-type soliton of equation (8)$\begin{eqnarray}w=\displaystyle \frac{L}{M},\end{eqnarray}$where L = 7405632t + 17485520x + 12777880y + 11432840z + 10760320, M = 1504269t2 + (3702816x − 3162822y + 1465698z + 2005692)t + 4371380x2 + (6388940y + 5716420z + 5380160)x + 14291050y2 + (8070240z + 5043900)y + 2185690z2 + 3698860z + 8819600.
When $t=0,\,z=5$, the graphical behavior of solution w exhibits a pattern with one maximal point and one minimal point, as shown in figures 1(a) and (c).
Figure 1.
New window|Download| PPT slide Figure 1.Perspective view of one-, two-lump soliton solutions w for equation (8), with the parameter (12), (17) respectively, at time t=−1 (a) 1-lump solution, (b) 2-lump solution, in the (x, y)-plane, with z=5, (c) 1-lump solution, (d) 2-lump solution, in the (x, z)-plane, with y=−5.
In order to construct the 2-lump soliton solutions of equation (8), we extend the formula of the function f as follows$\begin{eqnarray}f={C}^{4}+{A}^{2}+{B}^{2}+{b}_{1}{AB}+{a}_{11},\end{eqnarray}$with$\begin{eqnarray}C(x,y,z,t)={k}_{1}x+{k}_{2}y+{k}_{3}z+{k}_{4}t+{k}_{5},\end{eqnarray}$where ki(i=1, …, 5), and b1 are all undetermined real constants. Through the substitution of equation (14) with (15) and (10) into equation (6) and vanishing all the coefficients of different powers of x, y, z, t, we obtain more algebraic equations in the parameters ki(i=1, …, 5) and aj (j=1, …, 11). After accurate calculations, we can get$\begin{eqnarray}\begin{array}{rcl}{a}_{2} & = & \displaystyle \frac{3}{2}\displaystyle \frac{{a}_{3}{k}_{1}}{{k}_{4}},\ \ \ {a}_{4}=\displaystyle \frac{{a}_{1}{k}_{4}}{{k}_{1}},\ \ \ {a}_{7}=-\displaystyle \frac{3}{2}\displaystyle \frac{{a}_{3}{k}_{1}({a}_{6}{b}_{1}+2{a}_{1})}{{k}_{4}({a}_{1}{b}_{1}+2{a}_{6})},\\ {a}_{8} & = & -\displaystyle \frac{{a}_{3}({a}_{6}{b}_{1}+2{a}_{1})}{{a}_{1}{b}_{1}+2{a}_{6}},\\ {a}_{9} & = & \displaystyle \frac{{a}_{6}{k}_{4}}{{k}_{1}},\ \ \ {k}_{2}=0,\ \ \ {k}_{3}=0.\end{array}\end{eqnarray}$The remaining parameters are arbitrary constants with restricting condition ${k}_{1}{k}_{4}\ne 0,$ and ${a}_{1}{b}_{1}+2{a}_{6}\ne 0$, which guarantees the positiveness and well defined nature of the function f. We take the following choices for the parameters:$\begin{eqnarray}\begin{array}{rcl}{a}_{1} & = & -\displaystyle \frac{3}{2},\ \ \ {a}_{3}=-\displaystyle \frac{1}{5},\ \ \ {a}_{5}=5,\ \ \ {a}_{6}=1\ \ \ {a}_{10}=14,\\ {k}_{1} & = & \displaystyle \frac{1}{2},\ \ \ {k}_{4}=-\displaystyle \frac{1}{5},\\ {k}_{5} & = & -5,\ \ \ {k}_{10}=2,\ \ \ {b}_{1}=1,\ \ \ {a}_{11}=5.\end{array}\end{eqnarray}$The 2-lump soliton of equation (8)$\begin{eqnarray}w=\displaystyle \frac{{L}_{1}}{{M}_{1}},\end{eqnarray}$where ${L}_{1}=-0.008-2{t}^{3}+(0.06x-0.48){t}^{2}+(-0.15{x}^{2}\,+2.4x-10.3)t$ $+\,0.125{x}^{3}-3{x}^{2}+25.75x-65.5,$ ${M}_{1}\,=0.0004{t}^{4}+(0.032-0.004x){t}^{3}$ $+\,(1.03-0.24x\,+0.015{x}^{2}){t}^{2}+(-0.025{x}^{3}+0.6{x}^{2}-5.15x+13.1)t$ $+\,0.015625{x}^{4}\,-0.5{x}^{3}\,+\,6.4375{x}^{2}\,-\,32.75x\,+\,2.953125{y}^{2}$ $\,+\,(-1.575z\,+29.25)y+0.21{z}^{2}+138-7.8z$.
