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Construction of Lump-Type and Interaction Solutions to an Extended (3+1)-Dimensional Jimbo–Miwa-Like Equation

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Construction of Lump-Type and Interaction Solutions to an Extended (3+1)-Dimensional Jimbo–Miwa-Like Equation February 19, 2020 16:34 MPLB S0217984920501304 page 1 Modern Physics Letters B 2050130 (10 pages) c World Scientific Publishing Company DOI: 10.1142/S0217984920501304 Construction of lump-type and interaction solutions to an extended (3 + 1)-dimensional Jimbo{Miwa-like equation Feng-Hua Qi∗;x;yy and Wen-Xiu May;z;x;{;k;∗∗;xx, ∗School of Information, Beijing Wuzi University, Beijing 101149, China yDepartment of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China zDepartment of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia xDepartment of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA {School of Mathematics, South China University of Technology, Guangzhou 510640, China kCollege of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China ∗∗International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa [email protected] [email protected] Pan Wang Sports Business School, Beijing Sport University, Beijing 100084, China Qi-Xing Qu School of Information, University of International Business and Economics, Beijing 100029, China Received 7 October 2019 Revised 9 November 2019 Accepted 13 November 2019 Published 19 February 2020 We present lump-type solutions and interaction solutions to an extended (3 + 1)- dimensional Jimbo{Miwa-like equation. Three classes of lump-type solutions are ob- tained by the Hirota bilinear method. Interaction solutions are among lump-type solu- tions, two kink waves and periodic waves, and between two kink waves and a periodic wave are computed. Dynamical characters of the obtained solutions are graphically ex- hibited. These wave solutions enrich the dynamical theory of higher-dimensional non- linear dispersive wave equations. Keywords: Lump-type solution; interaction solution; kink solution; periodic solution. yy;xxCorresponding authors. 2050130-1 February 19, 2020 16:34 MPLB S0217984920501304 page 2 F.-H. Qi et al. 1. Introduction Nonlinear evolution equations model varieties of physical phenomena.1{14 Recently, Wazwaz presented two extended (3 + 1)-dimensional Jimbo{Miwa equations15{28: uxxxy + 3uxuxy + 3uyuxx + 2uyt − 3(uxz + uyz + uzz) = 0 ; (1) and uxxxy + 3uxuxy + 3uyuxx − 3uyt + 2(uxt + uyt + uzt) = 0 ; (2) which describe certain (3 + 1)-dimensional waves in plasmas and optics.15 Here, u(x; y; z; t) is a function of the spatial variables x; y; z and the temporal variable t, and the subscripts denote the corresponding partial derivatives.16 Exact solitary wave solutions,29 multiple soliton solutions,16,30 meromorphic exact solutions,31 resonant multi-soliton solutions,32 