Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 82, pp. 1–16. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.82 VARIETY OF SOLUTIONS AND DYNAMICAL BEHAVIOR FOR YTSF EQUATIONS WEI CHEN Abstract. We construct non-homogeneous polynomial lump wave solutions of the Yu-Toda-Sasa-Fukuyama (YTSF) equation, based on a bilinear ap- proach, enriching the formal diversity of lump waves. By studying the in- teraction between the lump solutions of the YTSF equation and the solitary wave solutions, we find a new aggregation effect and elastic collision effect. We obtain exact solutions, such as the solution of separated variables and periodic nonlinear wave solutions, by applying the Lie symmetry group method and the bilinear method. 1. Introduction The integrability and exact solutions of nonlinear evolution equations play an important role in the field of nonlinear science, such as condensed matter physics, solid state physics, aerodynamics, plasma physics, fluid dynamics and many other fields [7, 13, 17]. This article will focus on exact solutions to the (3+1)-dimensional Yu-Toda-Sasa-Fukuyama (YTSF) equation [1], as well as the interactions between these exact solutions, using the bilinear method and Lie symmetry method [11, 12]. The standard form of the YTSF equation is (−4ut + φ(u)uz)x + 3uyy = 0, φ(u) = ∂2x + 4u+ 2ux∂ −1 x , (1.1) in which u = u(t, x, y, z) is an analytic function of the scaled spatial coordinates x, y, z and the temporal coordinate t. The YTSF equation is derived by applying a strong symmetry to the 2-dimensional Bogoyavlenskii-Schiff equation, which de- scribes an elastic quasi-plane wave in a lattice or an interfacial wave in a two-layer liquid [20]. The typical way to explore exact solutions of the YTSF equation is to use classical methods such as the Bäcklund transformation method [21], the bilin- ear methods [10], elliptic function expansions [15] and Lie symmetry group method [16], which can bypass integration to derive explicit solutions. Yan [22] used the auto-Bäcklund transformation to study 1.1 and found a nonlinear wave solution to 1.1. Chen [2] applied Hirota bilinear formulation to study the lump solution of 1.1 2020 Mathematics Subject Classification. 35B35, 35B40. Key words and phrases. YTSF equation; symmetry reduction; two-lump wave; aggregation; interaction wave. ©2023. This work is licensed under a CC BY 4.0 license. Submitted April 27, 2023. Published December 10, 2023. 1 2 W. CHEN EJDE-2023/82 with the finding that the lump solution consists of a 2-fold homogeneous polyno- mial, but the lump solution in the form of non-homogeneous polynomial has not been found yet [1]-[14]. The rest of this article is organized as follows. In Section 2, we use the bi- linear method to construct the lump solution of the YTSF equation with a non- homogeneous multinomial function. After constructing the solution, we further study the interaction between lump solution and soliton solution. In Section 3, we apply the Lie symmetry method to obtain a new non-traveling wave solution with an arbitrary function. 