Group analysis of a class of nonlinear Kolmogorov equations

Group analysis of a class of nonlinear Kolmogorov equations

Olena Vaneeva Olena Vaneeva Institute of Mathematics of the National Academy of Sciences of Ukraine, 3 Tereshchenkivska Str., Kyiv-4, 01601 Ukraine 1Yuri Karadzhov Institute of Mathematics of the National Academy of Sciences of Ukraine, 3 Tereshchenkivska Str., Kyiv-4, 01601 Ukraine 2Christodoulos Sophocleous Department of Mathematics and Statistics, University of Cyprus, Nicosia CY 1678, Cyprus 3    Yuri Karadzhov and Christodoulos Sophocleous Olena Vaneeva Institute of Mathematics of the National Academy of Sciences of Ukraine, 3 Tereshchenkivska Str., Kyiv-4, 01601 Ukraine 1Yuri Karadzhov Institute of Mathematics of the National Academy of Sciences of Ukraine, 3 Tereshchenkivska Str., Kyiv-4, 01601 Ukraine 2Christodoulos Sophocleous Department of Mathematics and Statistics, University of Cyprus, Nicosia CY 1678, Cyprus 3
2email: vaneeva@imath.kiev.ua
6email: christod@ucy.ac.cym
Abstract

A class of -dimensional diffusion-convection equations (nonlinear Kolmogorov equations) with time-dependent coefficients is studied with Lie symmetry point of view. The complete group classification is achieved using a gauging of arbitrary elements (i.e. via reducing the number of variable coefficients) with the application of equivalence transformations. Two possible gaugings are discussed in order to show how equivalence groups serve in making the optimal choice.

1 Introduction

Second-order partial differential equations of the form

 ut=Duyy+ν[K(u)]x, (1)

where and are nonzero constants, and is a smooth nonlinear function of the dependent variable , appear in various applications. In particular, they describe diffusion-convection processes Escobedo , model an interaction of particles with two kinds of particles on a lattice Alexander , arise in mathematical finance, when studying agents’ decisions under risk Citti ; Pascucci . Equations (1) are called in the literature diffusion-advection equations, nonlinear ultraparabolic equations and nonlinear Kolmogorov equations. They were studied from various points of view. An important study of partial differential equations and especially nonlinear ones is finding Lie groups of point transformations that leave an equation under study invariant. Such symmetry transformations allow one to apply powerful, and what is most important, algorithmic methods for finding exact solutions of a given nonlinear equation. Moreover, Lie symmetries can serve as a selection criterion of physically important models among possible ones FN . Lie symmetries of equations (1) and the corresponding group invariant solutions were classified by Demetriou et al Demetriou . There are also studies on Lie symmetries of linear Kolmogorov equations Kov1 ; Kov2 and of constant coefficient nonlinear Kolmogorov equations of the form  Ser .

An attempt of group classification of a class of nonlinear Kolmogorov equations more general than (1), namely, such equations with time dependent coefficients,

 ut=f(t)uyy−g(t)[K(u)]x,fgKuu≠0, (2)

was recently made Kumar2014 . Here and are smooth nonvanishing functions of the variable and is a smooth nonlinear function of Nevertheless the complete classification of Lie symmetries of class (2) was not achieved in Kumar2014 , in particular, the case was missed and dimensions of maximal Lie symmetry algebras as well as some of their basis elements for the other cases of extensions were presented incorrectly. The case that is important for applications was not studied with Lie symmetry point of view at all.

In this paper we perform the complete group classification of equations (2). As class (2) is parameterized by three arbitrary elements, , and , the group classification problem appears to be too complicated to be solved completely without modern approaches based on the usage of point equivalence transformations. One of such tools is the gauging of arbitrary elements by equivalence transformations (i.e., reducing of a class to a subclass with fewer number of arbitrary elements). To use this technique, we firstly compute the equivalence group of class (2) in Section 2. A gauging of arbitrary elements is performed in the same section. In Section 3 Lie symmetries of the simplified class are exhaustively classified. In Section 4 we discuss how to choose an optimal gauging among possible ones. To illustrate that the chosen gauging is optimal, we also present results on group classification of class (2) carried out for an alternative gauging.