When $t=-1,\ \ z=5$, the graphical behavior of solution w exhibits one peak and one-lump wave in the plane (x, y) as shown in figures 1(b) and (d) in the plane (x, z).
2.2. Breather-type periodic soliton
Here, we discuss the periodic-kink wave solution for equation (8). Therefore, we use the form of an exponential function with the periodic one as follows$\begin{eqnarray}f=k{{\rm{e}}}^{A}+{{\rm{e}}}^{-A}+{G}_{1}\cos (B)+{G}_{2}\sin (C),\end{eqnarray}$where k, G1 and G2 are real constants to be determined. By the similar calculation of the above procedure, we obtain the following solutions:$\begin{eqnarray}{a}_{9}=\displaystyle \frac{{a}_{4}{a}_{6}}{{a}_{1}},\ \ \ {k}_{3}=\displaystyle \frac{2}{3}\displaystyle \frac{{a}_{4}{k}_{2}}{{a}_{1}},\end{eqnarray}$and a2, a3, a7, a8, k1, k4 are all zero, and a1, a4, a5, a6, a10, k2, k5 are arbitrary constants, which need to satisfy the condition ${a}_{1}\ne 0$. Combining the above values of constants in equation (19) then substituting into the equation (7), we get the following solution:$\begin{eqnarray}w=-\displaystyle \frac{2({G}_{1}\sin ({\theta }_{1}){a}_{6}-{{ka}}_{1}{{\rm{e}}}^{{\theta }_{2}}+{a}_{1}{{\rm{e}}}^{-{\theta }_{2}})}{k{{\rm{e}}}^{{\theta }_{2}}+{{\rm{e}}}^{-{\theta }_{2}}+{G}_{1}\cos ({\theta }_{1})+{G}_{2}\sin ({\theta }_{3})},\end{eqnarray}$where$\begin{eqnarray*}\left\{\begin{array}{l}{\theta }_{1}=\tfrac{{a}_{6}{{xa}}_{1}+{a}_{4}{a}_{6}t+{a}_{10}{a}_{1}}{{a}_{1}},\\ {\theta }_{2}={a}_{1}x+{a}_{4}t+{a}_{5},\\ {\theta }_{3}=\tfrac{3{k}_{2}{{ya}}_{1}+2{a}_{4}{k}_{2}z+3{k}_{5}{a}_{1}}{3{a}_{1}}.\end{array}\right.\end{eqnarray*}$Solution (21) describes a breather-type periodic-kink soliton, which results from the interaction between solitary and periodic waves; it has breathy-periodic features, but meanwhile takes on a kinky feature with space variable (x, y), and (x, z) as shown in figure 2 for fixed time t=1, with three-dimensional and density plots.
Figure 2.
New window|Download| PPT slide Figure 2.Perspective view of breathy-periodic kink soliton solutions w for equation (8), with the parameter ${G}_{1}=4,{G}_{2}=1,{k}_{2}=\tfrac{3}{2}$, ${k}_{4}=3,{a}_{1}=-\tfrac{1}{2},{a}_{4}=2,{k}_{5}=3$, ${a}_{5}=\tfrac{1}{2},k=5,{a}_{6}=1,{a}_{10}=3$, at time t=1, (a) plot of w, (b) Density plot of w, in the (x, y)-plane, with z=2, (c) plot of w, (d) Density plot of w, in the (x, z)-plane, with y=5.
