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LIE SYMMETRY ANALYSIS and CONSERVATION LAWS for the (2+1)-DIMENSIONAL MIKHALËV EQUATION 1. Introduction Searching for Solutions

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LIE SYMMETRY ANALYSIS and CONSERVATION LAWS for the (2+1)-DIMENSIONAL MIKHALËV EQUATION 1. Introduction Searching for Solutions Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 41, pp. 1{14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu LIE SYMMETRY ANALYSIS AND CONSERVATION LAWS FOR THE (2+1)-DIMENSIONAL MIKHALEV¨ EQUATION XINYUE LI, YONGLI ZHANG, HUIQUN ZHANG, QIULAN ZHAO Abstract. Lie symmetry analysis is applied to the (2+1)-dimensional Mikhal¨ev equation, which can be reduced to several (1+1)-dimensional partial differen- tial equations with constant coefficients or variable coefficients. Then we con- struct exact explicit solutions for part of the above (1+1)-dimensional partial differential equations. Finally, the conservation laws for the (2+1)-dimensional Mikhal¨evequation are constructed by means of Ibragimov's method. 1. Introduction Searching for solutions to partial differential equations (PDEs), which arise from physics, chemistry, economics and other fields, is one of the most fundamental and significant areas. A wealth of solving methods have been developed, such as the Lie symmetry analysis [5, 8, 11, 15], the homogeneous balance method [13, 18], Hirota's bilinear method [10], the Painlev's analysis method [6]. The Lie symmetry analysis is one of the most effective tools for solving partial differential equations and it was firstly traced back to the famous Norwegian mathematician Sophus Lie [12], who was influenced and inspired by the Galois theory founded in the early 18th century. Bluman and Cole proposed similarity theory for differential equations in 1970s [?]. Subsequently, the scope of application and theoretical depth of Lie symmetry analysis have been expanded. The (2+1)-dimensional Mikhal¨evequation reads [14] uyy + uxt + uxuxy − uyuxx = 0; (1.1) which was first derived by Mikhal¨evin 1992. He described a relationship between Poisson-Lie-Berezin-Kirillov brackets and the Mikhal¨evsystem uy = vx; vy + ut + uvx − vux = 0: (1.2) Pavlov adopts the method of extended Hodograph method to study integrability of exceptional hydrodynamic type systems. The corresponding particular solution of Mikhal¨evsystem [16] is constructed under the condition of three-component case. By constructing new integrable hydrodynamic chains, he describes and integrates all their fluid dynamics, and then extracts new (2+1) integrable hydrodynamic sys- tems from them [17]. Derchyi Wu discussed Cauchy problem of Pavlov's equation and solve the equation by using the backscattering method [19]. Grinevich and 2010 Mathematics Subject Classification. 35Q53, 37K30;,37K40. Key words and phrases. (2+1)-dimensional Mikhal¨evequation; Lie symmetry analysis; similarity reduction; conservation law; exact solution. c 2021 Texas State University. Submitted October 28, 2020. Published May 7, 2021. 