2 Equivalence transformations

Equivalence transformations are nondegenerate point transformations, that preserve the differential structure of the class under study, change only its arbitrary elements and form a group. There are several kinds of equivalence groups. The usual equivalence group, used for solving group classification problems since the late 50’s, consists of the nondegenerate point transformations of the independent and dependent variables and of the arbitrary elements of the class, where transformations for independent and dependent variables do not involve arbitrary elements of the class Ovsiannikov1982 . The notion of the generalized equivalence group, where transformations of variables of given DEs explicitly depend on arbitrary elements, appeared in the middle 90’s Meleshko1994 ; Meleshko1996 . The transformations from the extended equivalence group include nonlocalities with respect to arbitrary elements mogran . The generalized extended equivalence group possesses the properties of both generalized and extended equivalence groups. The group classification problems become simpler for solving if one use the widest possible equivalence group. Advantages of the usage of the generalized extended equivalence group in comparison with the usual one were shown, in particular, in VKS2015 . In some cases the usage of generalized extended equivalence groups is the only way to present the complete group classification, see, e.g., VPS2012 .

Equivalence transformations generate a subset of a set of admissible transformations which can be interpreted as triples, each of which consists of two fixed equations from a class and a point transformation that links these two equations popo2010a ; popo2012 . In this paper we restrict ourselves to the study of equivalence transformations.

To find the equivalence transformations we use the direct method Kingston&Sophocleous1998 . The details of calculations are skipped for brevity. As it is more convenient for the study of Lie symmetries to consider the equivalent form of the above class,

 ut=f(t)uyy−g(t)k(u)ux,fgku≠0, (3)

we present transformations for both and in the theorems below.

Theorem 2.1

The generalized extended equivalence group  of class (2) (resp. (3)) is formed by the transformations

 ~t=T(t),~x=δ1x+δ2∫g(t)dt+δ3,~y=δ4y+δ5,~u=δ6u+δ7, ~f(~t)=δ42Ttf(t),~g(~t)=ε1Ttg(t), ~K(~u)=δ6ε1(δ1K(u)+δ2u+ε2),(% resp.~k(~u)=1ε1(δ1k(u)+δ2),)

where and are arbitrary constants with , is an arbitrary smooth function with

The usual equivalence group of class (2) (resp. (3)) consists of the above transformations with

The group contains a subgroup of gauge equivalence transformations, i.e. the transformations that change only arbitrary elements while the independent and dependent variables remain unchanged popo2010a . This subgroup is formed by the transformations , , (resp. ). It is more convenient to consider class (3) than class (2) as in this case the dimension of the gauge equivalence subgroup reduces.

It appears that the subclass of equations (2) with quadratic in (resp. (3) with linear in ) admits a wider equivalence group. Up to the -equivalence we can consider the case (resp. ).

Theorem 2.2

The generalized extended equivalence group  of the class

 ut=f(t)uyy−g(t)uux,fg≠0, (4)

comprises the transformations

 ~t=T(t),~x=X(t)x+δ3∫g(t)X(t)2dt+δ4,~y=δ1y+δ2, ~u=δ5(uX(t)−δ6x+δ3),~f(~t)=δ21δ5Ttf(t),~g(~t)=X(t)2δ5Ttg(t),

where are arbitrary constants with , and is an arbitrary smooth function with

The usual equivalence group of class (4) consists of the above transformations with

As there is one arbitrary function, , in the transformations from the group , we can set one of the arbitrary elements or of the initial class equals to a nonzero constant value. We choose to perform the gauging by using the transformation

 ~t=∫g(t)dt,~x=x,~u=u. (5)

Then, any equation from class (2) (resp. (3)) is mapped to an equation from its subclass that is singled out by the condition . The detailed discussion on optimal choice of gauging is presented in Section 4. Without loss of generality, we can restrict ourselves to the study of class (2) with or, what is more convenient, its equivalent form

 ut=f(t)uyy−k(u)ux,fku≠0, (6)

since all results on symmetries, conservation laws, classical solutions and other related objects can be found for equations (3) using the similar results derived for (6).

The generalized extended equivalence groups of class (6) and its subclass with coincide with the usual equivalence groups of these classes.

Theorem 2.3

The usual equivalence group  of class (6) consists of the transformations

 ~t=ε1t+ε0,~x=δ1x+δ2t+δ3,~y=δ4y+δ5,~u=δ6u+δ7, ~f(~t)=δ42ε1f(t),~k(~u)=1ε1(δ1k(u)+δ2),

where , and are arbitrary constants with .

