Integrable hyperbolic equations of sin-Gordon type
aa r X i v : . [ n li n . S I] D ec Integrable hyperbolic equationsof sin-Gordon type.
A.G. Meshkov , V.V. Sokolov Oryol State Technical University Landau Institute for Theoretical Physics
E-mail : [email protected]
Abstract
A complete list of nonlinear one-field hyperbolic equations having generalized in-tegrable x- and y-symmetries of the third order is presented. The list includes bothsin-Gordon type equations and equations linearizable by differential substitutions.
Keywords: generalized symmetry, sin-Gordon type equation, Liouville type equation.MSC: 37K10, 35Q53, 35Q58
The symmetry approach to classification of integrable PDEs (see surveys [1–3] and referencesthere) is based on the existence of higher infinitesimal symmetries and/or conservation lawsfor integrable equations. This approach especially efficient for evolution equations with onespatial variable. In particular, all integrable equations of the form u t = u + F ( u , u , u ) , u i = ∂ i u∂x i (1.1)were found in [4, 5]. The following list of integrable equations List 1 : u t = u xxx + uu x , (1.2) u t = u xxx + u u x , (1.3) u t = u xxx + u x , (1.4) u t = u xxx − u x + ( c e u + c e − u ) u x , (1.5) u t = u xxx − u x u xx u x + 1) + c ( u x + 1) / + c u x , (1.6) u t = u xxx − u x u xx u x + 1) − ℘ ( u ) u x ( u x + 1) , (1.7) t = u xxx − u xx u x + 32 u x − ℘ ( u ) u x , (1.8) u t = u xxx − u xx u x , (1.9) u t = u xxx − u xx u x + c u / x + c u x , c = 0 or c = 0 , (1.10) u t = u xxx − u xx u x + cu, (1.11) u t = u xxx − u xx u x + 1 + 3 u xx u − ( √ u x + 1 − u x − − u − u x ( u x + 1) / + 3 u − u x ( u x + 1)( u x + 2) , (1.12) u t = u xxx − u xx u x + 1 − u xx ( u x + 1) cosh u sinh u + 3 u xx √ u x + 1sinh u − u x ( u x + 1) / cosh u sinh u + 3 u x ( u x + 1)( u x + 2)sinh u + u x ( u x + 3) , (1.13) u t = u xxx + 3 u u xx + 3 u u x + 9 uu x , (1.14) u t = u xxx + 3 uu xx + 3 u u x + 3 u x , (1.15) u t = u xxx . (1.16)can be derived from [4]. Here ( ℘ ′ ) = 4 ℘ − g ℘ − g , and k, c, c , c , g , g are arbitraryconstants. Equations (1.2)–(1.10) are integrable by the inverse scattering method whereas(1.11)–(1.15) are linearizable ( S and C -integrable in the terminology by F. Calogero). Theabove list is complete up to transformations of the form u → φ ( u ); t → t, x → x + ct ; x → αx, t → βt, u → λu ; u → u + γx + δt. (1.17)The latter transformation preserves the form (1.1) only for equations with ∂F∂u = 0. Moreoverthe linear equations admit the transformation: u → u exp( αx + βt ) . (1.18)Since the symmetry approach is pure algebraic, the function φ and the constants c, α, β, λ, γ and δ supposed to be complex-valued. Thus, we do not distinguish equations u t = u xxx − u x and u t = u xxx + u x and so on.For scalar hyperbolic equations of the form u xy = Ψ( u, u x , u y ) (1.19)the symmetry approach postulates the existence of both x -symmetries u t = A ( u, u x , u xx , . . . , ) , (1.20)2nd y -symmetries u t = B ( u, u y , u yy , . . . , ) . (1.21)Two equations (1.19) are called equivalent if they are related by transformations of the form x ↔ y ; u → φ ( u ); x → αx, y → βy, u → λu ; u → u + γx + δy, (1.22)and for the linear equations by u → u exp( αx + βy ); u → u + c x y. (1.23)Here, in