2. Lump solution and variable soliton for the YTSF equation In this section, we start with the lump solution of the special form of the YTSF equation, then we analyze the interaction between the lump solution and the soliton wave. 2.1. Two-lump solution to the YTSF equation. By the traveling wave trans- formation η = x+ cz − αt, the equation (1.1) reduces to 3uyy + (4αuη + 3(u2)η + cuηηη)η = 0. (2.1) It is easy to see that the real constant u0 is the trivial solution of the YTSF equation, where u0 can be arbitrary. By the second order logarithmic derivative transformation u = u0 + 2(lnF )ηη, (2.2) the equation (2.1) can be changed into (3D2 y + κD2 η + cD4 η)F · F = 0, (2.3) where κ = 2(2α+ 3cu0). In order to search the two-lump solution, we choose F = (λ2η 2 + λ3y 2)3 + λ4η 4 + λ5y 4 + λ6η 2y2 + λ7η 2 + λ8y 2 + λ1, (2.4) where λi (i = 1, 2, . . . , 8) are constants to be determined. Substituting (2.4) into (2.3), we obtain the nonlinear algebraic equations of undetermined parameters, from which we obtain λ1 = −15 (5c κ λ2 )3 , λ3 = κ 3 λ2, λ4 = −25c κ λ32, λ5 = −17κ 9 λ32, λ6 = −30cλ32, λ7 = −125 c2 κ2 λ32, λ8 = 19 κ (5c)2λ32, (2.5) where cλ2κ 6= 0. Substituting (2.5) into (2.4), we have F = λ32 [ (η2 + κ 3 y2)3 − 25c κ η4 − 17κ 9 y4 − 30cη2y2 − 125c2 κ2 η2 + 19 κ (5c)2y2 − 15 (5c κ )3] . (2.6) EJDE-2023/82 DYNAMICAL BEHAVIOR FOR YTSF EQUATIONS 3 Thus, from (2.2)-(2.6), we know that equation 1.1 has the rational solution of the form u = u0 + 2 ( 24 ( η2 + κ 3 y2 ) η2 + 6(η2 + κ 3 y2) 2 − 300c κ η2 − 60cy2 − 250c2 κ2 ) ÷ ( (η2 + κ 3 y2)3 − 25c κ η4 − 17cκ 9 y4 − 30cη2y2 − 125c2 κ2 η2 + 19 κ (5c) 2 y2 − 15( 5c κ ) 3) + 2 ( 6 ( η2 + κ 3 y2)2η − 100cη3 κ − 60cηy2 − 250c2η κ2 )2) ÷ ( (η2 + κ 3 y2)3 − 25c κ η4 − 17cκ 9 y4 − 30cη2y2 − 125c2 κ2 η2 + 19 κ (5c) 2 y2 − 15( 5c κ ) 3)2 . (2.7) 2.2. Aggregation effect and elastic collision of two-lump wave. To bet- ter understand these solutions, we now consider their dynamical behaivors. From graphs of solutions as shown in Figure 1-3, it is possible to see clearly the height and positional behaivor of the waves. The corresponding solutions u(x, y, z, t) in different planes are shown in Figure 1-3. We observe that the solution u contains three troughs and two peaks in non-standard formation. Therefore, this solution is also called two-lump solution. The physical behaivor influenced by the parameter u0 will be explored. Figure 1 shows the two-lump solution of the form (2.7) in the (t, y) plane. It can be seen that the middle trough changes correspondingly along the u axis with the change of the value of u0 and the distance between the left trough and the right trough changes symmetrically at the same time. In Figure 1(g) and (h), it can be seen that increasing the parameter u0 will cause the background plane wave to rise, which can be verified from the solution formula (2.7). In Figure 1(a), (b) and (c), it can be observed that increasing the value of u0 will increase the width of the wave when u0 < 0. When u0 > 0 , it appears that increasing the value of u0 leads to a decrease of the width of the wave. When u0 = 0, the distance between the left trough and the right trough is the widest. Figures 2 and 3 mainly reflect the influence of variable t upon the two-lump solution. Figure 2(a) shows the two-lump solution of the YTSF equation in the (x, y) plane when t = 0. One can also see that when (x, y)→ (∞,∞), u→ −1, and the two-lump solution is travelling horizontally on the (x, y) plane. The two-lump solution will not disappear as the time evolves. Figure 2(c) shows clearly how the waves move: the position of the two-lump solution changes correspondingly with time evolution while maintaining its structure. Figure 3 shows the two-lump solution of the YTSF equation in the (z, y) plane. The motion behaivor in Figure 3 is the same as the one observed in Figure 2. From Figures 2 and 3, we can see that the two-lump solution in the (x, y) and (z, y) planes always preserve their shape, amplitude and velocity while traveling. 