Nonlocal ansätze, reduction and some exact solutions for the system of the van der Waals equations

J. Math. Anal. Appl. 481 (2020) 123442

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Nonlocal ansätze, reduction and some exact solutions for the system of the van der Waals equations M.I. Serov, Yu.V. Prystavka ∗ Department of Higher and Applied Mathematics, Poltava National Technical Yuriy Kondratyuk University, 24, Pershotravnevyi Prospekt, Poltava, 36011, Ukraine

a r t i c l e

i n f o

Article history: Received 3 September 2018 Available online 28 August 2019 Submitted by G. Bluman Keywords: System of the van der Waals equations Nonlocal transformations of equivalence Nonlocal ansatz Exact solutions

a b s t r a c t In this paper two types of nonlocal equivalence transformations of the system of convection-diffusion equations are found. Using these transformations, we obtained two images of the van der Waals system. Lie symmetries of these images are used to construct nonlocal ansätze and reduce that system. Owing to this, it has become possible to have built classes of exact solutions of the van der Waals system. Lie symmetry of this system is used to construct nonlocal ansätze and reduce its both images. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Most mathematical models of physics, biology, chemistry, other natural sciences, as well as economics, financial mathematics, etc., are formulated with the help of differential equations. Therefore, research of special classes of differential equations and construction of their exact solutions is an integral part of solving many practical problems. Exact solutions of differential equations are very important in theoretical and applied researches. They are an effective tool for verifying an adequacy of mathematical models and an effectiveness of approximate methods. Many methods are used for constructing exact solutions of differential equations: the method of separation of variables, the method of special substitutions, the method of variation, Euler’s method, d’Alembert’s method, the method of characteristics (Monge’s method), the cascade (Laplace’s) method, Poisson method, the inverse scattering method and many others. Lie method [25] has a special place among the methods of solving differential equations because a significant number of these methods explicitly or implicitly rely on symmetric properties of corresponding differential equations. According to this method, differential equations with partial derivatives, which have a classical Lie symmetry, can be reduced to ordinary differential equations by using special substitutions * Corresponding author. E-mail addresses: [email protected] (M.I. Serov), [email protected] (Yu.V. Prystavka). https://doi.org/10.1016/j.jmaa.2019.123442 0022-247X/© 2019 Elsevier Inc. All rights reserved.

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M.I. Serov, Yu.V. Prystavka / J. Math. Anal. Appl. 481 (2020) 123442

(ansätze). Using solutions of reduced equations makes it possible to find exact solutions of an initial differential equation with partial derivatives. However, despite all the advantages of the classical Lie approach of finding solutions of differential equations and their systems, the class of solutions of equations that can be constructed within the framework of this method is limited by the number of symmetry operators of a certain equation or system. If an equation has a poor Lie symmetry or no symmetry at all, then application of this method to constructing solutions does not produce any desired result. These and some other problems became a reason to search for new approaches to constructing exact solutions of nonlinear differential equations. The basis of a number of such approaches is an idea of finding additional (non-Lie) symmetry operators. According to this idea, W. Fuschych and A. Nikitin [13] developed a new method to research invariant algebras of differential equations – non-Lie method to study symmetric properties of differential equations with partial derivatives. This method differs from Lie method by basic elements of invariant algebras of corresponding differential equations, which usually are integro-differential (pseudo-differential) operators. One of interesting methods of searching for solutions of differential equations is the method of additional conditions, which was described in [32], [33], [40], [9]. In papers [4] and [15] methods of researching nonclassical and disturbed symmetries were proposed. They were later developed into a conditional symmetry method (see [11], [19], [14], [16], [17], [12]). This method has been widely used in works [9], [41], [7], etc. Using this method allowed to obtain the additional symmetry that greatly expanded a possibility of finding classes of solutions of differential equations. Another direction of finding non-Lie symmetries is the method of nonlocal symmetries of differential equations. In paper [27] E. Noether suggested finding nonlocal transformations of invariance of differential equations. Her ideas were developed in other works. Thus, e.g., in works [2] and [3] G. Bluman and S. Kumei suggested algorithms for finding such transformations and the method of linearization of nonlinear differential equations with partial derivatives with the help of nonlocal transformations. Studying symmetric properties of a certain class of equations, it is important to know equivalence transformations of the given class. By means of transformations of equivalence, classes of differential equations can be divided into non-equivalent subclasses, marking out canonical equations in each of these subclasses. It is enough to investigate only canonical representatives of each subclass to conclude on symmetric properties of all equations of that class. L. Ovsiannikov ([29]) was the first scientist, who started to work in this area. The method of searching equivalence transformations was considered in papers by P. Olver [28], I. Akhatov, R. Gazizov and N. Ibragimov [1], V. Lahno, S. Spichak and V. Stognii [24]. This idea was developed in I. Lisle’s paper [26], in which he used equivalence transformations to study a number of specific nonlinear equations of mathematical physics. In papers by G. Bluman, A. Cheviakov and S. Anco [6] some results of researching nonlocal connections of systems of partial differential equations were suggested. A. Cheviakov in paper [10] proposed a practical algorithm of computation of Lie groups of point equivalence transformations and generalized equivalence transformations of families of differential equations, together with its symbolic implementation in GeM package for Maple. In the papers by M. Hashemi, M. Nucci [20], R. Popovych, O. Vaneeva and N. Ivanova [31], this method was used for studying nonlinear differential equations with partial derivatives. In paper [8] a complete group classification of a nonlinear reaction-convection-diffusion equation was carried out through equivalence transformations. Papers [34] and [5] give nonlocal transformations, which reduce nonlinear diffusion equation ut = ∂(u−2 ux ) to linear equation zt = zxx . In paper [23] those transformations were generalized. Besides, it was shown that using the transformations a nonlinear diffusion equation ut = ∂x [f (u)ux ], where u = u(t, x), ut = ∂u ∂t , ux = equation of the same class.

∂u ∂x ,

∂x =

∂ ∂x ,

(1)

f (u) — an arbitrary smooth function, is reduced to an

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In paper [18] the same transformations are used to construct nonlocal ansätze, which reduce Eq. (1) to ordinary differential equations, to linearize Eq. (1) and construct nonlocal formulae for generating of its solutions. The problem of generalizing the results of papers [23] and [18] in the case of a system of nonlinear diffusion equations was set and solved in papers [36] and [35]: Ut = ∂x [f (U )Ux ], 

u1 u2

where U =



 , f (U ) =

f 11 f 21

f 12 f 22

 , ua = ua (t, x), f ab = f ab (U ) are arbitrary smooth functions,

a, b = 1, 2. In paper [39] nonlocal equivalence transformations are used to extend classes of solutions of nonlinear convection-diffusion equations in the form ut = ∂x [f (u)ux + g(u)], where g(u) is an arbitrary smooth function. In paper [38] the maximal algebra of invariance was studied and some solutions of the system of van der Waals equations were found. This system belongs to the class of convection-diffusion systems. In this paper we generalize results of works [23], [18], [36], [35], [39] for a system of convection-diffusion equations: Ut = ∂x [F (U )Ux ] + K(U )Ux ,  where U =

u1 u2



 , F (U ) =

f 11 f 21

f 12 f 22



 , K(U ) =

k11 k21

k12 k22

(2)

 , ua = ua (t, x), t is a time variable, x

is a space variable; f ab = f ab (U ), kab = kab (U ) are coefficients of diffusion and convection respectively, a, b = 1, 2. This system is used as a model for describing various processes in mathematical physics, chemistry and biology. In the class of systems (2) there are systems that are widely used in the theory of processes of heat and mass transfer and diffusion, and describe an evolution of temperature and density in thermonuclear plasma. It describes a fluid movement in a porous medium, an energy transfer in plasma, a substances distribution in a soil and a lot of other physical and biochemical processes. Let us show how nonlocal equivalence transformations can be used for constructing nonlocal ansätze, reducing and getting exact “non-Lie” solutions of van der Waals equations, which belong to class (2). 2. Nonlocal equivalence transformations of the system of convection-diffusion equations Let the matrix K(U ) be such that its components satisfy the condition ku112 = ku121 ,

ku212 = ku221 .