general, the function φ and the constants supposed to be complex-valued.For the famous integrable sin-Gordon equation u xy = c e u + c e − u (1.24)the simplest x and y -symmetries are given by u t = u xxx − u x , u t = u yyy − u y . These evolution equations are integrable themselves (a special case of equation (1.5)).The general higher symmetry classification for equations (1.19) turns out to be very com-plicated problem, which is not solved till now. Some important special results have beenobtained in [6–8]. In general, all three functions Ψ , A, B should be found from the compati-bility conditions for equations (1.19), (1.20) and (1.19), (1.21). However, if the functions A and B are somehow fixed, then it is not difficult to verify whether the corresponding functionΨ exists or not and to find it.To describe all integrable equations (1.19) of the sin-Gordon type, we assume (see Dis-cussion) that both symmetries (1.20) and (1.21) are integrable equations of the form u t = u xxx + F ( u, u x , u xx ) , u t = u yyy + G ( u, u y , u yy ) . (1.25)It turns out that taken for x -symmetry one of the equations of the List 1, one can easily findthe corresponding equations (1.19) having y -symmetries or can prove that such equationsdo not exist. In the Section 2 we present all integrable hyperbolic equations thus obtained.The hyperbolic equations can be separated by presence or absence of x and y -integrals(see Discussion). Consider, for instance, the Liouville equation u xy = e u . We do not distinguish sin-Gordon and sinh-Gordon equations
3t is easy to verify that the function P = u xx − u x does not depend on y (i.e. is a function depending on x only) for any solution u ( x, y ) of theLiouville equation. Analogously, the function Q = u yy − u y does not depend on x on the solutions of the Liouville equation.A function w ( x, y, u, u y , u yy , . . . ) that does not depend on x for any solution of (1.19)is called x - integral . Similarly the y -integrals are defined. An equation of the form (1.19)is called equation of the Liouville type (or Darboux integrable equation), if the equationpossesses both nontrivial x - and y -integrals. Some of integrable hyperbolic equations foundin Section 2 are equations of the Liouville type.In contrast, the sin-Gordon equation (1.24) has no x - and y -integrals for generic valuesof the constants c i . There are two types of such equations. Equations of the first type canbe reduced to the linear Klein-Gordon equation u xy = cu by differential substitutions. If anequation with the third order symmetries has no integrals and linearizing substitutions, wecall it equation of sin-Gordon type . The following equations from the lists of Section 2 areequations of this kind: u xy = c e u + c e − u ; (1.26) u xy = f ( u ) q u x + 1 , f ′′ = cf ; (1.27) u xy = q u x q u y + 1; (1.28) u xy = p ℘ ( u ) − µ q u x + 1 q u y + 1 . (1.29)Here ( ℘ ′ ) = 4 ℘ − g ℘ − g , µ − g µ − g = 0 and c, c , c , a, µ, g , g are constants. Equa-tions (1.27), (1.29) are known. These equations are related to equation (1.26) via differentialsubstitutions [8, 10]. Equation (1.28) is probably new. The corresponding differential sub-stitution is given by v xy = 12 cosh v, u xy = √ u x q u y + 1 , v = ln( u y + q u y + 1) . Theorem 1.