3. Rational-exponential and rational-periodic solutions
3.1. Rational-exponential solutions
With the combination of an exponential function with the quadratic one, we derive the mixed solution consisting of one lump wave and one-kink soliton, we assume f as follows$\begin{eqnarray}f={A}^{2}+{B}^{2}+{a}_{11}+{b}_{1}{{\rm{e}}}^{C}.\end{eqnarray}$By inserting equation (22) with (15) and (10) into equation (6) and collecting all the coefficients of x, y, z, t, and the exponential functions, the parameters are determined$\begin{eqnarray}\begin{array}{rcl}{a}_{1} & = & \displaystyle \frac{{a}_{4}{k}_{1}}{{k}_{4}},\ \ \ {a}_{2}=-\displaystyle \frac{{a}_{7}{a}_{9}}{{a}_{4}},\ \ \ {a}_{3}=-\displaystyle \frac{2}{3}\displaystyle \frac{{a}_{7}{a}_{9}{k}_{4}}{{a}_{4}{k}_{1}},\\ {a}_{6} & = & \displaystyle \frac{{a}_{9}{k}_{1}}{{k}_{4}},\ \ \ {a}_{8}=\displaystyle \frac{2}{3}\displaystyle \frac{{a}_{7}{k}_{4}}{{k}_{1}},\ \ \ {k}_{2}=0,\ \ \ {k}_{3}=0,\end{array}\end{eqnarray}$where a4, a5, a7, a9, a10, a11, b1, k1, and k4 are arbitrary constants, which need to satisfy ${a}_{4}{k}_{1}\ne 0$, and ${k}_{4}\ne 0$, with a11, k>0. to assure the positiveness and well defined nature of the function f. Then the exact interaction solution w is summarized as follows:$\begin{eqnarray}w=\displaystyle \frac{4{a}_{1}A+4{a}_{6}B+2{b}_{1}{k}_{1}{{\rm{e}}}^{C}}{f}.\end{eqnarray}$We select the following parameters as$\begin{eqnarray}\begin{array}{rcl}{k}_{1} & = & 2,\ \ \ {k}_{4}=3,\ \ \ {a}_{9}=\displaystyle \frac{3}{2},\ \ \ {a}_{11}=1,\ \ \ {b}_{1}=1,\\ {a}_{5} & = & \displaystyle \frac{1}{2},\ \ \ {a}_{9}=3,\ \ \ {a}_{7}=\displaystyle \frac{3}{2},\ \ \ {a}_{10}=3,\end{array}\end{eqnarray}$the interaction phenomena consisting of a lump-type solution with a kink-soliton solution is shown for equation (8) in figure 3 with different time evolution.
Figure 3.
New window|Download| PPT slide Figure 3.Perspective view of interaction solution w for equation (8) consisting of one-lump and one-kink soliton with parameters (25), (a) t=−10, (b) t=1, (c) t=13, with z=−5 in the (x, y)-plane.
•Similarly, by adding sum of an exponential function to the previous formula of f, we can obtain the interaction solution with one-lump and two-kink soliton, the interaction with one-lump and three-kink soliton. Therefore, we suppose$\begin{eqnarray}f={A}^{2}+{B}^{2}+{a}_{11}+{b}_{1}{{\rm{e}}}^{C}+{b}_{2}{{\rm{e}}}^{-C},\end{eqnarray}$and$\begin{eqnarray}f={A}^{2}+{B}^{2}+{a}_{11}+{b}_{1}{{\rm{e}}}^{C}+{b}_{2}{{\rm{e}}}^{-C}+{b}_{3}{{\rm{e}}}^{D}+{b}_{4}{{\rm{e}}}^{-D},\end{eqnarray}$with$\begin{eqnarray}D(x,y,z,t)={k}_{6}x+{k}_{7}y+{k}_{8}z+{k}_{9}t+{k}_{10}.