(3)

Then there are such functions g 1 and g 2 that system (2) gets a form Ut = ∂x [F (U )Ux + G(U )],  where G(U ) =

 g 1 (U ) . g 2 (U )

(4)

M.I. Serov, Yu.V. Prystavka / J. Math. Anal. Appl. 481 (2020) 123442

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Let us consider a set of three transformations (see [36,35]): t = t,

ua = vxa ,

x = x,

(5)

where v a = v a (t, x) are new unknown functions, x = w1 ,

t = x0 ,

v 1 = x1 ,

v2 = w2 ,

(6)

where x0 , x1 are new independent variables, wa = wa (x0 , x1 ) are new dependent variables, x 0 = x0 ,

x1 = x1 ,

z a = z a (x0 , x1 ) are new dependent variables, w1a =

w1a = z a ,

(7)

∂wa ∂x1 .

Theorem 1. Transformations (5)–(7) are equivalence transformations of system (4), that is, they reduce it to the form (8)

Z0 = ∂1 [Φ(Z)Z1 + Ψ(Z)],  where Z =

z1 z2



 , Zμ =

∂Z ∂xμ ,

∂1 =

∂ ∂x1 ,

Φ(Z) =

ϕ11 ϕ21

ϕ12 ϕ22





, ϕab = ϕab (Z), Ψ(Z) =

ψ1 ψ2

 , ψa =

ψ a (Z), μ = 0, 1, and arbitrary elements ϕab , ψ a and f ab , g a are connected by the following interrelations: ϕ11 = (z 1 )−2 (f 11 + z 2 f 12 ), ϕ12 = −(z 1 )−1 f 12 , ϕ12 = (z 1 )−3 [(f 11 + z 2 f 12 )z 2 − (f 21 + z 2 f 22 )], ϕ22 = (z 1 )−2 (f 22 + z 2 f 12 ), ψ 1 = −z 1 g 1 , ψ 2 = −z 2 g 1 + g 2 , 2

(9)

2

where f ab = f ab ( z11 , zz1 ), g a = g a ( z11 , zz1 ). Proof. Let us apply a nonlocal transformation in form (5) to system (4). Substituting (5) into (4) and integrating the resulting system with respect to variable x, we obtain Ut = F (Vx )Vxx + G(Vx ),

(10)



 v1 where V = . v2 If we apply hodograph transformations (6) to system (10), then this system is reduced to the form w01 = w02 =

1 [(f 11 (w11 )2 1 [(f 11 (w11 )3

1 2 + w12 f 12 )w11 − w11 f 12 w11 ] − w12 g 1 ,

a

2

1 + w12 f 12 )w12 − (f 12 + w12 f 22 ]w11 + a

1 (f 22 (w11 )2

2 + w12 f 12 )w11 − w12 g 1 + g 2 , w2

w2

(11)

∂ w a ab where wμa = ∂w = f ab ( w11 , w11 ), g a = g a ( w11 , w11 ), a, b = 1, 2. ∂xμ , w11 = ∂x21 , μ = 0, 1, and in formulae (11) f 1 1 1 1 Differentiating system (11) with respect to variable x1 and applying transformations (7), we obtain the following system (8), and functions ϕab , ψ a are connected with functions f ab , g a by the interrelations (9). Thus, we have ascertained that the chain of substitutions (5)–(7) reduces system (4) to system of equations (8) of the same class. It is not difficult to make sure that system (8) with the above mentioned replacements is reduced to system (4). Theorem 1 is proved. 2

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Since system (4) contains two functions u1 and u2 , then hodograph transformation (6) is not the only one possible for this system. So, if in (6) we replace v 1 → v 2 , v 2 → v 1 , w1 → w2 , w2 → w1 , then instead of (6) we obtain t = x0 ,

x = w2 ,

v1 = w1 ,

v 2 = x1 .

(12)

Theorem 2. Transformations (5), (12) and (7) are equivalence transformations of system (4), that is, they reduce it to the form (8), and arbitrary elements ϕab , ψ a and f ab , g a are connected by the following interrelations: ϕ11 = (z 2 )−2 (f 11 − z 1 f 21 ), ϕ12 = (z 2 )−3 [−(z 1 f 11 + f 12 ) + z 1 (z 1 f 21 + f 22 )], ϕ21 = −(z 2 )−1 f 21 , ϕ22 = (z 2 )−2 (z 1 f 21 + f 22 ), ψ 1 = g 1 − z 1 g 2 , ψ 2 = −z 2 g 2 , 1

(13)

1

where f ab = f ab ( zz2 , z12 ), ϕab = ϕab (z 1 , z 2 ), g a = g a ( zz2 , z12 ), ψ a = ψ a (Z). Proof. By following the arguments similar to those suggested in the proof of Theorem 1, we conclude that the chain of substitutions (5), (12) and (7) also reduces system (4) to system of equations (8) of the same class. Vice versa, system (8) by the above mentioned replacements is reduced to system (4), and functions ϕab , ψ a are connected with functions f ab , g a by the relations (13). Theorem 2 is proved. 2 Let us call transformations (5)–(7) the transformations of the first type for system (4) and denote them by P1 , and transformations (5), (12) and (7) as the transformations of the second type and denote them by P2 . Remark. For the whole class of Eqs. (4), transformations P1 and P2 are equivalent, but for a particular system of form (4), these transformations give different results. Let us illustrate this fact using a concrete example. 3. The system of the van der Waals equations and its nonlocal images. Lie symmetries, Lie ansätze Let us consider the system of van der Waals equations u1t = λ1 u1xx − u1 u1x + μu2 u2x , u2t = λ2 u2xx − u1 u2x − u2 u1x ,

(14)

where x = (x0 , x1 ), ua = ua (x); λ1 is a coefficient of kinematic viscosity, λ2 is a diffusion coefficient, μ is a convection coefficient, a ∈ {1, 2}. This system is widely used in the molecular-kinetic theory of gases and liquids (see, for example, [21]). This system belongs to the class of systems of convection-diffusion equations. Its coefficients satisfy condition (3) and can be reduced to the form of system (4). If we apply transformation P1 to system (14), then we obtain the following system  Z0 = ∂1

1 (z 1 )3



0 λ1 z 1 (λ1 − λ2 ) z 2 λ2 z 1

 Z1 −



1 2

2 (z 1 )

which we call as the first image of system (14) and denote it by O1 .