periodic solitary wave solutions,33 lump solutions and the lump{kink solutions34 of Eqs. (1) and (2) have been obtained by various methods.35 Under the transformation u = 2(ln F )x, a Hirota bilinear form for Eq. (1) reads 3 2 (DxDy + 2DtDy − 3DxDz − 3DyDz − 3Dz )F · F = 0 : (3) Recently, a generalization of the bilinear differential operators has been pre- sented as36 M n Y @ @ i Dn1 :::DnM (f · g) = + α p;x1 p;xM @x @x0 i=1 i i 0 0 × f(x ; : : : ; x )g(x ; : : : ; x )j 0 0 ; (4) 1 M 1 M x1=x1;:::;xM =xM m r(m) in which n1; : : : ; nM are nonnegative integers, with an integer m, α = (−1) , if m ≡ r(m) with 0 ≤ r(m) < p. Taking p = 3, we have 2 3 4 5 6 α3 = −1; α3 = α3 = 1; α3 = −1; α3 = α3 = 1 ; and we can generalize the bilinear Eq. (3) into 3 2 (D3;xD3;y + 2D3;tD3;y − 3D3;xD3;z − 3D3;yD3;z − 3D3;z)F · F = 2(3 FxxFxy + 2FytF − 2FyFt − 3FxzF + 3FxFz 2 − 3FyzF + 3FyFz − 3FzzF + 3Fz ) = 0 ; (5) which is equivalently linked with an extended (3 + 1)-dimensional Jimbo{Miwa-like equation 3 3 9 3 2 3 3 2uyt + u uy + uuyux + u uxy + uxuxy + uyuxx 8 4 4 2 2 9 2 3 2 3 − 3 (uxz + uyz + uzz) + vu ux + vu + vuuxx = 0 ; (6) 8 4 x 4 where uy = vx. 2050130-2 February 19, 2020 16:34 MPLB S0217984920501304 page 3 Construction of lump-type and interaction solutions In this paper, we will consider Eq. (6). This paper will be structured as follows. In Sec. 2, we will present three classes of lump-type solutions of Eq. (6) through a generalized Hirota bilinear method. In Sec. 3, we will construct and analyze some interaction solutions. In Sec. 4, dynamical properties of the above presented solutions will be graphically discussed. Conclusions will be provided in Sec. 5. 2. Lump-Type Solutions To search for lump-type solutions of Eq. (6) (see, for example, Refs. 37{45 for other examples), we begin by assuming ( 2 2 F = G + H + a11;G = a1x + a2y + a3z + a4t + a5 ; (7) H = a6x + a7y + a8z + a9t + a10 ; where ai(1 ≤ i ≤ 11) are real parameters to be determined. Substituting such a function F in Eqs. (7) into the bilinear Eq. (5), we can, with Maple symbolic computation, get Case 1. 8 2 2 2 2 3(a3a2 + a3a2 − a8a2 + a1a3a2 − a6a8a2 + a3a7 > > > + a3a6a7 + a1a7a8 + 2a3a7a8) >a4 = ; > 2(a2 + a2) > 2 7 > < 3(a a2 + a a a + a a a + 2a a a + a a2 − a2a 8 2 3 6 2 1 8 2 3 8 2 7 8 3 7 (8) > − a a a a2a a a a > 1 3 7 + 7 8 + 6 7 8) >a9 = 2 2 ; > 2(a2 + a7) > > 2 2 2 2 > (a1 + a6)(a1a2 + a6a7)(a2 + a7) :>a11 = ; (a3a7 − a2a8)[(a1 + a3)a7 − a2(a6 + a8)] 2 2 where the involved constants should satisfy (a2 + a7)(a3a7 − a2a8)[(a1 + a3)a7 − a2(a6 + a8)] =6 0. Case 2. a3a7 3(a3a7 + a1a8 + a3a8) a2 = ; a4 = ; a8 2a7 (9) a1a3 3a8(a6 + a7 + a8) a8 = − ; a9 = ; a6 2a7 where a6a7a8 =6 0. 2050130-3 February 19, 2020 16:34 MPLB S0217984920501304 page 4 F.-H. Qi et al. Case 3. 