\end{eqnarray}$With direct substitution of equations (26) and (28) into equation (6), and gathering the coefficients of the variables $x,y,z,t$ and the exponential functions, we obtain the following parameter:$\begin{eqnarray}\begin{array}{rcl}{a}_{2} & = & -\displaystyle \frac{{a}_{7}{a}_{6}}{{a}_{1}},\ \ \ {a}_{3}=-\displaystyle \frac{2}{3}\displaystyle \frac{{a}_{7}{a}_{6}{a}_{4}}{{a}_{1}^{2}},\ \ \ {a}_{8}=\displaystyle \frac{2}{3}\displaystyle \frac{{a}_{4}{a}_{7}}{{a}_{1}},\\ {a}_{9} & = & \displaystyle \frac{{a}_{6}{a}_{4}}{{a}_{1}},\ \ \ {k}_{2}=0,\ \ \ {k}_{3}=0,\ \ \ {k}_{4}=\displaystyle \frac{{a}_{4}{k}_{1}}{{a}_{1}},\end{array}\end{eqnarray}$where a1, a4, a5, a6, a7, a10, a11, k1, k5, b1, and b2 are all real arbitrary constants, with ${a}_{1}\ne 0,{a}_{11},{b}_{1}\gt 0$, b2>0. The function f is well defined. In this situation, a mixed solution consisting of one-lump and two-kink soliton is expressed as:$\begin{eqnarray}w=\displaystyle \frac{4{a}_{1}A+4{a}_{6}B+2{b}_{1}{k}_{1}{{\rm{e}}}^{C}-2{b}_{2}{k}_{1}{{\rm{e}}}^{-C}}{f}.\end{eqnarray}$To illustrate the interaction phenomena represented by one-lump and two-kink soliton solutions, we take the following choices for the parameters:$\begin{eqnarray}\begin{array}{rcl}{a}_{1} & = & 10,\ \ \ {b}_{1}=4,\ \ \ {b}_{2}=4,\ \ \ {a}_{4}=15,\\ {k}_{1} & = & -1,\ \ \ {a}_{11}=1,\ \ \ {k}_{6}=-2,\\ {k}_{10} & = & 1,\ \ \ {a}_{5}=-9,\ \ \ {k}_{5}=1,\\ {a}_{6} & = & 6,\ \ \ {a}_{7}=-9,\ \ \ {a}_{10}=-6,\end{array}\end{eqnarray}$as shown in figure 4 with different values of space z in the (x, y)-plane.
Figure 4.
New window|Download| PPT slide Figure 4.Perspective view of interaction solution w for equation (8) consisting of one-lump and two-kink soliton with parameters (31), (a) z=−25, (b) z=−5, (c) z=20, with t=2 in the (x, y)-plane.
In the same way by substituting equation (27) with (28) into equation (6) and gathering the coefficients of x, y, z, t and the exponential functions, we obtain the following parameter:$\begin{eqnarray}\begin{array}{rcl}{a}_{2} & = & -\displaystyle \frac{{a}_{6}{a}_{7}}{{a}_{1}},\ \ \ {a}_{3}=-\displaystyle \frac{2}{3}\displaystyle \frac{{a}_{4}{a}_{6}{a}_{7}}{{a}_{1}^{2}},\ \ \ {a}_{8}=\displaystyle \frac{2}{3}\displaystyle \frac{{a}_{7}{a}_{4}}{{a}_{1}},\\ {a}_{9} & = & \displaystyle \frac{{a}_{4}{a}_{6}}{{a}_{1}},\ \ \ {k}_{1}=\displaystyle \frac{{a}_{1}{k}_{4}}{{a}_{4}},\ \ \ {k}_{9}=\displaystyle \frac{{k}_{6}{a}_{4}}{{a}_{1}},\end{array}\end{eqnarray}$where k2, k3, k7, k8=0, and the other parameters not expressed are real arbitrary constants, with the conditions ${a}_{1},{a}_{4}\ne 0$, and a11, bi (i=1, 2, 3)>0, which assure the positiveness and well defined nature of the function f. Then the mixed solution composed of a lump wave and three-kink soliton is expressed as$\begin{eqnarray}w=\displaystyle \frac{4{a}_{1}A+4{a}_{6}B+2{b}_{1}{k}_{1}{{\rm{e}}}^{C}-2{b}_{2}{k}_{1}{{\rm{e}}}^{-C}+2{b}_{3}{k}_{6}{{\rm{e}}}^{D}-2{b}_{4}{k}_{6}{{\rm{e}}}^{-D}}{f},\end{eqnarray}$we take the following choices for the parameters:$\begin{eqnarray}\begin{array}{rcl}{a}_{1} & = & \displaystyle \frac{3}{2},\ \ \ {b}_{1}=4,\ \ \ {b}_{2}=4,\ \ \ {b}_{3}=5,\ \ \ {b}_{4}=5,\\ {a}_{4} & = & -2.6,\ \ \ {k}_{4}=-1.9,\ \ \ {a}_{11}=3,\\ {k}_{6} & = & -1.9,\ \ \ {k}_{10}=2,\ \ \ {a}_{5}=5,\ \ \ {k}_{5}=-17,\\ {a}_{6} & = & 64,\ \ \ {a}_{7}=-35,\ \ \ {a}_{10}=15,\end{array}\end{eqnarray}$the interaction solution consisting of one-lump and three-kink soliton is illustrated in figure 5 with various values of space z=−25, z=−2, z=20 in the (x, y)-plane.