μ(z 1 )3 − z 1 μ(z 2 )3 + z 2

 ,

(15)

M.I. Serov, Yu.V. Prystavka / J. Math. Anal. Appl. 481 (2020) 123442

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If we apply transformation P2 to system (14), then we get the following system  Z0 = ∂1

1 (z 2 )3



λ1 z 2 (λ2 − λ1 ) z 1 0 λ2 z 2

 Z1 +



1 2

2 (z 2 )

(z 1 )2 + μ 2z 1 z 2

 ,

(16)

which we call as the second image of system (14) and denote it by O2 . Let us set a problem to apply nonlocal equivalence transformations of the first and second types to find nonlocal ansätze, reduce and construct exact solutions for the van der Waals equations and their images O1 and O2 . In paper [37] it was established that system (14) is invariant under a generalized Galilean algebra ∂ AG2 (1, 1) =< ∂t , ∂x , G = t∂x + ∂u1 , D = 2t∂t + x∂x − I, Π = t2 ∂t + tx∂x − tI + x∂u1 >, where ∂t = ∂t , ∂ ∂ ∂ 1 2 ∂x = ∂x , ∂u1 = ∂u1 , ∂u2 = ∂u2 , I = u ∂u1 + u ∂u2 . If we make a complete analysis of its symmetric properties, then we receive the following statement. Theorem 3. The maximal algebra of the invariance of system (14) with respect to the relations between the constants is the following algebras 1) 2) 3) 4)

AG2 (1, 1), if λ1 = λ2 , μ = 0; < AG2 (1, 1), u2 ∂u2 >, if λ1 = λ2 , μ = 0; < AG2 (1, 1), u1 ∂u1 >, if λ1 = λ2 = 1, μ = 0; < AG2 (1, 1), u2 ∂u1 , u2 ∂u2 >, if λ1 = λ2 = 1, μ = 0.

Proof. Let us study the symmetry of system (14) by the Lie method (see, for example, [25], [28], [29]). We obtain the infinitesimal operator of the algebra of invariance in the form X = ξ μ (x, U )∂μ + η a (x, U )∂ua ,

(17)

where ξ μ , η a are unknown functions, μ = 0, 1. On the condition that system (14) is invariant with respect to operator (17), we obtain a system of defining equations for coordinates ξ μ and η a of operator (17): ξ10 = ξu0a = ξu1a = ηuab uc = 0, ξt0 = 2ξx1 , (λ1 − λ2 )ηu1 2 = 0, (λ1 − λ2 )ηu2 1 = 0, 1 2 η 1 = −ξ11 u1 − ξ01 − λ1 ηxu 1 − λ2 ηxu2 , 2 2 1 1 2 2 η = (ηu2 − ηu1 − ξ1 )u − 2λ2 ηxu1 , 2 − ηt2 = 0, ηx2 u1 + ηx1 u2 + λ2 ηxx 2 2 1 1 1 μηx u + ηx u + λ1 ηxx − ηt1 = 0, 1 2 μηu2 1 u2 − λλ12 ηu1 2 u2 + λ1 ηxu 1 − λ2 ηxu2 = 0, 1 μη 2 + μ(ηu2 2 − ηu1 1 + ξ11 )u2 + 2λ1 ηxu 2 = 0.

(18)

The solution to system (18) is functions 1) ξ 0 = c1 x20 + 2κx0 + d0 , ξ 1 = (c1 x0 + κ)x1 + gx0 + d1 , η 1 = −(c1 x0 + κ)x1 u1 − x1 − g, η 2 = (c1 x0 + κ)x1 u2 , if λ1 = λ2 , μ = 0; 2) ξ 0 = c1 x20 +2κx0 +d0 , ξ 1 = (c1 x0 +κ)x1 +gx0 +d1 , η 1 = −(c1 x0 +κ)x1 u1 −x1 −g, η 2 = (c1 x0 +κ +c2 )x1 u2 , if λ1 = λ2 , μ = 0; 3) ξ 0 = c1 x20 +2κx0 +d0 , ξ 1 = (c1 x0 +κ)x1 +gx0 +d1 , η 1 = −(c1 x0 +κ +c2 )x1 u1 −x1 −g, η 2 = (c1 x0 +κ)x1 u2 , if λ1 = λ2 = 1, μ = 0;

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4) ξ 0 = c1 x20 + 2κx0 + d0 , ξ 1 = (c1 x0 + κ)x1 + gx0 + d1 , η 1 = −(c1 x0 + κ)x1 u1 + c2 u2 − x1 − g, η 2 = (c1 x0 + κ + c3 )x1 u2 , if λ1 = λ2 = 1, μ = 0. For each case, the relevant algebra of invariance is obtained by a standard method. Theorem 3 is proved. 2 Let us study the maximal algebra of invariance of images (15) and (16). The following statement is true. Theorem 4. The maximal algebra of invariance of system (15) is the algebra 1) Abas =< ∂0 , ∂1 , D = 2x0 ∂0 + z 1 ∂z1 >, if μ = 0, λ1 = λ2 ; 2) A =< Abas , z 2 ∂z2 >, if μ = 0, λ1 = λ2 ; x1 3) A =< Abas , z 2 ∂z2 , e 2 ∂z2 >, if μ = 0, λ1 = λ2 = 1, where ∂0 =

∂ ∂x0 ,

∂1 =

∂ ∂x1 ,

∂z1 =

∂ ∂z 1 ,

∂z2 =

∂ ∂z 2 .

Theorem 5. The maximal algebra of invariance of system (16) is the algebra 1) 2) 3) 4) 5)

Abas =< ∂0 , ∂1 , D1 = 2x0 ∂0 + z 2 ∂z2 , Q = z 2 ∂z1 >, if μ = 0, λ1 = λ2 ; A =< Abas , D2 = x1 ∂1 − I >, if μ = 0, λ1 = λ2 ; A =< Abas , D2 = x1 ∂1 − I, K = x21 ∂1 − 2x1 I + 2∂z1 >, if μ = 0, λ1 = λ2 = 1; A =< Abas , Q1 = ex1 (∂1 + ∂z1 − I), Q2 = e−x1 (∂1 + ∂z1 + I) >, if μ = −1, λ1 = λ2 = 1; A =< Abas , Q1 = cos x1 (∂1 − ∂z1 ) + sin x1 I, Q2 = sin x1 (∂1 − ∂z1 ) − cos x1 I >, if μ = 1, λ1 = λ2 = 1,

where ∂0 =

∂ ∂x0 ,

∂1 =

∂ ∂x1 ,

∂z1 =

∂ ∂z 1 ,

∂z2 =

∂ ∂z 2 ,

I = z 1 ∂z1 + z 2 ∂z2 .

Theorems 4, 5 are proved in the same way as Theorem 3. Let us use the Lie symmetry of system (14) to construct invariant ansätze. The solution to system (14) will be sought in the form (see [19], [36], [29]) Ja = ϕa (ω),

a = 1, 2,

where ϕa = ϕa (ω) are arbitrary smooth functions, ω, J1 , J2 are the first integrals of the system of ordinary differential equations: dx du1 du2 dt = 1 = 1 = 2 = dτ. 0 ξ ξ η η The coordinates of the infinitesimal operator for the case μ = 0, λ1 = λ2 are given by the formulae ξ 0 = c5 t2 + 2c4 t + c1 , ξ 1 = (c5 t + c4 )x + c3 t + c2 , η 1 = −(c5 t + c4 )u1 + c5 x + c3 , η 2 = −(c5 t + c4 )u2 , where c1 , c2 , c3 , c4 , c5 are group parameters. System (19) has the form dx du1 du2 dt = = = = dτ, c5 t2 + 2c4 t + c1 (c5 t + c4 )x + c3 t + c2 −(c5 t + c4 )u1 + c5 x + c3 −(c5 t + c4 )u2

(19)

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M.I. Serov, Yu.V. Prystavka / J. Math. Anal. Appl. 481 (2020) 123442

or t˙ = c5 t2 + 2c4 t + c1 , x˙ = (c5 t + c4 )x + c3 t + c2 , u˙ 1 = −(c5 t + c4 )u1 + c5 x + c3 , u˙ 2 = −(c5 t + c4 )u2 .