2.3. Interaction between two-lump wave and soliton wave. In this section we will study the interaction between two-lump solution and soliton wave solution 4 W. CHEN EJDE-2023/82 (a) u0 = −2 (b) u0 = −1.5 (c) u0 = −1 (d) u0 = 0 (e) u0 = 1 (f) u0 = 1.5 (g) u0 = −2,−1.5,−1 (h) u0 = 0, 1, 1.5 (i) u0 = −1.5, 1.5 Figure 1. Two-lump solution of the form (2.7) in the (t, y) plane with y = z = 1 and c = −1, λ1 = λ2 = 1 with (a) u0 = −2, (b) u0 = −1.5, (c) u0 = −1, (d) u0 = 0, (e) u0 = 1, (f) u0 = 1.5. The black dashed line, the blue line and the magenta point of (g) curves correspond to (a), (b) and (c) respectively. The red line, the blue line and the black line of (h) curves correspond to (d), (e), and (h) respectively. The red and black lines of (i) curves correspond to (b) and (f). by choosing F = λ1 + (λ2η 2 + λ3y 2)3 + λ4η 4 + λ5y 4 + λ6η 2y2 + λ7η 2 + λ8y 2 + λ9e λ10η+λ11y+λ12 , (2.8) EJDE-2023/82 DYNAMICAL BEHAVIOR FOR YTSF EQUATIONS 5 (a) t = 0 (b) t = 0 (c) t = −10, 0, 10 Figure 2. (a) Two-lump solution of the YTSF equation in the (x, y) plane given by (2.7) with parameters λ2 = λ3 = 1, c = u0 = −1, and z = 1. (b) is the density plot corresponding to (a). (c) is the evolution of the two-lump solution when t = −10 (red), t = 0(blue), t = 10(black). (a) t = 0 (b) t = 0 (c) t = −10, 0, 10 Figure 3. (a) Two-lump solution of the YTSF equation in the (z, y) plane given by (2.7) with λ2 = λ3 = 1, c = u0 = −1, and z = 1. (b) is the density plot corresponding to (a). (c) is the evolution of the two-lump solution when t = −10 (red), t = 0 (blue), t = 10 (black). where λi (i = 1, 2, . . . , 12) are constants to be determined. Substituting (2.8) into (2.3), we obtain the nonlinear algebraic equations with undetermined parameters, from which we have c = λ3 = λ5 = λ6 = λ8 = λ11 = 0, u0 = −2 3 α. (2.9) Substituting (2.9) in (2.8) yields F = λ32η 6 + λ4η 4 + λ7η 2 + λ1 + λ9e λ10η+λ12 . (2.10) 6 W. CHEN EJDE-2023/82 Substituting (2.10) into (2.2) with η = x+ cz − αt, we obtain a hybrid solution to equation 1.1 as u = −2 3 α+ 2 ( 30λ32(−αt+ x) 4 + 12λ4(−αt+ x) 2 + 2λ7 + λ9λ 2 10e λ10(−αt+x)+λ12 ) ÷ ( λ32(−αt+ x) 6 + λ4(−αt+ x) 4 + λ7(−αt+ x) 2 + λ1 + λ9e λ10(−αt+x)+λ12 ) − 2 ( 6λ32(−αt+ x) 5 + 4λ4(−αt+ x) 3 + 2λ7(−αt+ x) + λ9λ10e λ10(−αt+x)+λ12 )2 ÷ ( λ32(−αt+ x) 6 + λ4(−αt+ x)4 + λ7(−αt+ x)2 + λ1 + λ9e λ10(−αt+x) + λ12 ) . (2.11) The expression 2.11 is a linear superposition of two-lump wave and soliton wave, which is also an interaction solution to equation 1.1. (a) (b) Figure 4. Interaction between two-lump wave and soliton wave of the YTSF equation given by (2.11). (a) α = −1, λ1 = λ2 = λ4 = λ7 = λ9 = λ10 = λ12 = 1. (b) α = λ4 = −1, λ1 = λ2 = λ7 = λ9 = λ10 = λ12 = 1. Figure 4 illustrates the interaction between two-lump wave and soliton wave. As the exponential changes from negative to positive, the sum of the exponential function and the polynomial changes from a polynomial control to an exponential control. The results obtained here show that the two-lump soliton degenerates to a rouge wave and then to a soliton and finally disappears with a spatio-temporal variation, which has not been found in the previous literature. 