Figure 5.
New window|Download| PPT slide Figure 5.Perspective view of interaction solution w for equation (8) consisting of one-lump and three-kink soliton with parameters (34), (a) z=−25, (b) z=−2, (c) z=20, with t=−1 in the (x, y)-plane.
3.2. Rational-periodic solutions
For interaction solutions among lump waves and triangular periodic waves, we merge the form of a quadratic function with sine and cosine functions$\begin{eqnarray}f={A}^{2}+{B}^{2}+{a}_{11}+{b}_{1}\cos (C)+{b}_{2}\sin (D).\end{eqnarray}$The parameters are determined$\begin{eqnarray}\begin{array}{rcl}{a}_{2} & = & -\displaystyle \frac{{a}_{6}{a}_{7}}{{a}_{1}},\ \ \ {a}_{3}=-\displaystyle \frac{{k}_{3}{a}_{6}{a}_{7}}{{a}_{1}{k}_{2}},\ \ \ {a}_{4}=\displaystyle \frac{3}{2}\displaystyle \frac{{a}_{1}{k}_{3}}{{k}_{2}},\\ {a}_{8} & = & \displaystyle \frac{{a}_{7}{k}_{3}}{{k}_{2}},\ \ \ {a}_{9}=\displaystyle \frac{3}{2}\displaystyle \frac{{a}_{6}{k}_{3}}{{k}_{2}},\ \ \ {k}_{9}=\displaystyle \frac{3}{2}\displaystyle \frac{{k}_{3}{k}_{6}}{{k}_{2}},\end{array}\end{eqnarray}$where k1, k4, k7, and k8 are all null, and a1, a5, a6, a7, a10, a11, k2, k3, k5, k6, and k10 are all real arbitrary constants, with the condition ${a}_{1},{k}_{2}\ne 0$, then the function f is well defined. The interaction between the lump and breather-type periodic soliton is expressed as follows:$\begin{eqnarray}w=\displaystyle \frac{4{a}_{1}A+4{a}_{6}B-2{b}_{1}{k}_{1}\sin (C)+2{b}_{2}{k}_{6}\cos (D)}{f},\end{eqnarray}$where$\begin{eqnarray}\begin{array}{rcl}f & = & {A}^{2}+{B}^{2}+{a}_{11}+{b}_{1}\cos (C)+{b}_{2}\sin (D),\\ A & = & {a}_{1}x-\displaystyle \frac{{a}_{6}{a}_{7}y}{{a}_{1}}-\displaystyle \frac{{k}_{3}{a}_{6}{a}_{7}z}{{a}_{1}{k}_{2}}+\displaystyle \frac{3}{2}\displaystyle \frac{{a}_{1}{k}_{3}t}{{k}_{2}}+{a}_{5},\\ B & = & {a}_{6}x+{a}_{7}y+\displaystyle \frac{{a}_{7}{k}_{3}z}{{k}_{2}}+\displaystyle \frac{3}{2}\displaystyle \frac{{a}_{6}{k}_{3}t}{{k}_{2}}+{a}_{10},\\ C & = & {k}_{2}y+{k}_{3}z+{k}_{5},\ \ \ D={k}_{6}x+\displaystyle \frac{3}{2}\displaystyle \frac{{k}_{3}{k}_{6}t}{{k}_{2}}+{k}_{10}.\end{array}\end{eqnarray}$The specific parameters are chosen as$\begin{eqnarray}\begin{array}{rcl}{a}_{1} & = & \displaystyle \frac{1}{4},\ \ \ {a}_{5}=-4,\ \ \ {a}_{7}=-\displaystyle \frac{1}{5},\\ {a}_{6} & = & -\displaystyle \frac{1}{7},{a}_{10}=-3,\ \ \ {a}_{11}=2,\ \ \ {k}_{6}=-1.5,\\ {k}_{2} & = & 1,\ \ \ {k}_{3}=-9,\ \ \ {k}_{5}=-\displaystyle \frac{1}{9},\\ {k}_{10} & = & -1.3,\ \ \ {b}_{1}=1.5,\ \ \ {b}_{2}=1,\end{array}\end{eqnarray}$then the mixed solution w composed of a lump and the breather-type periodic waves is represented in figure 6, in (x, y)- and (x, z)-planes, with three-dimensional and density plots.