(20)

Before we solve system of equations (20), let us reduce it to a simpler form by transformations, which are generated by algebra AG2 (1, 1): t t = e2θ4 1−θ + θ1 , 5t x t + θ3 e2θ4 1−θ + θ2 , x = eθ4 1−θ 5t 5t 

u1 = e−θ4 [(1 − θ5 t)u1 + θ5 x],

(21)



u2 = e−θ4 (1 − θ5 t)u2 , where θi are arbitrary constants. Lemma. Transformations (21) are the equivalence transformations of system of equations (20), that is, they reduce it to the form t˙ = C5 t2 + 2C4 t + C1 , x˙ = (C5 t + C4 )x + C3 t + C2 , u˙ 1 = −(C5 t + C4 )u1 + C5 x + C3 , u˙ 2 = −(C5 t + C4 )u2 ,

(22)

and arbitrary elements c1 , c2 , c3 , c4 , c5 and C1 , C2 , C3 , C4 , C5 are connected by the following interrelations: C1 = (c5 θ12 + 2c4 θ1 + c1 )e−2θ4 , C2 = −θ3 C1 + (c5 θ1 θ2 + c4 θ2 + c3 θ1 + c2 )e−θ4 , C3 = θ3 θ4 C1 + (c5 θ2 + c3 )eθ4 − θ3 (c5 θ1 + c4 ) − θ5 (c5 θ1 θ2 + c4 θ2 + c3 θ1 + c2 )e−θ4 , C4 = −θ5 C1 + c5 θ1 + c4 , C5 = θ52 C1 − 2θ5 (c5 θ1 + c4 ) + c5 e2θ4 . The lemma is proved by a direct substitution of transformations (22) into system (21). Let us analyze if we can choose parameters θi to reduce to zero some constants Ci . By elementary transformations, constants C1 and C5 are reduced to the form c4 2 1 ) + (c1 c5 − c24 )], c5 c5 c4 θ5 + θ5 (θ1 + ))2 + (c1 c5 − c24 )]. c5 c5

C1 = c5 e−2θ4 [(θ1 + C5 = c5 e−2θ4 [(e2θ4

In the case when c1 c5 − c24 > 0 it is impossible to select the parameters θi to reduce to zero constants C1 and C5 . It is easy to make sure that in this case the parameters θi can be chosen so that C2 = C3 = C4 = 0, C1 = C5 = 1. In the case when c1 c5 − c24 ≥ 0 the parameters θi can be chosen so that C5 = 0. Let us consider these cases separately.

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In the first case, we write system (20) as follows t˙ = t2 + 1, x˙ = tx, u˙ 1 = tu1 , u˙ 2 = tu2 . Solving this system, we get the form of an invariant ansatz u1 = (t2 + 1)− 2 (ψ 1 (ω) + tω), 1

u2 = (t2 + 1)− 2 ψ 2 (ω), 1

ω = (t2 + 1)− 2 x. 1

In the second case, for the construction of invariant ansätze of system (14) it is necessary to integrate the system of ordinary differential equations of the form t˙ = 2c4 t + c1 , x˙ = c4 x + c3 t + c2 , u˙ 1 = −c4 u1 + c3 , u˙ 2 = −c4 u2 .

(23)

To facilitate integration of the system, we reduce it to a simpler form. This can be done by applying transformations that are generated by AG1 (1, 1): t = e2θ4 t + θ1 , x = eθ4 x + θ3 e2θ4 t + θ2 ,  u1 = e−θ4 u1 + θ3 ,  u2 = e−θ4 u2 .

(24)

Applying transformation (24) to system (23), we reduce it to the form t˙ = 2C4 t + C1 , x˙ = C4 x + C3 t + C2 , u˙ 1 = −C4 u1 + C3 , u˙ 2 = −C4 u2 , where C1 C2 C3 C4

= (2c4 θ1 + c1 )e−2θ4 , = (c4 θ2 + c3 θ1 − 2c4 θ1 θ3 − c1 θ3 + c2 )e−2θ4 , = (−c4 θ3 + c3 )eθ4 , = c4 .

(25)

After analyzing possibility of choosing parameters θi in such a way as to reduce to zero some of constants (25), we come to the following nonequivalent cases 2.1) C4 = 1, C1 = C2 = C3 = 0; 2.2) C4 = C2 = 0, C3 = 1, C1 − ∀; 2.3) C4 = C3 = 0, C1 , C2 − ∀.

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Solving system (23) for each of the above cases, we find the corresponding ansätze u1 = t− 2 ψ 1 (ω), u2 = t− 2 ψ 2 (ω), ω = t− 2 x; u1 = ψ 1 (ω) − 2kt, u2 = ψ 2 (ω), ω = x + kt2 ; u1 = ψ 1 (ω), u2 = ψ 2 (ω), ω = kt + mx. 1

1

1

Making the considerations, the same as for the system of the van der Waals equations, we have the nonequivalent Lie ansätze for system (15): for the case μ = 0, λ1 = λ2 z 1 = ψ 1 (ω), z 2 = ψ 2 (ω), ω = x1 + mx0 ; √ z 1 = x0 ψ 1 (ω), z 2 = ψ 2 (ω), ω = x1 + m ln x0 ;

(26)

z 1 = ψ 1 (ω), z 2 = ex0 ψ 2 (ω), ω = x1 + mx0 ; √ z 1 = x0 ψ 1 (ω), z 2 = x0 ψ 2 (ω), ω = x1 + m ln x0 ;

(28)

(27)

in the case μ = 0, λ1 = λ2

(29)

in the case μ = 0, λ1 = λ2 = 1 z 1 = ψ 1 (ω), z 2 = ex0 ψ 2 (ω), ω = x1 + mx0 ; √ z 1 = x0 ψ 1 (ω), z 2 = x0 ψ 2 (ω), ω = x1 + m ln x0 . x1 2

z 1 = ψ 1 (ω), z 2 = ψ 2 (ω) + e , ω = x1 + 2mx0 ; x1 √ z 1 = x0 ψ 1 (ω), z 2 = ψ 2 (ω) + e 2 , ω = x1 + 2m ln x0 .