1 2 X. Y. LI, H. Q. ZHANG, Y. L. ZHANG, Q. L. ZHAO EJDE-2021/41 Santini investigated nonlocality and the inverse scattering transformation for the Mikhal¨evequation [9]. Dunajski [7] presented a twistor description of (1.2) and demonstrated that the solutions of (1.2) could be used to construct Lorentzian Einstein-Weyl structures in three dimensions. In this paper, we apply Lie sym- metry analysis to the (2+1)-dimensional Mikhal¨evequation to present its exactly explicit solutions and construct its conservation laws. The concept of conserva- tion laws is important in nonlinear science. The famous Noether's theorem [1] provides a systematic and effective way of determining conservation laws for Euler- Lagrange differential equations once their Noether symmetries are known. Later, researchers made various generalizations of Noether's theorem. Among these ex- tended methods, the new conservation theorem, also called nonlocal conservation theorem, introduced by Ibragimov, is one of the most frequently used approaches. In this paper we will apply the Ibragimov's method to construct conservation laws for the (2+1)-dimensional Mikhal¨evequation. The paper is organized as follows. In Section 2, we will apply Lie symmetry analysis to the (2+1)-dimensional Mikhal¨evequation. In Section 3, we will study some exact explicit solutions for the (2+1)-dimensional Mikhal¨evequation based on the similarity reductions. In Section 4, the conservation laws for the (2+1)- dimensional Mikhal¨evequation will be established by using Ibragimov's method. In Section 5, we will give some conclusions and discussions. 2. Lie symmetry analysis for the (2+1)-dimensional Mikhalev¨ equation First of all, let us consider an one-parameter group of infinitesimal transforma- tion, x ! x + εξ(x; y; t; u) + O("2); t ! t + ετ(x; y; t; u) + O("2); (2.1) y ! y + εη(x; y; t; u) + O("2); u ! u + εφ(x; y; t; u) + O("2); where " 1 is a group parameter. The vector field associated with the above group of transformation (2.1) is presented @ @ @ @ V = ξ(x; y; t; u) + η(x; y; t; u) + τ(x; y; t; u) + φ(x; y; t; u) : (2.2) @x @y @t @u Thus, the second prolongation pr(2) V is (2) @ @ @ @ @ @ Pr V = V + φx + φy + φyy + φxt + φxy + φxx ; (2.3) @ux @uy @uyy @uxt @uxy @uxx where y φ = Dy(φ − ξux − ηuy − τut) + ξuxy + ηuyy + τuty; x φ = Dx(φ − ξux − ηuy − τut) + ξuxx + ηuyx + τutx; yy 2 φ = Dy(φ − ξux − ηuy − τut) + ξuxyy + ηuyyy + τutyy; xy (2.4) φ = DyDx(φ − ξux − ηuy − τut) + ξuxxy + ηuxyy + τuxty; xx 2 φ = Dx(φ − ξux − ηuy − τut) + ξuxxx + ηuxxy + τuxxt; xt φ = DtDx(φ − ξux − ηuy − τut) + ξuxxt + ηuxyt + τuxtt; EJDE-2021/41 (2+1)-DIMENSIONAL MIKHALEV¨ EQUATION 3 and the operators Dx;Dy;Dt are the total derivatives with respect to x; y; t respec- tively. The determining equation of (1.1) arises from the invariance condition (2) pr V ∆=0 = 0; (2.5) where ∆ = uyy + uxt + uxuxy − uyuxx = 0. Furthermore, we have yy xt x xy y xx φ + φ + φ uxy + φ ux − φ uxx − φ uy = 0; (2.6) where the coefficient functions φy; φx, φyy, φxy; φxx and φxt are determined in (2.4). Then, the forms of the coefficient functions by calculating the standard symmetry group are obtained 1 1 ξ = (F (t) + 2c )x − F (t)y2 + (−2F (t) + c )y − F (t) + c ; 1t 1 2 1tt 2 2t 2 3 3 η = (F1t(t) + c1)y + F2(t); τ = F1(t); (2.7) 1 1 φ = (F (t) + 3c )u − (F (t)y − c + F (t))x + F (t)y3 + F (t)y2 1t 1 1tt 2 2t 6 1ttt 2 2tt + F3t(t)y + F4(t); where ci (i = 1; 2; 3) are arbitrary constants