Figure 6.
New window|Download| PPT slide Figure 6.Perspective view of rational-periodic solutions w for equation (8), with the parameter (39), (a) plot of w, (b) Density plot of w, at time t=0 in the (x, y)-plane, with z=1, (c) plot of w, (b) Density plot of w, at time t=−1 in the (x, z)-plane, with $y=\tfrac{1}{2}$.
4. Interaction solutions between lump, triangular periodic waves, and multi-kink soliton
4.1. Interaction solution with a lump, periodic-solitary waves, and one-kink soliton
To obtain the mixed solution consisting of lump, triangular periodic waves, and one-kink soliton, we combine the exponential, sine, and cosine functions with the quadratic one. Therefore, we assume f as follows:$\begin{eqnarray}f={A}^{2}+{B}^{2}+{a}_{11}+{b}_{1}{{\rm{e}}}^{C}+{b}_{2}\cos (D)+{b}_{3}\sin (E),\end{eqnarray}$with$\begin{eqnarray}E(x,y,z,t)={p}_{1}x+{p}_{2}y+{p}_{3}z+{p}_{4}t+{p}_{5}.\end{eqnarray}$By the similar calculation of the above procedure, we get the following interaction solution:$\begin{eqnarray}w=\displaystyle \frac{4{a}_{1}A+4{a}_{6}B+2{b}_{1}{k}_{1}{{\rm{e}}}^{C}-2{b}_{2}{k}_{6}\sin (D)+2{b}_{3}{p}_{1}\cos (E)}{f},\end{eqnarray}$where$\begin{eqnarray}\begin{array}{rcl}f & = & {A}^{2}+{B}^{2}+{a}_{11}+{b}_{1}{{\rm{e}}}^{C}+{b}_{2}\cos (D)+{b}_{3}\sin (E),\\ A & = & -\displaystyle \frac{{a}_{9}{k}_{6}{a}_{7}x}{{k}_{9}{a}_{2}}+{a}_{2}y+\displaystyle \frac{2{a}_{2}{k}_{9}z}{3{k}_{6}}-\displaystyle \frac{{a}_{7}{a}_{9}t}{{a}_{2}}+{a}_{5},\\ B & = & \displaystyle \frac{{a}_{9}{k}_{6}x}{{k}_{9}}+{a}_{7}y+\displaystyle \frac{2{a}_{7}{k}_{9}z}{3{k}_{6}}+{a}_{9}t+{a}_{10},\\ C & = & {k}_{1}x+\displaystyle \frac{{k}_{9}{k}_{1}t}{{k}_{6}}+{k}_{5},\\ D & = & {k}_{6}x+{k}_{9}t+{k}_{10},\ \ \ E=\displaystyle \frac{{k}_{6}{p}_{4}x}{{k}_{9}}+{p}_{4}t+{p}_{5},\end{array}\end{eqnarray}$where a2, a5, a7, a9, a10, a11, k1, k5, k6, k9, k10, p4, and p5 are all real arbitrary constants, the conditions a2, k6, and ${k}_{9}\ne 0$. We select the following parameter:$\begin{eqnarray}\begin{array}{rcl}{b}_{1} & = & 5,\ \ \ {b}_{2}=9,\ \ \ {b}_{3}=6,\ \ \ {a}_{2}=3,\ \ \ {a}_{4}=\displaystyle \frac{1}{8},\\ {a}_{5} & = & \displaystyle \frac{1}{2},\ \ \ {a}_{7}=-5,\ \ \ {a}_{9}=\displaystyle \frac{1}{5},\ \ \ {a}_{10}=1,\\ {a}_{11} & = & 6,\ \ \ {k}_{1}=-2,\ \ \ {k}_{5}=6.5,\ \ \ {k}_{6}=-9,\\ {k}_{9} & = & -1.19,\ \ \ {k}_{10}=15,\ \ \ {p}_{4}=9,\ \ \ {p}_{5}=3,\end{array}\end{eqnarray}$then the mixed solution w composed of lump, triangular-periodic waves, and one-kink soliton is shown in figures 7(a), (b), in (x, y)-plane for fixed time t=0, and space z=2 with three-dimensional and density plots. Figure 7(c) exhibits the wave along the x-axis with different values of space z=−100, z=2, z=40, and y=0.
Figure 7.