Theorem 2.4

The usual equivalence group  of the class

 ut=f(t)uyy−uux,f≠0, (7)

is formed by the transformations

 ~t=αt+βγt+δ,~x=κx+μt+νγt+δ,~y=λy+ε, ~u=1Δ(κ(γt+δ)u−κγx+δμ−γν),~f(~t)=λ2Δ(γt+δ)2f(t),

where and are arbitrary constants defined up to a nonzero multiplier with , ; and are arbitrary constants, .

Theorem 4 implies that any equation (7) with , where and are constants, is mapped by a point transformation to a constant-coefficient equation from the same class.

We also present equivalence transformations for the subclass of class (3) singled out by the condition , which we will use for the comparison of the cases and in Section 4.

Theorem 2.5

The generalized extended equivalence group  of the class

 ut=uyy−g(t)k(u)ux,gku≠0, (8)

comprises the transformations

 ~t=δ24t+δ0,~x=δ1x+δ2∫g(t)dt+δ3,~y=δ4y+δ5,~u=δ6u+δ7, ~g(~t)=ε1δ24g(t),~k(~u)=1ε1(δ1k(u)+δ2),

where and are arbitrary constants with .

Theorem 2.6

The generalized extended equivalence group  of the class

 ut=uyy−g(t)uux,g≠0, (9)

consists of the transformations

 ~t=δ21t+δ2,~x=x+δ4γ1∫g(t)dt+γ2+δ5,~y=δ1y+δ3, ~u=δ6((γ1∫g(t)dt+γ2)u−γ1(x+δ4)),~g(~t)=g(t)δ21δ6(γ1∫g(t)dt+γ2)2,

where and are arbitrary constants with .

3 Lie symmetries

The group classification problem for class (3) up to -equivalence reduces to the similar problem for class (6) up to -equivalence (resp. the group classification problem for class (4) up to -equivalence reduces to such a problem for class (7) up to -equivalence).

To solve the group classification problem for class (6) we use the classical approach based on integration of determining equations implied by the infinitesimal invariance criterion Ovsiannikov1982 . We search for symmetry operators of the form generating one-parameter Lie groups of transformations that leave equations (6) invariant Olver1986 ; Ovsiannikov1982 . It is required that the action of the second prolongation of the operator  on (6) vanishes identically modulo equation (6),

 Q(2){ut−f(t)uyy+k(u)ux}|ut=f(t)uyy−k(u)ux=0. (10)

The infinitesimal invariance criterion (10) implies the determining equations, simplest of which result in

 τ=τ(t),ξ=ξ(t,x),η=η1(t)y+η0(t),θ=φ(t,x,y)u+ψ(t,x,y),

where , , , , and are arbitrary smooth functions of their variables. Then the rest of the determining equations are

 τft=(2η1−τt)f,2fφy=−η1ty−η0t, (11) (φu+ψ)ku+(τt−ξx)k=ξt, (12) (φxu+ψx)k+(φt−fφyy)u+ψt−fψyy=0. (13)

Firstly we integrate equations (12) and (13) for up to the -equivalence taking into account that . The method of furcate split Nikitin&Popovych2001 ; Ivanova&Popovych&Sophocleous2006I is further used. For any operator equation (12) gives equations on of the general form

 (au+b)ku+ck=d, (14)

where and are constants. The number of such independent equations is not greater than two, otherwise they form incompatible system for If , then (14) is not an equation on but an identity, this corresponds to the case of arbitrary . If , then the integration of (14) up to the -equivalence gives three different cases: (i) , (ii) (iii) If then the function is linear in ,

The determining equation (13) implies that there exist two essentially different cases of classification: I. , and II. , i.e.

Consider firstly the case of arbitrary function . In this case equations (12) and (13) should be split with respect to and . The splitting results in the equations Therefore . As , the second equation of (11) implies i.e. and Here are arbitrary constants. Then the general form of the infinitesimal generator is and the first equation of (11) takes the form

 (c1t+c2)ft=(2c4−c1)f. (15)

This is the classifying equation for If is an arbitrary nonvanishing smooth function, then the latter equation should be split with respect to and its derivative, which results in Therefore, the kernel of the maximal Lie invariance algebras of equations from class (6) is (Case 1 of Table 1). To perform the further classification we integrate equation (15) up to the -equivalence. All -inequivalent values of that provide Lie symmetry extensions for equations from class (6) with arbitrary are exhausted by the following values: The corresponding bases of maximal Lie invariance algebras are presented by Cases 2–4 of Table 1.