(30) (31) (32) (33)

Using the Lie symmetry, we obtain the following nonequivalent Lie ansätze for system (16). We will give only those ansätze, which by transformations (5), (12) and (7) are converted into nonlocal ansätze of system (14). When μ = 0, λ1 = λ2 = 1 z1 =

ψ 1 (ω)+2x1 , x21 +1

z1 =

z2 =

1

ψ (ω)+2x1 , x21 +1

√

2

x0 ψx2 (ω) , +1 1

z2 =

2

ψ (ω) , x21 +1

ω = ln x0 + m arctan x1 ;

(34)

ω = x0 + m arctan x1 ;

(35)

when μ = −1, λ1 = λ2 = 1 z1 =

ψ 1 (ω)+σ  , σ

z2 =

ψ 2 (ω) σ ,

ω = x0 + k arctan mex1 ;

(36)

z1 =

ψ 1 (ω)+σ  , σ

z2 =

ψ 2 (ω) σ ,

ω = x0 − k arctan nex1 ;

(37)

z 1 = ex1 ψ 1 (ω) − 1, z1 =

ψ 1 (ω)+σ  , σ

z2 =

z1 =

ψ 1 (ω)+σ  , σ

z2 =

z 1 = ex1 ψ 1 (ω) − 1, where σ = c1 ex1 + c2 e−x1 ;

√

z 2 = ex1 ψ 2 (ω), 2

x0 ψ σ(ω) ,

ω = x0 + lex1 ;

(38)

ω = ln x0 + k arctan mex1 ;

(39)

x0 ψ σ(ω) , ω = ln x0 − k arctan nex1 ; √ z 2 = x0 ex1 ψ 2 (ω), ω = ln x0 + lex1 ,

(40)

√

2

(41)

M.I. Serov, Yu.V. Prystavka / J. Math. Anal. Appl. 481 (2020) 123442

11

when μ = 1, λ1 = λ2 = 1 z1 = z1 =

ψ 1 (ω)+cos x1 , sin x1

1

ψ (ω)+cos x1 , sin x1

2

z 2 = ψsin(ω) ω = x0 + ln tan x21 , x1 , 2 √ z 2 = x0 ψsin(ω) ω = ln x0 + ln tan x21 , x1 ,

(42) (43)

where k, l, m are arbitrary constants. 4. Nonlocal ansätze and reduction of the system of the van der Waals equations and its images To find the nonlocal ansätze of system (14), we apply the composition of nonlocal transformations (5)–(7)) to the found Lie ansätze of system (15) and (16). Let us illustrate the process of finding nonlocal ansätze using the example of the transformation of ansatz (32), which we shall rewrite one more time z 1 = ψ 1 (ω),

z 2 = ψ 2 (ω) + e

x1 2

,

(44)

ω = x1 + 2mx0 .

Applying transformation (5) to (44), first, we obtain v 1 = Ψ1 (ω),

v 2 = Ψ2 (ω) + 2e

x1 2

,

ω = x1 + 2mx0 ,

where Ψ1 , Ψ2 are initial for functions ψ 1 , ψ 2 . After hodograph transformation (6) this ansatz gets the form x = Ψ1 (ω),

w2 = Ψ2 (ω) + 2e

w1 2

,

ω = w1 + mt.

By interchanging invariant variables Ψ1 and ω, and, then introducing a substitution Ψ1 = 2 ln ϕ1 ,

Ψ2 = Φ2 ,

we obtain the following ansatz w1 = 2 ln ϕ1 (ω) − 2mt,

w2 = Φ2 (ω) + 2e−mt ϕ1 (ω),

ω = x.

(45)

Differentiating ansatz (45) by variable x and using transformation (7), we have 1

˙ (ω) u1 = 2 ϕ ϕ1 (ω) ,

u2 = ϕ2 (ω) + 2e−mt ϕ˙ 1 (ω),

(46)

ω = x,

where ϕ2 (ω) = Φ˙ 2 (ω). Similarly, we get the other images of the Lie ansätze of system (15). As a result, we obtain that ansätze (26)–(31) will transform into the Lie ansätze for system (14), and ansätze (32)–(33) will transform into nonlocal ansätze (46) and u1 =

1 2 ϕ˙ (ω) √ , t ϕ1 (ω)

u2 =

1 √ [ϕ2 (ω) t

+ 2t−m ϕ˙ 1 (ω)],

ω=

x √ . t

(47)

We apply the composition of nonlocal transformations (5), (12) and (7) to the already found Lie ansätze of system (16). As a result, we obtain that the ansätze (34)–(43) will transform into the following nonlocal ansätze: for the case: μ = 0, λ1 = λ2 = 1 u1 =

1 √ [ϕ1 (ω) t

+ 2ϕ˙ 2 (ω) tan α],

u1 = ϕ1 (ω) + 2ϕ˙ 2 (ω) tan α,

u2 =

1 √ ϕ˙ 2 (ω) cos−2 t

u2 = ϕ˙ 2 (ω) cos−2 α,

α,

ω= ω = x,

x √ ; t

(48) (49)

M.I. Serov, Yu.V. Prystavka / J. Math. Anal. Appl. 481 (2020) 123442

12

where α = ϕ2 (ω) + q ln t; in the case μ = 1, λ1 = λ2 = 1 2

u1 = ϕ1 (ω) − 2 ϕ

(ω)ϕ˙ 2 (ω)) , A

2

u2 = 2et ϕ˙

(ω)) A ,

(50)

ω = x,

where A = (ϕ2 )2 + e2t ; u1 =

1 √ [ϕ1 (ω) t

2

− 2ϕ

(ω)ϕ˙ 2 (ω) ], B

√ 2 u2 = 2 t ϕ˙ B(ω) ,

ω=

x √ , t

(51)

where B = (ϕ2 )2 + t2 ; in the case μ = −1, λ1 = λ2 = 1 u1 = ϕ1 (ω) − ϕ˙ 2 (ω) cot β, u1 = ϕ1 (ω) − ϕ˙ 2 (ω) coth β,

u2 = ϕ˙ 2 (ω) sin−1 β, u2 = ϕ˙ 2 (ω) sinh

−1

(52)

ω = x;

β,

(53)

ω = x,

where β = ϕ2 (ω) + qt; u1 = ϕ1 (ω) − u1 = u1 =

u1 =

u2 =

ϕ˙ 2 (ω) ϕ2 (ω)+t ,

u2 =

1 √ ϕ˙ 2 (ω) sin−1 t

− ϕ˙ 2 (ω) coth α],

u2 =

1 √ ϕ˙ 2 (ω) sinh−1 t

1 √ [ϕ1 (ω) t

−

ϕ˙ 2 (ω) ϕ2 (ω)+t ],

u2 =

(54)

ω = x;

− ϕ˙ 2 (ω) cot α],

1 √ [ϕ1 (ω) t

1 √ [ϕ1 (ω) t

ϕ˙ 2 (ω) ϕ2 (ω)+t ,

2 1 ϕ˙ (ω) √ , t ϕ2 (ω)+t

α, α,

ω=

ω=

x √ ; t

ω= x √ . t

x √ ; t

(55) (56) (57)

Substituting ansätze (46) and (47) into system (14), we obtain the following reduced systems of the equations: in the case μ = 0, λ1 = λ2 = 1 1 2

˙ ) ϕ¨1 = 2 (ϕϕ − mϕ1 , 1 1

1

˙ ϕ ¨ ϕ¨2 = 2 ϕ ˙ 2 + 2( ϕ 1 − ϕ1 ϕ

(ϕ˙ 1 )2 2 (ϕ1 )2 )ϕ ;

(58)

1

˙ ω ϕ¨1 = (2 ϕ ˙ 1 − mϕ1 , ϕ1 − 2 )ϕ 1

˙ ω ϕ¨2 = (2 ϕ ˙2 + ϕ1 − 2 )(ϕ

ϕ˙ 1 2 ϕ1 ϕ )

− (2m + 12 )ϕ2 ;

(59)

ϕ¨1 = ϕ1 ϕ˙ 1 + 4ϕ˙ 2 ϕ¨2 − 12 (ω ϕ˙ 1 + ϕ1 ), ϕ¨2 = (ϕ1 − 12 ω)ϕ˙ 2 ;

(60)

ϕ¨1 = ϕ1 ϕ˙ 1 + 4ϕ˙ 2 ϕ¨2 , ϕ¨2 = ϕ1 ϕ˙ 2 ;