8 a2a − a a2 >a = 3 7 7 8 ; > 2 > 2a3a8 > > 3a (2a3 + 2a a2 + a a2 + 2a2a + 2a a a + a2a ) >a 3 8 6 8 7 8 3 8 1 3 8 3 7 ; > 4 = 2 2 < 2a7(a3 + a8) (10) 2 2 2 > 3a8(2a6a3 + a7a3 − 2a1a8a3 + a7a8) >a9 = 2 2 ; > 2a7(a3 + a8) > > 2 2 2 2 2 2 > (a1 + a6)a7(a3 + a8)(a1a3 + 2a6a8a3 − a1a8) :>a11 = 2 3 2 2 2 ; 2a3a8(a8 + a6a8 + a3a8 + 2a1a3a8 − a3a6) 3 2 2 2 where a3a7a8(a8 + a6a8 + a3a8 + 2a1a3a8 − a3a6) =6 0. All the above cases of parameter selections result in three classes of solutions to Eq. (7); and then through the transformation u = 2(ln F )x we get three classes of lump-type solutions of Eq. (6), which can be presented as 4a1(a4t + a1x + a2y + a3z + a5) + 4a6(a9t + a6x + a7y + a8z + a10) u = 2 2 ; (11) (a4t + a1x + a2y + a3z + a5) + (a9t + a6x + a7y + a8z + a10) + a11 in which it is sufficient for u to be analytic if a11 > 0. For example, choosing exact values of parameters for the three cases above, we can get three kinds of lump-type solutions separately. (1) Taking a1 = 1; a2 = 2; a3 = 1; a5 = 1; a6 = 1; a7 = 1; a8 = 1 and a10 = 1 for Case 1, we can get 2(4x + 6y + 4z + 66t + 4) u 5 = 39t 2 27t 2 ; (12) (x + y + z + 10 + 1) + (x + 2y + z + 10 + 1) + 15 (2) Taking a1 = 1; a3 = 1; a5 = 1; a6 = 1; a7 = 1; a10 = 1 and a11 = 1 for Case 2, we can get 8x − 12t + 8 u = 3t 2 3t 2 ; (13) (x + y − z − 2 + 1) + (x − y + z − 2 + 1) + 15 (3) Taking a1 = 1; a3 = 1; a5 = 1; a6 = 1; a7 = 1; a8 = 1 and a10 = 1 for Case 3, we can get 4(2x + y + 2z + 9t + 2) u : = 3t 2 15t 2 (14) (x + y + z + 2 + 1) + (x + z + 2 + 1) + 1 3. Interaction Solutions First, in order to construct interaction solutions (see, for example, Refs. 41, 46, 47 and Refs. 48{52 for other examples of nonlinear PDEs and linear PDEs, 2050130-4 February 19, 2020 16:34 MPLB S0217984920501304 page 5 Construction of lump-type and interaction solutions respectively), we assume that 8F = G2 + H2 + L + b ;G = b x + b y + b z + b t + b ; > 11 1 2 3 4 5 > <>H = b6x + b7y + b8z + b9t + b10 ; (15) k1x+k2y+k3z+k4t+k5 −k1x−k2y−k3z−k4t−k5 >L = v1e + v2e > :> + v3 cos(k6x + k7y + k8z + k9t + k10) ; where bi(1 ≤ i ≤ 11), kj(1 ≤ j ≤ 10), v1, v2 and v3 are all real parameters to be determined. Substituting such a function F by Eqs. (15) into the bilinear Eq. (5), we can, through Maple, obtain Case 1. 8 2 2 2 3b3(b1 + b2 + b3) b1b2 b3b7 + b1(b2 + b7) >b4 = ; b6 = − ; b8 = ; > 2b2 b7 b2b7 > > > (b1 + b3)k2 >k1 = 0; k3 = ; > b2 > > 2 2 2 3 2 2 2 2 < 3(b2 + b7)b1 + 3(b2 + b3b2 + b7b2 + 2b3b7)b1 + 3b3(b2 + b3)b7 b9 = 2 ; (16) > 2b2b7 > > > (b1 + b3)k7 3(b1 + b3)(b1 + b2 + b3)k2 >k8 = ; k4 = 2 ; > b2 2b2 > > > 3(b1 + b3)(b1 + b2 + b3)k7 :>k6 = 0; k9 = 2 ; 2b2 which needs to satisfy b2b7 =6 0; to make F to be analytic. Case 2. 8 3 >b ; b −b ; b k k2 − ; b ; b −b ; > 2 = 0 3 = 1 4 = 1( 6 1) 7 = 0 8 = 6 > 2 <> 3b b = − 6 ; k = 0 ; (17) > 9 2 > 2 > 3 2 3k6 :>k3 = −k1; k4 = k1(k6 − 1); k7 = 0; k8 = −k6; b9 = − : 2 2 With u = 2(ln F )x, a kind of interaction solutions of Eq.
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