and Fi(t)(i = 1; 2; 3; 4) are arbitrary functions with regard to t. For convenience, we assume that F1(t) = c4t + c8;F2(t) = c5t + c9;F3(t) = c6t + c10;F4(t) = c7t + c11: (2.8) Therefore, the Lie algebra of infinitesimal symmetries of equation (1.1) is spanned by the vector field @ @ @ 1 @ @ V = 2x + y + 3u ;V = y + x ; 1 @x @y @u 2 2 @x @u @ @ @ @ @ V = ;V = x + y + t + u ; 3 @x 4 @x @y @t @u (2.9) @ @ @ @ @ V = −y + t − x ;V = −t + y ; 5 @x @y @u 6 @x @u @ @ @ @ @ V = t ;V = ;V = ;V = − ;V = : 7 @u 8 @t 9 @y 10 @x 11 @u We apply the Lie bracket [Vi;Vj] = ViVj −VjVi, with the (i; j)-th entry representing [Vi;Vj] to get the commutator table listed in Table 1. Table 1. Lie bracket of equation (1.1) Lie V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V1 0 −V2 −2V3 0 −V5 −2V6 −3V7 0 −V9 −2V10 −3V11 1 1 V2 V2 0 −V11 0 2 V6 V7 0 0 2 V10 V11 0 V3 2V3 V11 0 −V10 −V11 0 0 0 0 0 0 V4 0 0 −V3 0 0 0 −V7 −V8 −V9 −V10 −V11 1 V5 V5 − 2 V6 V11 0 0 0 0 −V9 −V10 −V11 0 V6 2V6 −V7 0 0 0 0 0 −V10 −V11 0 0 V7 3V7 0 0 V7 0 0 0 −V11 0 0 0 V8 0 0 0 V8 V9 V10 V11 0 0 0 0 1 V9 V9 − 2 V10 0 V9 V10 V11 0 0 0 0 0 V10 −2V3 −V11 0 V10 V11 0 0 0 0 0 0 V11 3V3 0 0 V11 0 0 0 0 0 0 0 4 X. Y. LI, H. Q. ZHANG, Y. L. ZHANG, Q. L. ZHAO EJDE-2021/41 Next, using Table 1 and the Lie series 1 Ad(exp("V ))V = V − "[V ;V ] + "2[V ; [V ;V ]] − :::; (2.10) i j j i j 2 i i j where " is a real number and [·; ·] is the Lie bracket. The adjoint representation is shown in Table 2. Table 2. Adjoint representation of equation (1.1). Ad V1 V2 V3 V4 V5 V6 " 2" " 2" V1 V1 V2e V1e V4 V5e V6e " "2 V2 V1 − "V2 V2 V3 + "V11 V4 V5 − 2 V6 + 4 V7 V6 − "V7 V3 V1 − 2"V3 V2 − "V11 V3 V4 + "V10 V5 + "V11 V6 " V4 V1 V2 V3e V4 V5 V6 " V5 V1 − "V5 V2 + 2 V6 V3 − "V11 V4 V5 V6 V6 V1 − 2"V6 V2 + "V7 V3 V4 V5 V6 V7 V1 − 3"V7 V2 V3 V4 − "V7 V5 V6 V8 V1 V2 V3 V4 − "V8 V5 − "V9 V6 + "V3 1 V9 V1 − "V9 V2 − 2 "V3 V3 V4 − "V9 V5 + "V3 V6 − "V11 V10 V1 − 2"V10 V2 + "V11 V3 V4 − "V10 V5 − "V11 V6 −3" V11 V1e V2 V3 V4 − "V11 V5 V6 Ad V7 V8 V9 V10 V11 3" " 2" 3" V1 V7e V8 V9e V10e V11e " "2 V2 V7 V8 V9 − 2 V10 + 4 V11 V10 − "V11 V11 V3 V7 V8 V9 V10 V11 " " " " " V4 V7e V8e V9e V10e V11e "2 "3 "2 V5 V7 V8 + "V9 + 2 V10 + 3! V11 V9 + "V10 + 2 V11 V10 + "V11 V11 V6 V7 V8 + "V10 V9 + "V11 V10 V11 V7 V7 V8 + "V11 V9 V10 V11 V8 V7 − "V11 V8 V9 V10 V11 V9 V7 V8 V9 V10 V11 V10 V7 V8 V9 V10 V11 V11 V7 V8 V9 V10 V11 The one-parameter symmetry groups gi (1 ≤ i ≤ 11) generated by the corre- sponding infinitesimal generators Vi (1 ≤ i ≤ 11) will be obtained 2" " 3" g1 :(x; y; t; u) ! (e x; e y; t; e u); 1 1 g :(x; y; t; u) ! ( y" + x; y; t; y"2 + x" + u); 2 2 4 " " " " g3 :(x; y; t; u) ! (x + "; y; t; u); g4 :(x; y; t; u) ! (e x; e y; e t; e u); "2 "3 "2 (2.11) g :(x; y; t; u) ! (− t − "y + x; "t + y; t; t + y − "x + u); 5 2 6 2 g6 :(x; y; t; u) ! (x − t"; y; t; u + "y); g7 :(x; y; t; u) ! (x; y; t; u + "t); g8 :(x; y; t; u) ! (x; y; t + "; u); g9 :(x; y; t; u) ! (x; y + "; t; u); g10 :(x; y; t; u) ! (−" + x; y; t; u); g11 :(x; y; t; u) ! (x; y; t; u + "); where g3; g9 are space translations, g8 is a time translation, g11 is a dependent variable translation, g4 is a scaling transformation, and g5 is a generalized Galilean transformation.
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