(61)

ϕ¨1 = ϕ1 ϕ˙ 1 , ϕ¨2 = ϕ1 ϕ˙ 2 − ϕ2 ;

(62)

ϕ¨1 = [ϕ1 − 12 (ϕ1 − ω)]ϕ˙ 1 , ϕ¨2 = (ϕ1 − 12 ω)ϕ˙ 2 − 12 ϕ2 ;

(63)

ϕ¨1 = ϕ1 ϕ˙ 1 − ϕ˙ 2 ϕ¨2 , ϕ¨2 = ϕ1 ϕ˙ 2 + q;

(64)

in the case μ = 1, λ1 = λ2 = 1

in the case μ = −1, λ1 = λ2 = 1

M.I. Serov, Yu.V. Prystavka / J. Math. Anal. Appl. 481 (2020) 123442

13

ϕ¨1 = ϕ1 ϕ˙ 1 + ϕ˙ 2 ϕ¨2 , ϕ¨2 = ϕ1 ϕ˙ 2 + q;

(65)

ϕ¨1 = ϕ1 ϕ˙ 1 , ϕ¨2 = ϕ1 ϕ˙ 2 + 1;

(66)

ϕ¨1 = −ϕ˙ 2 ϕ¨2 + (ϕ1 − 12 ω)ϕ˙ 1 − 12 ϕ1 , ϕ¨2 = (ϕ1 − 12 ω)ϕ˙ 2 + q;

(67)

ϕ¨1 = ϕ˙ 2 ϕ¨2 + (ϕ1 − 12 ω)ϕ˙ 1 − 12 ϕ1 , ϕ¨2 = (ϕ1 − 12 ω)ϕ˙ 2 + q;

(68)

ϕ¨1 = (ϕ1 − 12 ω)ϕ˙ 1 − 12 ϕ1 , ϕ¨2 = (ϕ1 − 12 ω)ϕ˙ 2 .

(69)

The problem of finding nonlocal ansätze and reducing can be solved for images O1 and O2 as well by using the Lie ansätze of the system of the van der Waals equations and transformations P1 and P2 . To find nonlocal ansätze of the first image of system of the van der Waals equations we apply the composition of nonlocal transformations (5)–(7) to the already found Lie ansätze of system (14). As a result, we obtain that the part of the ansätze of system (14) will transform into the Lie ansätze for system (15), and the other ansätze will transform into the following nonlocal ansätze 1 ϕ1 (ω)−2kx0 ,  x20 +1 , z 1 = ϕ1 (ω)+x 0ω

z1 =

z2 = z2 =

ϕ2 (ω) ϕ1 (ω)−2kx0 , 2

ϕ (ω) ϕ1 (ω)+x0 ω ,

ω = τ + kx20 , ω=

 τ , x20 +1

τ1 = z 1 ; τ1 = z 1 .

(70) (71)

Similarly, we obtain a nonlocal ansatz for system (16): z 1 = ϕ1 (ω) + x0 ϕ2 (ω)ϕ˙ 2 (ω),

z2 =



x20 + 1ϕ˙ 2 (ω),

ω = x1 .

(72)

Substituting ansätze (70) and (71) into system (15), we will obtain the following reduced systems of equations: λ1 ϕ¨1 − ϕ1 ϕ˙ 1 + μϕ2 ϕ˙ 2 + 2k = 0, λ2 ϕ¨2 − ϕ1 ϕ˙ 2 − ϕ˙ 1 ϕ2 = 0;

(73)

λ1 ϕ¨1 − ϕ1 ϕ˙ 1 + μϕ2 ϕ˙ 2 − ω = 0, λ2 ϕ¨2 − ϕ1 ϕ˙ 2 − ϕ˙ 1 ϕ2 = 0.

(74)

Substituting ansatz (72) into system (16), we obtain the following reduced system of equations: 1 (2λ1 ϕ˙ 1 − λ2 −2λ (ϕ1 )2 + μ)ϕ¨2 − (λ1 ϕ¨1 − λ2 −(ϕ˙ 2 )4 ϕ2 = 0, λ2 ϕ¨2 + ϕ1 ϕ˙ 2 = 0.

λ2 −2λ1 1 1 ϕ ϕ˙ )ϕ˙ 2 − λ2

(75)

5. Exact solutions to the system of the van der Waals equations If we solve the reduced systems (58)–(69) and use the corresponding ansätze, then we can construct exact solutions to the system of the van der Waals equations. Here are some of them. The solutions of reduced system (58) are the functions ϕ1 = cω1 , ϕ2 = c2 ω +

c3 ω2 ,

(76)

M.I. Serov, Yu.V. Prystavka / J. Math. Anal. Appl. 481 (2020) 123442

14

c1 ϕ1 = cos ω, 2 ϕ = −2 ln cos ω + c2 ω + c3 ,

(77)

c1 ϕ1 = cosh ω, 2 ϕ = −2 ln cosh ω + c2 ω + c3 ,

(78)

where c1 , c2 , c3 are arbitrary constants. Using ansatz (46), multiplied by transformations t → t,

x → x + θt,

u1 → u1 − θ,

u 2 → u2 ,

(79)

and solutions (76)–(78), we find solutions of system (14): 2 u1 = − x+θt + θ,

u2 = c2 (x + θt) +

(80)

c3 −2c1 e−mt (x+θt)2 ;

u1 = 2 tan(x + θt) + θ,

(81)

u2 = −2 ln cos(x + θt) + 2c1 e−mt tan(x+θt) cos(x+θt) + c2 (x + θt) + c3 ; u1 = −2 tanh(x + θt) + θ,

(82)

u2 = −2 ln cosh(x + θt) − 2c1 e−mt tanh(x+θt) cosh(x+θt) + c2 (x + θt) + c3 . The solutions of reduced system (62) are the following functions (see [22], [30]) ϕ1 = − ω2 , ϕ2 = cosω ω ;

(83)

ϕ1 = tan ω2 ,√ ϕ2 =

c1 cos

5 2 ω+c2 cos ω 2

√

sin

ω ϕ1 = − tanh , √ 2

ϕ2 =

c1 cos

3 2 ω+c2 cosh ω 2

5 2

(84)

;

√

sin

3 2

(85)

,

where c1 , c2 are arbitrary constants. Using ansatz (50), which was generated by Galilean transformations (79) and solutions (83)–(85), we find solutions of system (14): u1 = u2 =

cos(x+θt)[(x+θt) sin(x+θt)+cos(x+θt)] 2 x+θt [ cos2 (x+θt)+e2t (x+θt)2 t (x+θt) sin(x+θt)+cos(x+θt) 2e cos2 (x+θt)+e2t (x+θt)2 ;

u1 = tan x+θt 2 − u2 = et

5 2 (x

x+θt C[2Cx cos x+θt 2 +C sin 2 ] C 2 +e2t cos2 x+θt 2

1

5 2 (x

+ θ, (87)

x+θt 2 ]

+ θ, (88)

,

√

+ θt) + c2 sin

(86)

;

D[2Dx cosh x+θt 2 −D sinh D 2 +e2t coth2 x+θt 2

cosh x+θt 2 x+θt 2Dx cosh x+θt 2 −D sin 2 D 2 +e2t cosh2 x+θt 2

√

where C = c1 cos

x+θt 2

x+θt 2Cx cos x+θt 2 +C sin 2 C 2 +e2t cos2 x+θt 2

u1 = − tan x+θt 2 − u2 = et

1 cos

− 1] + θ,

√

+ θt), D = c1 cos

3 2 (x

√

+ θt) + c2 sin

3 2 (x

+ θt).