3. Various solutions and dynamical behaivor of the YTSF equation Consider the potential form of the TYSF equation 1.1 vxxxz + 4vxvxz + 2vxxvz + 3vyy − 4vxt = 0, u = vx. (3.1) EJDE-2023/82 DYNAMICAL BEHAVIOR FOR YTSF EQUATIONS 7 This equation has a movable logarithmic branch point in the sense of WTC method [18]. Assume that equation (3.1) has the traveling wave solution in the form v = w(η), η = px+ qy + rz − st, (3.2) where p, q, r and s are real constants to be determined. Substituting (3.2) into equation (3.1) and integrating once leads to p3rwηηη + 3p2rw2 η + 3q2wη + 4pswη + C1 = 0. (3.3) Settin wη = f(η), then (3.2) is reduced to a second-order nonlinear ordinary differential equation, p3rfηη + 3p2rf2 + 3q2f + 4psf + C1 = 0. (3.4) Multiplying (3.4) by fη, integrating once with respect to η and taking the integra- tion constant to be C2, we have p3rf2η + 2p2rf3+4pf2s+ 3f2q2 + 2C1f + 2C2 = 0. (3.5) Using the Jacobi elliptic function expansion method [15] to find the periodic wave solution of (3.5), we can expand the solution to equation (3.5) in the form f(η) = a1sn 2(bη,m) + a0, (3.6) where sn(bη,m) is the Jacobi elliptic sine function with the modulus m ∈ (0, 1), and a0, a1 and b are non-zero constants to be determined. Substituting (3.6) into (3.5) leads to an algebraic system. By assuming the coefficients of snk (k = 0, 2, 4, 6) to be zero, we obtain p2ra21(2b2m2p+ a1) = 0, a21(4b2m2p3r + 4b2p3r − 4p3ra0 − 4ps− 3q2) = 0, a1(2b2p3ra1 + 3p2ra20 + 4psa0 + 3q2a0 + C1) = 0, 2p2ra30 + 4psa20 + 3q2a20 + 2C1a0 + 2C2 = 0. (3.7) If 16b4p6r2(m4−m2+1)−16p2s2−24pq2s−9q4)+12C1p 2r = 0 and (4b2p3r(m2+ 1)−4ps−3q2)(4b2p3r(2m2−1)+4ps+3q2)(4b2p3r(m2−2)−4ps−3q2)−216C2p 4r2 = 0, then the coefficients are a0 = 4b2p3r(m2 + 1)− 4ps− 3q2 6p2r , a1 = −2b2m2p. (3.8) Combining (3.6) and (3.8), we have f1(η) = −2b2m2psn2(bη,m) + 4b2p3r(m2 + 1)− 4ps− 3q2 6p2r . (3.9) When m→ 1, equation (3.9) reduces to the shock wave solution f2(η) = −2b2p tanh2(bη) + 8b2p3r − 4ps− 3q2 6p2r . (3.10) In view of wη = f(η), we deduce w1 = −2bpE(sn(bη,m),m) + ρ1η, (3.11) where E is an elliptic integral of the second kind and w2 = bp ln(tanh2(bη)− 1) + ρ2η, (3.12) where ρ1 = 4b2p3r(m2−2)−4ps−3q2 6p2r and ρ2 = 8b2p3r−4ps−3q2 6p2r . 8 W. CHEN EJDE-2023/82 In (3.7), replacing sn(bη,m) by cn(bη,m) and repeating the same process as above, we find a new triangular periodic solution w3 = 2bpE(sn(bη,m),m) + ρ3η (3.13) and a new Jacobi doubly periodic solution w4 = 2bp arctan(sinh (bη)2) + ρ4η (3.14) where ρ3 = 2pm2(3pr−2s)−3q2m2−6p2r+4b2p3rm2(1−2m2) 6p2rm2 and ρ4 = − 4b2p3r+4ps+3q2 6p2r . As m → 1, the Jacobi biperiodic solution (3.13) degenerates to the triangular periodic solution (3.14). From (3.11)-(3.14), we obtain the following solutions to equation 3.1: v1 = −2bpE(sn(b(px+ qy + rz − st),m),m) + ρ1(px+ qy + rz − st), v2 = bp ln(tanh2b(px+ qy + rz − st)− 1) + ρ2(px+ qy + rz − st), v3 = 2bpE(sn(b(px+ qy + rz − st),m),m) + ρ3(px+ qy + rz − st), v4 = 2bp arctan(sinh (b(px+ qy + rz − st))2) + ρ4(px+ qy + rz − st), where E is an elliptic integral of the second kind. 3.1. Lie symmetry reduction of the YTSF equation 3.1. In this subsection, we concentrate on finding the exact solution to the YTSF equation using the Lie symmetry method. According to the Lie group theory [14], we know that σ must satisfy the equation σx3z + 4σxvxz + 4σxzvx + 2σxxvz + σzvxx + σyy − 4σxt = 0. (3.15) To seek symmetry of (3.1), we take the function σ in the form σ = f1vx + f2vy + f3vz + f4vt + f5v + f6, (3.16) where fi (i = 1, 2, . . . , 6) are functions to be determined and v(t, x, y, z) is the solution to 3.1. Substituting (3.16) and (3.15) into 