New window|Download| PPT slide Figure 7.Perspective view of interaction solution w for equation (8), with the parameter (44), (a) plot of w, (b) Density plot of w, at time t=0 in the (x, y)-plane, with z=2, (c) plot of w along the x-axis with different values of space z.
4.2. Interaction solution consisting of lump, periodic-solitary waves, and two-kink soliton
We assume the function f as follows:$\begin{eqnarray}f={A}^{2}+{B}^{2}+{a}_{11}+{b}_{1}{{\rm{e}}}^{C}+{b}_{2}{{\rm{e}}}^{-C}+{b}_{3}\sin (D),\end{eqnarray}$in a similar way, we get the following interaction solution:$\begin{eqnarray}w=\displaystyle \frac{4{a}_{1}A+4{a}_{6}B+2{b}_{1}{k}_{1}{{\rm{e}}}^{C}-2{b}_{2}{k}_{1}{{\rm{e}}}^{-C}+2{b}_{3}{k}_{6}\cos (D)}{f},\end{eqnarray}$where$\begin{eqnarray}\begin{array}{rcl}f & = & {A}^{2}+{B}^{2}+{a}_{11}+{b}_{1}{{\rm{e}}}^{C}+{b}_{2}{{\rm{e}}}^{-C}+{b}_{3}\sin (D),\\ A & = & {a}_{1}x-\displaystyle \frac{{a}_{6}{a}_{7}y}{{a}_{1}}-\displaystyle \frac{2}{3}\displaystyle \frac{{a}_{6}{a}_{7}{a}_{4}z}{{a}_{1}^{2}}+{a}_{4}t+{a}_{5},\\ B & = & {a}_{6}x+{a}_{7}y+\displaystyle \frac{2}{3}\displaystyle \frac{{a}_{4}{a}_{7}z}{{a}_{1}}+\displaystyle \frac{{a}_{4}{a}_{6}t}{{a}_{1}}+{a}_{10},\\ C & = & {k}_{1}x+\displaystyle \frac{{a}_{4}{k}_{1}t}{{a}_{1}}+{k}_{5},\ \ \ D={k}_{6}x+\displaystyle \frac{{a}_{4}{k}_{6}t}{{a}_{1}}+{k}_{10},\end{array}\end{eqnarray}$where ${a}_{1},{a}_{4},{a}_{5},{a}_{6},{a}_{7},{a}_{10},{a}_{11},{k}_{1},{k}_{5},{k}_{6},$ and k10 are all real arbitrary constants, and the condition ${a}_{1}\ne 0$. The new interaction solution with lump, periodic wave, and two-kink soliton of the function w (figure 8) are found by choosing$\begin{eqnarray}\begin{array}{rcl}{b}_{1} & = & 5,\ \ \ {b}_{2}=5,\ \ \ {b}_{3}=4,\ \ \ {a}_{1}=\displaystyle \frac{1}{12},\\ {a}_{4} & = & 1,\ \ \ {a}_{5}=4,\ \ \ {a}_{6}=0.75,\\ {a}_{7} & = & 0.35,\ \ \ {a}_{10}=3,\ \ \ {a}_{11}=2,\ \ \ {k}_{1}=0.45,\\ {k}_{5} & = & \displaystyle \frac{1}{2},\ \ \ {k}_{6}=3,\ \ \ {k}_{10}=4.\end{array}\end{eqnarray}$
Figure 8.
New window|Download| PPT slide Figure 8.Perspective view of interaction solution w for equation (8), with the parameter (48), (a) plot of w, (b) density plot of w, at time t=1 in the (x, y)-plane, with z=0, (c) plot of w along the x-axis with various values of space z.