If , then splitting equations (12) and (13) with respect to different powers of  leads to the system , These equations together with (11) imply , , where , are arbitrary constants. The classifying equation for takes the form (15). Therefore, the cases of Lie symmetry extensions are given by the same forms of as in previous case, namely, arbitrary, power, exponential and constant. See Cases 5–8 of Table 1. The dimensions of the respective Lie symmetry algebras increase by one in comparing with the case of arbitrary The highest dimension is five, not six as it was stated in the paper by Kumar et al Kumar2014 .

The consideration of the cases and is rather similar to the case of with therefore, we omit the details of calculations. The classification results are presented in Cases 9–16 of Table 1.

Consider the case of linear , then up to the equivalence we can assume . We substitute to equations (12) and (13) and further split them with respect to different powers of . This leads to the system and We differentiate the first and the second equation of this system with respect to the variable  and get the additional conditions Then also and the second equation of (11) gives The general form of the infinitesimal operator is where are arbitrary constants. The classifying equation for is

 (c2t2+c1t+c0)ft=(2c6−c1−2c2t)f. (16)

If this is not an equation on but an identity, then Therefore, the constants appearing in the infinitesimal generator are arbitrary and the maximal Lie invariance algebra of the equations (7) with arbitrary is the four-dimensional algebra (Case 1 of Table 2).

The further group classification of equations (6) with , i.e. equations (7), is equivalent to the integration of the following equation on

 (at2+bt+c)ft=(d−2at)f, (17)

where and are arbitrary constants with Up to -equivalence the parameter quadruple  can be assumed to belong to the set where , are nonzero constants, . The proof is similar to ones presented in Vaneeva et al. VPS2012 ; VSL2015 . It is based on the fact that transformations from the equivalence group can be extended to the coefficients , , and as follows

 ~a=μ(aδ2−bγδ+cγ2),~b=μ(−2aβδ+b(αδ+βγ)−2cαγ),~c=μ(aβ2−bαβ+cα2),~d=μ(dΔ+2aβδ−2bβγ+2cαγ),

where and is an arbitrary nonzero constant.

Integration of the equation (17) for four inequivalent cases of the quadruple gives respectively , and We further substitute the obtained inequivalent values of into equation (16) and find the corresponding values of constants and, therefore, the general forms of the infinitesimal generators. The results of the group classification of class (7) are presented in Table 2.

The classification lists presented in Tables 1 and 2 give the exhaustive group classification of the class of variable coefficient nonlinear Kolmogorov equations (3) with nonlinear and of the class of equations (4) up to the - and -equivalences, respectively.

4 Discussion on the choice of the optimal gauging

Appropriate choice of gauging of the arbitrary elements is a crucial step in solving group classification problems. The gauging could seem more convenient if one look for the determining equations for finding Lie symmetries. For class (8) they have the form

 2ηy=τt,ηyy−ηt=2φy,(φu+ψ)gku+[τgt+(τt−ξx)g]k=ξt, (φxu+ψx)gk+(φt−φyy)u+ψt−ψyy=0.

For the case the difference in classification is not so crucial (cf. Table 1 with Table 3). Though one can see that for the operator appearing in Cases 13–16 of Table 1 transforms to various forms in the respective cases of Table 3. For the case the difficulty of group classification of the class (3) with increases essentially in comparison with the gauging . Solving the determining equations results in the following form of the infinitesimal generator

 Q=(c1t+c0)∂t+[(c2x+c3)∫g(t)dt+c4x+c5]∂x+ (12c1y+c6)∂y+[(c7−c2∫g(t)dt)u+c2x+c3]∂u,

where , are arbitrary constants. The classifying equation for is the integro-differential equation (cf. with the classifying equation (16) for that is much simpler). The results of group classification for class (9) are presented in Table 4. Comparing Tables 2 and 4 one can conclude that forms of the basis operators of the maximal Lie invariance algebras are more cumbersome in Table 4.

The links between equations of the form (9) are also more tricky than those between equations from class (7). For example, the equation

 ut=uyy−1tcosh2(νlnt)uux,

where the variable coefficient can be rewritten as , admits the five-dimensional maximal Lie invariance algebra with the basis operators and The equivalence of this equation and the equation

 ~u~t=~u~y~y−~t2ν−1~u~u~x

from the same class does not seem obvious. Nevertheless, there exists the transformation from the equivalence group ,

 ~t=t,~x=14x(t2ν+1),~y=y,~u=ut2ν+1+