Suppose both x - and y -symmetry of a hyperbolic equation of the form (1.19) belong to the list (1.2)–(1.16) up to transformations (1.17), (1.18). Then this equation belongs o the following list: u xy = f ( u ) q u x + 1 , f ′′ = cf, (2.1) u xy = ae u + be − u , (2.2) u xy = q u x q u y + 1 , (2.3) u xy = q u x + 1 q u y + 1 , (2.4) u xy = p ℘ ( u ) − µ q u x + 1 q u y + a, (2.5) u xy = 2 uu x , (2.6) u xy = 2 u x √ u y , (2.7) u xy = u x q u y + 1 . (2.8) u xy = p u x u y , (2.9) u xy = u x ( u y + a ) u , a = 0 , (2.10) u xy = ( ae u + be − u ) u x , (2.11) u xy = u y η sinh − u (cid:0) η e u − (cid:1) , (2.12) u xy = 2 u y η sinh u ( η cosh u − , (2.13) u xy = 2 ξη sinh u (cid:0) ( ξη + 1) cosh u − ξ − η (cid:1) , (2.14) u xy = u − u y η ( η −
1) + c u η ( η + 1) , (2.15) u xy = 2 u − u y η ( η − , (2.16) u xy = 2 u − ξ η ( ξ − η − , (2.17) u xy = u − u x u y − u u y , (2.18) u xy = u − u x ( u y + a ) − uu y (2.19) u xy = √ u y + au y , (2.20) u xy = cu, (2.21) up to transformations (1.22), (1.23). Here ℘ is the Weierstrass function: ( ℘ ′ ) = 4 ℘ − g ℘ − g , µ − g µ − g = 0 , ξ = p u y + 1 , η = √ u x + 1; a, b, c, µ, g , g are constants. Proof.
If (1.25) is an x -symmetry for (1.19), then d dxdy ( u xxx + F ) = ∂ Ψ ∂u x ddx ( u xxx + F ) + ∂ Ψ ∂u y ddy ( u xxx + F ) + ∂ Ψ ∂u ( u xxx + F ) . (2.22)Eliminating all mixed derivatives in virtue of (1.19), we arrive at a defining relation, whichhas to be fulfilled identically with respect to the variables u, u y , u x , u xx , u xxx . Comparingthe coefficients at u xxx in this relation, we get ddy ∂F∂u xx + 3 ddx ∂ Ψ ∂u x = 0 . (2.23)5f some equation from the list (1.2)–(1.16) is taken for the x -symmetry then the function F is known and the defining relation can also be splited with respect to u xx .For example, let equation (1.7) be an x -symmetry for (1.19). Then the u xx -splitting of(2.23) gives rise to:( u x + 1) ∂ Ψ ∂u x − u x ( u x + 1) ∂ Ψ ∂u x + ( u x − , ( u x + 1) (cid:18) Ψ ∂ Ψ ∂u x ∂u y + u x ∂ Ψ ∂u∂u x (cid:19) − u x ∂ Ψ ∂u − u x Ψ ∂ Ψ ∂u y = 0 . The general solution of this system is given byΨ = p u x + 1 (cid:16) g ( u, u y ) + C ln( u x + p u x + 1) (cid:17) . Substituting this expression into (2.22) and finding the coefficient at u xx , we obtain C = 0and therefore Ψ = g ( u, u y ) p u x + 1 . (2.24)Splitting (2.22) with respect to u xx and u x , we obtain that (2.22) is equivalent to a systemconsisting of (2.24) and equations g ∂ g∂u∂u y − ∂g∂u ∂g∂u y = 0 , ℘ ′ ( u ) u y = 2 ∂g∂u ∂g∂u y ,g ∂ g∂u y + g (cid:18) ∂g∂u y (cid:19) − g℘ + ∂ g∂u = 0 , (2.25)where ( ℘ ′ ) = 4 ℘ − g ℘ − g . Since ℘ ′ = 0 we have g u = 0 and g u y = 0 . It follows fromthe first two equations (2.25) that g = p ℘ ( u ) − µ p u y + a , where µ and a are constants ofintegration. The third equation is equivalent to the algebraic equation 4 µ − g µ − g = 0for µ . Thus, we get equation (2.5).To prove the theorem we perform similar computations for each equation from the list(1.2)–(1.15) taken for x -symmetry. For equations (1.2), (1.4) and (1.8) the correspondinghyperbolic equation does not exist. In contrast, equation (1.12) is an x -symmetry for severaldifferent hyperbolic equations. Indeed, in this case calculating the coefficient at u xx in (2.23),we get 2( u x + 1) ∂ Ψ ∂u x − ( u x + 1) ∂ Ψ ∂u x + Ψ = 0 , which implies Ψ = f ( u, u y )( u x + 1) + f ( u, u y ) √ u x + 1. Substituting this into (2.23), weobtain (cid:18) u ∂f ∂u y − (cid:19) (cid:18) u f ∂f ∂u y − uf + 2 u y (cid:19) = 0 ,u (cid:18) f ∂f ∂u y + ∂f ∂u (cid:19) − uf + 2 u y = 0 ,f = 2 f − u u y − uf ∂f ∂u y .