3.1, we have f1zvx4 + f2zvx3y + f4zvx3t − (3f1xz + f5z)vx3 + 3f4xvx2zt + 3f2xvx2yz + 3f3xvx2z2 + · · · = 0. (3.17) Because of the linear independence of the derivatives of v in (3.17), we obtain f1 = 2 3 p2 ′(t)y + p′1(t), f2 = p2(t), f3 = p3(t), f4 = λ, f5 = 0, f6 = ( 4 3 p′′2(t)y + 2p′′1(t))z + p′3(t)x+ 2 3 p′′3(t)y2 + p′4(t)y + p′5(t), (3.18) where pi(t) (i = 1, 2, 3, 4, 5) are arbitrary functions of t. Substituting (3.18) into (3.16) yields σ = (2 3 p2 ′(t)y + p′1(t) ) vx + p2(t)vy + p3(t)vz + λvt + ( 4 3 p′′2(t)y + 2p′′1(t))z + p′3(t)x+ 2 3 p′′3(t)y2 + p′4(t)y + p′5(t). (3.19) We can obtain many symmetries of (3.1). On the basis of the integrability of the reduced equation, we find two types of solutions λ = 1, p2(t) = p3(t) = 0; (3.20) λ = 1, p1(t) = p2(t) = p4(t) = p5(t) = 0, p3(t) = p′(t), (3.21) EJDE-2023/82 DYNAMICAL BEHAVIOR FOR YTSF EQUATIONS 9 where p(t) = ∫ p3(t)d(t) and p3(t) is an arbitrary functions of t. Substituting (3.20) into (3.19) yields σ = p′1vx + vt + 2p′′1z + p′4y + p′5. (3.22) Letting σ be 0 leads to v = −2p′1(t)z − p4(t)y − p5(t) + f(ξ, y, z), ξ = x− p(t). (3.23) Substituting (3.23) into (3.1), we obtain a symmetry reduction as 4fξfξz + 2fξξfz + fzξξξ + 3fyy = 0. (3.24) Repeating the above process, from (3.1), (3.19), and (3.21), we obtain another symmetry reduction: v = −p′(t)x− 2 3 p′′(t)y2 + f(x, y, ξ), ξ = z − p(t), 2fxxfξ + 4fξxfx + 3fyy + fξxxx = 0, (3.25) where p(t) = ∫ p3(t)d(t) and p3(t) is an arbitrary functions of t. 3.2. Solution to the symmetry reduction equation. In this subsection, we use the consistent Riccati expansion (CRE) method [4] to solve the symmetry reduction equation 3.24. Balancing the highest derivative term with nonlinear terms in 3.24 gives the expansion f = a0 + a1U(η), (3.26) where a0, a1 and η are to be determined later and U = U(η) is a solution to the Riccati equation U ′η = µ0 + µ1U 2. (3.27) This equation has the following three sets of traveling wave solutions U1 = 1 C − µ1η , µ0 = 0, U2 = √ µ0 µ1 tan( √ µ0µ1η + C), µ0µ1 > 0, U3 = − √ −µ0 µ1 tanh( √ −µ0µ1η + C), µ0µ1 < 0, (3.28) where µ0 and µ1 are constants. Substituting (3.28) with (3.26) into (3.24), we have 12ηzη 2 ξa1µ 3 1 (2ηξµ1 + a1)U5 + 2µ3 1(3a1zη 3 ξµ1 + 9a1ηzξη 2 ξµ1 + 9a1ηξξηξηzµ1 + 9ηξξηza1ξµ1 + 4a1zηξξa1 + 2ηξηzξa 2 1 + ηzηξξa 2 1 + 8ηzηξa1ξa1)U4 + . . . = 0. (3.29) By eliminating all the coefficients of U i (i = 0, 1, 2, . . . , 5), we obtain η = (ξC1 + C2)k a0 = h(ξ)y + q(ξ) a1 = −3 4 kC1µ1(ξC1 + C2)k−1, (3.30) where C1 and C2 are constants and function h(ξ), q(ξ) are arbitrary smooth func- tion of ξ. Substituting (3.30) into (3.26) yields f(ξ, y, z) = h(ξ)y + q(ξ)− 3 4 kC1µ1(ξC1 + C2)k−1U((ξC1 + C2)k). (3.31) 10 W. CHEN EJDE-2023/82 Lemma 3.1 ([4]). If U(η) is the solution to the Riccati equation (3.27), then the function defined in equation (3.26) is the solution to the symmetry reduction equation (3.19). Applying Lemma 3.1, the solution of the symmetry reduction equation (3.19) is f1 = h(ξ)y + q(ξ)− 3 4 kC1µ1 (ξC1 + C2) k−1 C − µ1(ξC1 + C2) k , f2 = h(ξ)y + q(ξ)− 3 4 kC1 √ µ0µ1(ξC1 + C2)k−1 tan( √ µ0µ1(ξC1 + C2)k + C), f3 = h(ξ)y + q(ξ) + 3 4 kC1 √ −µ0µ1(ξC1C2)k−1 tanh( √ −µ0µ1(ξC1 + C2)k + C). By applying the symmetric transformation (3.23), the corresponding solution to the YTSF equation 3.1 is v1 = −2p′1(t)z − p4(t)y − p5(t) + h(ξ)y + q(ξ)− λµ1 (ξC1 + C2) k−1 C − µ1(ξC1 + C2) k , v2 = −2p′1(t)z − p4(t)y − p5(t) + h(ξ)y + q(ξ) − λκ1(ξC1 + C2)k−1 tan(κ1(ξC1 + C2)k + C), v3 = −2p′1(t)z − p4(t)y − p5(t) + h(ξ)y + q(ξ) + λκ2(ξC1 + C2)k−1 tanh(κ2(ξC1 + C2)k + C), where ξ = x− p(t), λ = 3 4kC1, κ1 = √ µ0µ1, and κ2 = √ −µ0µ1. We now solve the symmetry reduction equation (3.25). 