M.I. Serov, Yu.V. Prystavka / J. Math. Anal. Appl. 481 (2020) 123442

15

The solution of reduced system (66) is the functions ϕ1 = − ω2 , ϕ2 = cω1 + c2 +

(89)

ω2 6 ,

where c1 , c2 are arbitrary constants. Using ansatz (54), which was generated by transformations (79) and solution (89), we find solutions of system (14): 2 u1 = − x+θt −

2(x+θt)3 −c1 (x+θt)4 +(6t+c2 )(x+θt)2 +c1 (x+θt)

+ θ,

2(x+θt) −c1 . u2 = − (x+θt)4 +(6t+c 2 2 )(x+θt) +c1 (x+θt) 3

(90)

Firstly, we would like to know if it is possible to get solutions (80), (81), (82), (86), (87), (88), (90), using only a certain Lie symmetry of system (14). The conditions of invariance of solution Φa (t, x, u1 , u2 ) = 0 with respect to algebra A =< X1 , X2 , ..., Xn > has the form: XΦa |Φ1 =0

Φ2 =0

= 0,

(91)

where X = ck Xk (k = 1, n), ck – are arbitrary constants, Xk – are basic generators of algebra A. We check the fulfillment of conditions (91) for solution (86) at θ = 0, which can be represented in the form x Φ1 ≡ u1 + M M = 0, K 2 2 = 0, Φ ≡ u − 2et M

(92)

where K = x sin x + cos x, M = cos2 x + e2t x2 . The infinitesimal operator of algebra of invariance < AG2 (1, 1), u1 ∂u1 > has the form X = c1 ∂t + c2 ∂x + c3 G + c4 D + c5 Π + c6 u1 ∂u1 = = (c5 t2 + 2c4 t + c1 )∂t + (c5 tx + c4 x + c3 t + c2 )∂x + +[(−c5 t + c6 − c4 )u1 + c5 x + c3 ]∂u1 − (c5 t + c4 )u2 ∂u2 .

(93)

Substituting (92) and (93) into (91), we get: (c5 t − c6 + c4 )M Mx + (c5 x + c3 )M 2 + (c5 t2 + 2c4 t + c1 )(M Mtx − −Mt Mx ) + (c5 tx + c4 x + c3 t + c2 )(M Mxx − Mx2 ) = 0, (c5 t + c4 )KM + (c5 t2 + 2c4 t + c1 )(2 cos2 x − M )K+ +(c5 tx + c4 x + c3 t + c2 )[x cos xM + 2K(sin x cos x− − x1 (M − cos2 x))] = 0.

(94)

Since equality (94) must be performed at arbitrary values of variables t and x, then, making equal to zero coefficients of various functions of these variables, we get c1 = c2 = c3 = c4 = c5 = c6 = 0. This means that the solution (86) does not satisfy conditions (92) for any Lie invariance operator of system (14), and, consequently, it is a “non-Lie” solution. Also it can be shown that solutions (80), (81), (82), (87), (88), (90) are “non-Lie” solutions too. The obtained solutions of the system of the van der Waals equations can be studied in terms of a possibility of their physical application. Here is an example of solution (86). It is easy to see that when t → ∞ or x → ∞ then u1 → θ, u2 → 0. This suggests that this solution can be physically applied.

16

M.I. Serov, Yu.V. Prystavka / J. Math. Anal. Appl. 481 (2020) 123442

Solving reduced systems (73)–(75) and using the corresponding ansätze, we can find exact solutions of both images of the system of the van der Waals equations. Since systems (15) and (16) are not sufficiently studied in terms of physical application and it is not known if they describe specific physical processes, we do not provide their solutions. Presence of nonlocal ansätze means presence of nonlocal symmetries of this system. Each nonlocal anzatz corresponds to a nonlocal operator. Let us find nonlocal operators that correspond to nonlocal ansatz of van der Waals equations. Firstly we represent operators, which generate Lie ansätze of the first and second images: X1 = 2∂0 − 2m∂1 − me

x1 2

(95)

∂z2 ;

X2 = 4m∂1 − 2x0 ∂0 − z ∂z1 + 2me 1

x1 2

(96)

∂z2 ;

X3 = km∂0 − m2 ex1 (∂1 + ∂z1 − z 1 ∂z1 − z 2 ∂z2 )− −e−x1 (∂1 + ∂z1 + z 1 ∂z1 + z 2 ∂z2 );

(97)

X4 = kn∂0 + n2 ex1 (∂1 + ∂z1 − z 1 ∂z1 − z 2 ∂z2 )− −e−x1 (∂1 + ∂z1 + z 1 ∂z1 + z 2 ∂z2 );

(98)

X5 = l∂0 − e−x1 (∂1 + ∂z1 + z 1 ∂z1 + z 2 ∂z2 );

(99)

X6 = km(2x0 ∂0 + z 2 ∂z2 ) − 2m2 ex1 (∂1 + ∂z1 − z 1 ∂z1 − z 2 ∂z2 )− −2e−x1 (∂1 + ∂z1 + z 1 ∂z1 + z 2 ∂z2 );

(100)

X7 = kn(2x0 ∂0 + z 2 ∂z2 ) − 2n2 ex1 (∂1 + ∂z1 − z 1 ∂z1 − z 2 ∂z2 )+ +2e−x1 (∂1 + ∂z1 + z 1 ∂z1 + z 2 ∂z2 );

(101)

X8 = l(2x0 ∂0 + z 2 ∂z2 ) + 2ex1 (∂1 + ∂z1 − z 1 ∂z1 − z 2 ∂z2 );

(102)

X9 = ∂0 − sin x1 (∂1 − ∂z1 ) + cos x1 (z 1 ∂z1 + z 2 ∂z2 );

(103)

X10 = 2x0 ∂0 + z ∂z2 − 2 sin x1 (∂1 − ∂z1 ) + 2 cos x1 (z ∂z1 + z ∂z2 );

(104)

X11 = m∂0 − ∂1 − (x21 ∂1 − 2x1 (z 1 ∂z1 + z 2 ∂z2 ) + 2∂z1 );

(105)

X12 = 2∂1 − m(2x0 ∂0 + z 2 ∂z2 ) + 2x21 ∂1 − 4x1 (z 1 ∂z1 + +z 2 ∂z2 ) + 4∂z1 .