4.3. Interaction solution among lump, periodic-solitary waves, and three-kink soliton
Similarly, we can derive the interacted lump with periodic-solitary waves and three-kink soliton by assuming the function f as follows:$\begin{eqnarray}\begin{array}{rcl}f & = & {A}^{2}+{B}^{2}+{a}_{11}+{b}_{1}{{\rm{e}}}^{C}+{b}_{2}{{\rm{e}}}^{-C}\\ & & +{b}_{3}\cos (D)+{b}_{4}{{\rm{e}}}^{E}+{b}_{5}{{\rm{e}}}^{-E},\end{array}\end{eqnarray}$the parameters are determined:$\begin{eqnarray}\begin{array}{rcl}{a}_{3} & = & \displaystyle \frac{2}{3}\displaystyle \frac{{a}_{2}{k}_{4}}{{k}_{1}},\ \ \ {a}_{4}=\displaystyle \frac{{a}_{1}{k}_{4}}{{k}_{1}},\ \ \ {a}_{6}=-\displaystyle \frac{{a}_{1}{a}_{2}}{{a}_{7}},\\ {a}_{8} & = & \displaystyle \frac{2}{3}\displaystyle \frac{{a}_{7}{k}_{4}}{{k}_{1}},\ \ \ {a}_{9}=-\displaystyle \frac{{a}_{1}{a}_{2}{k}_{4}}{{a}_{7}{k}_{1}},\\ {k}_{6} & = & \displaystyle \frac{{k}_{1}{k}_{9}}{{k}_{4}},\ \ \ {p}_{1}=\displaystyle \frac{{k}_{1}{p}_{4}}{{k}_{4}},\end{array}\end{eqnarray}$and k2, k3, k7, k8, p2, and p3=0, with a7, k1, and ${k}_{4}\ne 0$, a1, a2, a5, a7, a10, a11, k1, k4, k5, k9, k10, p4, and p5 are all real arbitrary constants, then the function f is well defined, we get the following interaction solution:$\begin{eqnarray}w=\displaystyle \frac{4{a}_{1}A+4{a}_{6}B+2{b}_{1}{k}_{1}{{\rm{e}}}^{C}-2{b}_{2}{k}_{1}{{\rm{e}}}^{-C}-2{b}_{3}{k}_{6}\sin (D)+2{b}_{4}{p}_{1}{{\rm{e}}}^{E}-2{b}_{5}{p}_{1}{{\rm{e}}}^{-E}}{f}.\end{eqnarray}$The new interaction solutions with lump, periodic wave, and three kink-soliton of the function w (figure 9) are derived by selecting the following parameter:$\begin{eqnarray}\begin{array}{rcl}{b}_{1} & = & 9,\ \ \ {b}_{2}=9,\ \ \ {b}_{3}=450,\ \ \ {b}_{4}=59,\\ {b}_{5} & = & 59,\ \ \ {a}_{1}=1,\ \ \ {a}_{2}=39,\ \ \ {a}_{4}=-19,\\ {a}_{5} & = & 2.9,\ \ \ {a}_{7}=9,\ \ \ {a}_{10}=-\displaystyle \frac{4}{9},\ \ \ {a}_{11}=9,\\ {k}_{1} & = & -\displaystyle \frac{1}{2},\ \ \ {k}_{4}=-3.1,\ \ \ {k}_{5}=3,\\ {k}_{9} & = & 45.6,\ \ \ {k}_{10}=3,\ \ \ {p}_{4}=1.8,\ \ \ {p}_{5}=5.\end{array}\end{eqnarray}$
Figure 9.
New window|Download| PPT slide Figure 9.Perspective view of interaction solution w for equation (8), with the parameter (52), (a) plot of w, (b) density plot of w, at time $t=-1$ in the (x, y)-plane, with $z=-\tfrac{3}{2}$, (c) plot of w along the x-axis with various values of the time.
5. Conclusions
In this paper, based on the generalized bilinear form and symbolic calculation, lump-types and breather-type periodic soliton of the new (3+1)-dimensional nonlinear evolution equation were presented, with the nonsingularity conditions being given. By assuming the auxiliary function as a mixture of a quadratic with multi-exponential functions, or a quadratic with two periodic functions sine and cosine, two kinds of interaction solution among lump-multi-kink solution, and lump-breather type periodic solution are derived. A new type of interaction solution consisting of lump, periodic waves, and one-, two- and even three-kink solitons were found by introducing a new collection consisting of quadratic, exponential, and trigonometric functions, which is explicitly expressed and plotted. The results presented show that equation (8) may have very rich dynamical behavior, where the interaction solution with a lump, periodic-solitary waves, and three-kink soliton has not been constructed before, and this direct method is a powerful technique even for other high-dimensional NPDEs. It is interesting to find more interaction solutions based on generalized bilinear or trilinear differential equations to the nonlinear evolution equations, which will be studied in future works.
Acknowledgments
The work was supported by the National Natural Science Foundation of China No. 11835011 and No. 11675146.
,1,∗, Ji Lin
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