3.2.1. Periodic solution of (3.25). Using the wave transform f = p(η), η = αx+ βy + γξ, (3.32) where α, β, γ are real constants and substituting (3.32) into (3.25), we find that the function p(η) satisfies the fourth-order nonlinear ordinary differential equation 6α2γp′′p′+α3γp(4) + 3β2p′′ = 0. (3.33) Integrating (3.33) once with respect to η and taking the integration constant to be B, we obtain 3α2γp′(η)2+α3γp(3)(η) + 3β2p′(η) +B = 0. (3.34) By setting dp(η) dη = q(η), the second order nonlinear ordinary differential equation reduces to 3α2γq2+α3γq′′ + 3β2q +B = 0. (3.35) Multiplying the equation (3.35) by q′ and integrating with respect to η, we will have 1 2 α3γq ′2+α2γq3 + 3 2 β2q2 +Bq + C = 0. (3.36) Here we use the same idea as that in the method of expanding the elliptic function mentioned above to obtain the following two types of solutions to equation (3.36). EJDE-2023/82 DYNAMICAL BEHAVIOR FOR YTSF EQUATIONS 11 (1) If the arbitrary constants B and C satisfy 9β4 − 16α6b4γ2(m4 −m2 + 1)−B12α2r = 0,( 4α3b2γ(m2 + 1)− 3β2 ) ( 4α3b2γ(m2−2)− 3β2 ) = 216Bγ2α4 (4α3b2γ(2m2−1)+3β2) , then the generalized periodic solution can be expressed by the elliptic Jacobi sine function in the form q1(η) = −2αb2m2sn2(bη,m) + 4α3b2γ(m2 + 1)− 3β2 6α2r . (3.37) When m→ 1, the solution (3.37) reduces to the following shock wave solution q2(η) = −2αb2tanh2(bη) + 8α3b2γ − 3β2 6α2r . (3.38) In particular, letting b = bi with i2 = −1, the periodic solution is q3(η) = 2αb2 tan2 (bη) + 8α3b2γ + 3β2 6α2r . (3.39) (2) If the arbitrarily integral constant satisfies 9β4 − 16α6b4γ2(m4 −m2 + 1)−B12α2r = 0,( 4α3b2γ(m2 + 1)− 3β2 ) ( 4α3b2γ(m2−2)− 3β2 ) = 216Cγ2α4 (4α3b2γ(2m2 − 1) + 3β2) , then the generalized periodic solution in terms of the Jacobi elliptic cosine function is q4(η) = 2αb2m2cn2(bη,m)− 4α3b2γ(2m2 − 1) + 3β2 6α2r . (3.40) When m→ 1, solution (3.40) reduces to the traveling solitary wave solution q5(η) = 2αb2sech2(bη)− 4α3b2γ + 3β2 6α2r . (3.41) Let b = bi with i2 = −1, the periodic solution of traveling wave is given by q6(η) = −2αbsec2(bη) + 4α3b2γ − 3β2 6α2r . (3.42) Substituting (3.37)-(3.42) with p(η) = ∫ q(η)dη into (3.23), we deduce the corre- sponding solution to YTSF equation 3.1 as v5 = 2αbE(sn(b(αx+ βy + γ(z − p(t))),m),m))− p′(t)x− 2 3 p′′(t)y2 + ρ5(αx+ βy + γ(z − p(t))), (3.43) v6 = αb(2 tanh b(αx+ βy + γ(z − p(t))) + ln tanh b(αx+ βy + γ(z − p(t)))− 1 tanh b(αx+ βy + γ(z − p(t))) + 1 ) − p′(t)x− 2 3 p′′(t)y2 + ρ6(αx+ βy + γ(z − p(t))), (3.44) v7 = 2αb tan b(αx+ βy + γ(z − p(t)))− p′(t)x − 2 3 p′′(t)y2 + ρ7(αx+ βy + γ(z − p(t))), (3.45) 12 W. CHEN EJDE-2023/82 v8 = 2αbE(sn(b(αx+ βy + γ(z − p(t))),m),m)− p′(t)x − 2 3 p′′(t)y2 − ρ8(αx+ βy + γ(z − p(t))), (3.46) v9 = 2αb tanh b(αx+ βy + γ(z − p(t)))− p′(t)x − 2 3 p′′(t)y2 − ρ9(αx+ βy + γ(z − p(t))), (3.47) v10 = −2α tan b(αx+ βy + γ(z − p(t)))− p′(t)x − 2 3 p′′(t)y2 + ρ10(αx+ βy + γ(z − p(t))), (3.48) (3.49) where E is an elliptic integral of the second kind, p(t) = ∫ p3(t)d(t), and ρ5 = 4α3b2(γ(m2 + 1)− 3r)− 3β2 6α2r , ρ6 = 8α3b2γ − 3β2 6α2r , ρ7 = 4α3b2(2γ − 3r) + 3β2 6α2r , ρ8 = 4b2α3(m2(2γ − 3r)− γ + 3r) + 3β2 6α2r , ρ9 = 4α3b2γ + 3β2 6α2r , ρ10 = 4α3b2γ − 3β2 6α2r . 