6f the first factor in the first equation is equal to zero, we arrive at (2.15). If the secondfactor equals zero, then we get f = 2 u y p au y + 1 u (1 + p au y + 1) , where a is a constant. The case a = 0 corresponds to (2.17), while a = 0 leads to equation(2.15) with c = 0. The limit a → ∞ gives us equation (2.16).The computations for remaining x -symmetries from the list except for the Swartz-KdVequation (1.9) are very similar and we do not demonstrate them here.Consider the Swartz-KdV equation (1.9). Equation (1.9) is exceptional because there isa wide class of hyperbolic equations with this x -symmetry. Not all equations from this classare integrable and we derive those of them that have y -symmetries.It is easy to verify that equation u xy = f ( u, u y ) u x (2.26)has the following symmetry u t = u xxx − u xx u x + q ( u ) u x , (2.27)where (cid:18) ∂∂u + f ∂∂u y (cid:19) f + 2 qf + q ′ u y = 0 . (2.28)The function q ( u ) can be eliminated by an appropriate transformation u → ϕ ( u ), but weprefer to use transformations of this type for bringing the y -symmetry to one of equations(1.2)–(1.15). Here and in the sequel we make the transformation x ↔ y in formulas (1.2)–(1.15) as well as in other formulas we need.Any of y -symmetries has the form u t = u + A ( u, u ) u + A ( u, u ) u + A ( u, u ) , u n = ∂ n u∂u ny . Equation (2.23) with x ↔ y is equivalent to3 ∂ f∂u y + 2 ∂ ( f A ) ∂u y + 2 ∂A ∂u = 0 , u y ∂ f∂u∂u y + 3 f ∂f∂u y + 2 A u y ∂f∂u + f ∂A ∂u y + 2 A f + ∂A ∂u = 0 . (2.29)Equations (2.28) and (2.22) give rise to additional restrictions for the functions f and q .For symmetries (1.2)–(1.5) we have A = A = 0 and equations (2.29) imply f = u y g ( u ) + h ( u ), gh = 0 , g ′ + g = 0. In the case g = 0 we get (2.10) with a = 0 . For g = 0 it7ollows from (2.28) that q ′ = 0 and f ′′ + 2 qf = 0. If q = 0, then without loss of generality wetake q = − and arrive at equation (2.11). In the case g = 0 , q = 0 we get equation (2.6).For symmetries (1.6) and (1.7) A = 0 , A = − / u y ( u y + 1) − . It follows from (2.28)and (2.29) that f = h ( u ) u x p u y + 1 , h ′′ = 2 h ( h + c ) , q = c − / h . If h ′ = 0, then weput h = 1 and obtain equation (2.8). In the case h ′ = 0 we get h = √ ℘ − µ , q = − / ℘ .The hyperbolic equation is given by (2.5) with x ↔ y and a = 0.For symmetries (1.8), (1.9) A = − u − y , A = 0 . It follows from (2.29) that f = g ( u ) u y . So, we obtain the equation u xy = g ( u ) u x u y . Both x - and y -symmetries of the equation takethe form (2.27), where q = C exp (cid:18) − Z g ( u ) du (cid:19) − g ′ − g . The equation is equivalent to the D’ Alembert equation u xy = 0 under the following trans-formation: ¯ u = Z du exp (cid:18) − Z g ( u ) du (cid:19) . For symmetries (1.10) and (1.11) A = − u − y , A = 0. It follows from (2.29), (2.28)that f = g ( u ) u y + C √ u y , gC = 0 , qC = 0 , g ′ + g = 0 , q ′ + 2 qg = 0. If C = 0, then q = g = 0. Taking C = 2, we get (2.7). If C = 0, then g = u − , q = c u − , and we obtain(2.10) with a = 0.If the y -symmetry has the form (1.12), then it follows from (2.28), (2.29) that f = ku − ( u y + 1 − p u y + 1), ( k − k −