(106)

2

1

2

Operators (95)–(96) generate ansätze (32)–(33), and operators (97)–(106) generate ansätze (34)–(43) respectively. We obtain following nonlocal operators for van der Waals system, acting by transformations P1 on operators (95)–(96) and transformations P2 on operators (97)–(106): Q1 = ∂t −

1

m v2 2e

u1 ∂u2 ;

(107)

Q2 = 2t∂t + x∂x − u1 ∂u1 − u2 ∂u2 − 2me

v1 2

u1 ∂u2 ;

(108)

Q3 = km∂t + (m2 ev + e−v )u2 ∂u1 + (m2 ev − e−v )u2 ∂u2 ;

(109)

Q4 = kn∂t − (n2 ev − e−v )u2 ∂u1 − (n2 ev + e−v )u2 ∂u2 ;

(110)

−v 2 2

(111)

2

2

2

2

2

Q5 = l∂t − e

2

2

2

u (∂u1 − ∂u2 );

Q6 = 2kmt∂t + kmx∂x + [kmu1 + 2(m2 ev + e−v )u2 ]∂u1 + 2 2 +[kmu2 + 2(m2 ev − e−v )u2 ]∂u2 ;

(112)

Q7 = 2knt∂t + knx∂x − [knu1 + 2(n2 ev − e−v )u2 ]∂u1 − 2 2 −[knu2 + 2(n2 ev + e−v )u2 ]∂u2 ;

(113)

2

2

2

2

2

Q8 = 2l∂t + lx∂x − (lu1 − 2ev u2 )(∂u1 + ∂u2 );

(114)

M.I. Serov, Yu.V. Prystavka / J. Math. Anal. Appl. 481 (2020) 123442

Q9 = ∂t + u2 sin v 2 ∂u1 − u2 cos v 2 ∂u2 ;

17

(115)

Q10 = 2t∂t + x∂x − (u − 2u sin v )∂u1 − (u + 2u cos v ∂u2 );

(116)

Q11 = m∂t − 2u2 ∂u1 − 2v 2 u2 ∂u2 ;

(117)

Q12 = 2t∂t + mx∂x − (mu1 + 4u2 )∂u1 − (mu2 + 4v 2 u2 )∂u2 ,

(118)

1

2

2

1

2

2

  where v 1 = u1 dx, v 2 = u2 dx. Operators (107)–(108) generate ansätze (46)–(47), and operators (109)–(118) generate ansätze (48)–(57) respectively. With the help of operators (95) and (97) we will provide obtaining nonlocal operators (107)–(118) from Lie operators (95)–(106) by using transformations P1 and P2 . Operator (95) corresponds to the next system of differential equations 2z01 − 2mz11 = 0, x1 2z02 − 2mz11 + me 2 = 0.

(119)

Firstly, acting by transformation (7) on system (119) and then integrating with respect to variable x1 , we get 2w01 − 2mw11 = 0, x1 2w02 − 2mw11 + 2me 2 = 0.

(120)

Acting by a hodograph transformation (6) on system (120) and using formulae v1

w01 = − vt1 , x

w02 = vt2 −

2 vt1 vx 1 , vx

w11 =

1 1, vx

w12 =

2 vx 1, vx

we get vt1 + m = 0, vt2 + me

v1 2

(121)

= 0.

Differentiating system (121) with respect to x and using transformation (5), we get u1t = 0, u2t +

1

m v2 2e

(122)

u1 = 0.

Operator (107) corresponds to system (122). Consequently, we can see that Lie operator (95) transfers into nonlocal operator (107). Operator (97) is corresponded to the system kmz01 − m2 ex1 (z11 + z 1 − 1) − e−x1 (z11 − z 1 − 1) = 0, kmz02 − m2 ex1 (z12 + z 2 ) − e−x1 (z12 − z 2 ) = 0.

(123)

Acting by transformation (123) on system (7) and then integrating with respect to variable x1 , we get kmw01 − m2 ex1 (w11 − 1) − e−x1 (w11 + 1) = 0, kmw02 − m2 ex1 w12 − e−x1 w12 = 0. Acting by a hodograph transformation (12) on this system and using formulae w01 = vt1 −

1 vt2 vx 2 , vx

v2

w02 = − vt2 , x

w11 =

1 vx 2, vx

w12 =

1 2, vx

M.I. Serov, Yu.V. Prystavka / J. Math. Anal. Appl. 481 (2020) 123442

18

we get kmvt1 + m2 ev − e−v = 0, 2 2 kmvt2 + m2 ev + e−v = 0. 2

2

(124)

Differentiating system (124) with respect to x and using transformation (5), we get kmu1t + (m2 ev − e−v )u2 = 0, 2 2 kmu2t + (m2 ev + e−v )u2 = 0. 2

2

(125)

System (125) is corresponded to operator (109). Finally we can see that Lie operator (97) transfers into nonlocal operator (109). We can also show how Lie operators (96), (98)–(106) transfer into nonlocal ones. Consequently, we showed that coordinates ξ μ and η a of Lie operators (95)–(106) depend on variables t, x, z 1 , z 2 , and coordinates of nonlocal operators (107)–(118) depend on t, x, u1 , u2 , v 1 , v 2 , where v 1 =  1  u dx, v 2 = u2 dx. 6. Conclusions In this paper, nonlocal equivalence transformations of the first and second types are used to find images of the system of the van der Waals equations. The symmetric properties of the first and second images of system (14) are used for constructing nonlocal ansätze, reducing and finding some exact solutions to this system. Nonlocal equivalence transformations were used to find nonlocal ansätze and to reduce both images of the system of the van der Waals equations. By using Lie ansätze of the system of the van der Waals equations and nonlocal equivalence transformations P1 and P2 , nonlocal ansätze were constructed and images O1 and O2 were reduced. It is shown that nonlocal operators of invariance of a given system correspond to nonlocal ansätze of van der Waals equations. References [1] I.S. Akhatov, R.K. Gazizov, N.K. Ibragimov, Nonlocal symmetries. Heuristic approach, J. Sov. Math. 55 (1) (1991) 1401–1450, https://doi.org/10.1007/BF01097533. [2] G. Bluman, S. Kumei, Symmetry-based algorithms to relate partial differential equations. I. Local symmetries, European J. Appl. Math. 1 (1990) 189–216. [3] G. Bluman, S. Kumei, Symmetry-based algorithms to relate partial differential equations. II. Linearization by nonlocal symmetries, European J. Appl. Math. 1 (1990) 217–223. [4] G.W. Bluman, J.D. Cole, The general similarity solution of the heat equation, J. Math. Mech. 18 (1968/69) 1025–1042. [5] G.W. Bluman, G.J. Reid, S. Kumei, New classes of symmetries for partial differential equations, J. Math. Phys. 29 (4) (1988) 806–811, https://doi.org/10.1063/1.527974. [6] G.W. Bluman, A.F. Cheviakov, S.C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Applied Mathematical Sciences, vol. 168, Springer, New York, 2010. [7] V.M. Boyko, R.O. Popovych, Conditional symmetries of the linear rod equation, Dopov. Akad. Nauk Ukr. 9 (2013) 7–15 (in Ukrainian). [8] R. Cherniga, M. Serov, I. Rassokha, Lie symmetries and form-preserving transformation of reaction-diffusion-convection equation, J. Math. Anal. Appl. 342 (2) (2008) 1363–1379. [9] R. Cherniha, M. Serov, O. Pliukhin, Nonlinear Reaction-Diffusion-Convection Equations: Lie and Conditional Symmetry, Exact Solutions and Their Applications, Chapman & Hall/CRC Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, Florida, 2018, https://www.taylorfrancis.com/books/9781498776196. [10] A.F. Cheviakov, Symbolic computation of equivalence transformations and parameter reduction for nonlinear physical models, Comput. Phys. Commun. 220 (2017) 56–73. [11] W.I. Fushchych, Conditional symmetry of equations of nonlinear mathematical physics, Ukraïn. Mat. Zh. 43 (11) (1991) 1456–1470 (in Russian). [12] W.I. Fushchych, V.¯I. Chopyk, Conditional symmetry and new images of Galilean algebra for nonlinear parabolic equations, Ukraïn. Mat. Zh. 45 (10) (1993) 1433–1443 (in Ukrainian). [13] W.I. Fushchych, A.G. Nikitin, Symmetry of the Maxwell’s Equations, Naukova Dumka, Kyiv, 1983 (in Russian).

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