3.2.2. Variable separation solution to (3.25). Suppose that (3.25) has a solution in the form f(ξ, x, y) = g(ξ) + h(x, y), (3.50) where g(ξ) and h(x, y) are functions to be undetermined. Substituting (3.50) into (3.25) yields 2hxxgξ + 3hyy = 0, (3.51) from (3.51) we can obtain 2gξ = −3 hyy hxx . (3.52) Take 2gξ = −α. Then (3.52) will reduce into 3hyy − αhxx = 0, where α is an arbitrarily non-zero real number. After finding the results of gξ and h(x, y) in (3.50), we obtain g(ξ) = −1 2 αξ + C1, h(x, y) = r (y√α+ √ 3x√ α ) + s (y√α−√3x√ α ) , (3.53) where r (y√α+√3x√ α ) is an arbitrary smooth function of y √ α+ √ 3x√ α and s (y√α−√3x√ α ) is an arbitrary smooth function of y √ α− √ 3x√ α . Substituting (3.53) into (3.50) reduces to f(x, y, ξ) = −1 2 αξ + C1 + r (y√α+ √ 3x√ α ) + s (y√α−√3x√ α ) . (3.54) Substituting (3.54) into (3.25), we obtain v11 = −p′(t)x− 2 3 p′′(t)y2− 1 2 α(z−p(t)) +C1 + r (y√α+ √ 3x√ α ) + s (y√α−√3x√ α ) . EJDE-2023/82 DYNAMICAL BEHAVIOR FOR YTSF EQUATIONS 13 3.2.3. Two-wave solution to (3.25). We suppose that f(x, y, ξ) = g(η, y), η = kx− cξ. (3.55) Then the second equation in (3.25) is reduced to 3gyy − 3ck2(g2η)η − ck3gηηηη = 0. (3.56) We make the variable transformation g = 2k(lnϕ)η, (3.57) where ϕ = ϕ(η, y) is a function to be undetermined. Substituting (3.57) into (3.56), we obtain the Hirota bilinear form (3D2 yy − ck3D4 η + 2B)(ϕ · ϕ) = 0, (3.58) where B is an integral arbitrary constant. Then we will seek the two-wave solution to (3.58) in the form ϕ = e−p1(λ1η+λ2y+λ3) + δ1 cos(p2(λ4η + λ5y + λ6)) + δ2e p1(λ1η+λ2y+λ3), (3.59) where δi, pi (i = 1, 2) are real constants. If we set the integral constant B = 0, then we obtain p1p2δ1δ2(2ck3λ1λ4(λ21p 2 1 − λ44p22)− 3λ2λ5) = 0, δ1δ2(ck3(λ21p 2 1(λ21p 2 1 − 6λ24p 2 2) + λ44p 4 2)− 3λ22p 2 1 + 3λ25p 2 2) = 0, (δ21p 2 2(4ck3λ44p 2 2 + 3λ25) + 4δ2p 2 1(4ck3λ41p 2 1 − 3λ22)) = 0, δ1p1p2(2ck3λ31λ4p 2 1 − 2ck3λ1λ 3 4p 2 2 − 3λ2λ5) = 0, δ1(ck3(λ41p 4 1 − 6λ21λ 2 4p 2 1p 2 2 + λ44p 4 2)− 3λ22p 2 1 + 3λ25p 2 2) = 0, δ1p1p2(2ck3λ31λ4p 2 1 − 2ck3λ1λ 3 4p 2 2 − 3λ2λ5) = 0, δ1(ck3(λ41p 4 1 − 6λ21λ 2 4p 2 1p 2 2 + λ44p 4 2)− 3λ22p 2 1 + 3λ25p 2 2) = 0. (3.60) Solving (3.60), we find c = 12−2k−3λ−41 p−61 and δ2 = δ21τ, λ2 = √ 3(18 + τ 18 ), λ4 = √ 3λ1p1, λ5 = 1 6p1 , p2 = √ 1− 36p21, (3.61) with τ = 1− (6p1)−2 6= 0. If |p1| > 1 6 , we derive an exact solution to (3.60) as ϕ1 = δ1 cosh(h1( √ 3λ1p1η + 1 6p1 y + λ6)) + 2δ1 √ τ cosh(p1(λ1η + λ2y + λ3) + ω1). (3.62) If |p1| < 1/6, we derive an exact solution to (3.60) as ϕ2 = δ1 cos(h2( √ 3λ1p1η + 1 6p1 y + λ6)) − 2δ1 √ −τ sinh(p1(λ1η + λ2y + λ3) + ω2), (3.63) 14 W. CHEN EJDE-2023/82 where ω1 = 1 2 ln(δ21τ), ω2 = 1 2 ln(−δ21τ), h1 = √ 36p21 − 1, and h2 = √ 1− 36p21, λ2 are given in equation (3.61). Substituting (3.62) and (3.63) into (3.57) we deduce g1 = A (√ 3h1 sinh(h1( √ 3λ1p1η + 1 6p1 y + λ6)) + 2 √ τ sinh(p1(λ1η + λ2y + λ3) + ω1) ) ÷ ( cosh(h1( √ 3λ1p1η + 1 6p1 y + λ6)) + 2 √ τ cosh(p1(λ1η + λ2y + λ3) + ω1) ) , (3.64) g2 = −A (√ 3h2 sin(h2( √ 3λ1p1η + 1 6p1 y + λ6)) + 2 √ −τ cosh(p1(λ1η + λ2y + λ3) + ω2) ) ÷ ( cos(h2( √ 3λ1p1η + 1 6p1 y + λ6)) − 2 √ −τ sinh(p1(λ1η + λ2y + λ3) + ω2) ) , (3.65) where A = 2kλ1p1. Substituting (3.64) and (3.65) into (3.57), we obtain two-wave solution of YTSF equation 3.1 as v12 = −p′(t)x− 2 3 p′′(t)y2 +A √ 3h1 sinh(M) + 2 √ τ sinh(N1) cosh(M)) + 2 √ τ cosh(N1) , (3.66) v13 = −p′(t)x− 2 3 p′′(t)y2 −A √ 3h2 sin(M)) + 2 √ −τ cosh(N2) cos(M))− 2 √ −τ sinh(N2) , (3.67) where M = h1( √ 3λ1p1(kx−c(z−p(t)))+ 1 6p1 y+λ6, N1 = p1(λ1(kx−c(z−p(t)))+ λ2y+ λ3) +ω1, N2p1(λ1(kx− c(z− p(t))) + λ2y+ λ3) +ω2, p(t) = ∫ p3(t)d(t), and p3(t) is an arbitrary function of t. Equations (3.66) and (3.67) are quasi traveling wave solutions with respect to the spatial variable x. Figure 5 demonstrates the evolutionary patterns of the interaction between the kink wave with periodic wave and the kink wave with Gaussian wave described by equation 3.66. Selecting p(t) = 1√ 2π e− 1 2 t 2 in 3.66, we obtain a composite solution with the Gauss wave and the kink wave, as is shown in Figure 5(a). The Gauss wave is combined with the kink wave in the x-direction, and the Gauss wave travel along the t-direction. If we choose p(t) = cos t in 3.66, we obtain a composite solution with the periodic wave and the kinked wave, as is shown in Figure 5(b), in which the periodic wave is combined with the kink wave in the x-direction, and the periodic wave moves along the t-direction. 4. Conclusion In this study, we constructed a series of exact solutions such as the two-wave solution, the periodic solution, and the lump solution by using the Lie symmetry method, the Jacobi elliptic function expansion and the bilinear method. Among them, the lump solution has a non-singular structure and we have simulated its dynamical behaviors with the help of computer graphics (see Figures 1–3). From exact solutions we have found, there are two special interactions: one of which is EJDE-2023/82 DYNAMICAL BEHAVIOR FOR YTSF EQUATIONS 15 Figure 5. Kink wave solution (3.66) with k = 0.8, δ1 = 6 √ 35 35 , λ1 = λ6 = p1 = 1, λ3 = −1. the aggregation effect and elastic collision effect between the lump solution and the soliton wave (see Figure 4), and the other is the interaction between the kink wave with the periodic wave and the kink wave with the Gaussian wave (see Figure 5). Both of them are of great interest in the community of physics and engineering [16]–[23]. Acknowledgments. This work is partially supported by the National Natural Science Foundation of China (Grant No 11771414), by the Ministry of Education of the P.R.C. (N0.2020ITA02019), and by the Longshan Academic Talent Research Supporting Program of SWUST (Grant No 17LZXJ05,17LZXY04). References [1] M. J. Ablowitz, A. I. Nachman; Multidimensional nonlinear evolution equations and inverse scattering, Phys. D., 18 (1986), 223-241. [2] S. J. Chen, Y. H. Yin, W. X. Ma, X. Lü; Abundant exact solutions and interaction phenomena of the (2+1)-dimensional YTSF equation, Anal. Math. Phys., 9 (2019), 2329-2344. [3] Z. D. Dai, J. Liu, D. L. Li; Applications of HTA and EHTA to YTSF Equation, Appl. Math. Comput., 207 (2009), 360-364. [4] E. G. Fan; Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A., 277 (2000), 212-218. [5] M. Foroutan, J. Manafian, A. Ranjbaran; Lump solution and its interaction to (3+1)-D potential-YTSF equation, Nonlinear Dynam., 92 (2018), 2077-2092. 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School of Mathematics and Physics, Southwest University of Science and Technology, Mianyang, Sichuan, 621010, China Email address: chenwei7@mail.ustc.edu.cn 1. Introduction 2. Lump solution and variable soliton for the YTSF equation 2.1. Two-lump solution to the YTSF equation 2.2. Aggregation effect and elastic collision of two-lump wave 2.3. Interaction between two-lump wave and soliton wave 3. Various solutions and dynamical behaivor of the YTSF equation 3.1. Lie symmetry reduction of the YTSF equation 3.1 3.2. Solution to the symmetry reduction equation 4. Conclusion Acknowledgments References