2) = 0 , q = 3(2 − k ) / (8 u ). If k = 1 we get (2.15) with x ↔ y . The case k = 2 leads to (2.16).In the case of y -symmetry (1.13) the system of equations (2.28), (2.29) has two solutionscorresponding to equations (2.12), (2.13) with x ↔ y .Symmetry (1.14) gives rise to equation (2.18) with x ↔ y .Symmetry (1.15) corresponds to the following equation u xy = u x u y u + a − ( u + a ) u x . The shift u → u − a brings it to a special case of equation (2.19).Considering the linear x -symmetry (1.16), we obtain equation (2.10) with arbitrary pa-rameter a , equation (2.21), and u xy = a u x + f ( u y − a u ) , (2.30)where f satisfies some nonlinear third order ODE. The requirement of the existence of a y -symmetry leads to (2.20). 8ore detailed information of each equation of the list (2.1)–(2.21) can be found in Ap-pendix 1. Discussion.
The hyperbolic equations of the form (1.19) having both x and y -integrals were describedin [8]. In particular, it was shown that any such equation possesses both x and y highersymmetries depending on arbitrary functions. Although not all of these symmetries areintegrable, usually some integrable symmetries exist for such equations.There are integrable equations having only y -integrals (or only x -integrals). An exampleof such equation is given by (2.5), where a = 0. Namely, the equation u xy = ξ ′ ( u ) u y p u x + 1 , (2.31)where ξ ′ ( u ) = √ ℘ − µ , has the following first order y -integral I = ( u x + p u x + 1 ) e − ξ and has no x -integrals for the generic Weierstrass function ℘ . Notice that the same formulagives an y -integral for (2.31) with arbitrary function ξ .In some sense equations (1.19) having integrals can be reduced to ODEs. If weare looking for equations (1.19) integrable by the inverse scattering transform method,we should concentrate on integrable equations (1.19) without integrals. There aretwo classes of such equations. The first one consists of the Klein-Gordon equation u xy = cu, c = 0 and equations related through differential substitutions to the Klein-Gordon equation. The symmetries for such equations are C -integrable (in terminology by F.Calogero). The second class of hyperbolic integrable equations having no integrals containsequations that can not be reduced to a linear form by differential substitutions. This themost interesting class consists of hyperbolic equations admitting only S -integrable highersymmetries. Such equations can be regarded as S - integrable hyperbolic equations.For the first glance the anzats (1.25) seems to be very restrictive if we want to describe all S -integrable equations (1.19). The first question is: why only third order equations are takenfor symmetries? We can justify it in the following way. All known S -integrable hierarchiesof evolution equations (1.20) contain either a third order or a fifth order equation. Forpolynomial equations this is not an observation but a rigorous statement [9]. That is why itis enough to consider hyperbolic equations with symmetries of third order (sin-Gordon typeequations) and hyperbolic equations with fifth order symmetries (Tzitzeica type equations).9he following Tzitzeica type S -integrable equations are known [8, 11]: u xy = c e u + c e − u , (2.32) u xy = S ( u ) f ( u x ) g ( u y ) , (2.33) u xy = ω ′ + 3 cω ( u ) f ( u x ) g ( u y ) , (2.34) u xy = h ( u ) g ( u y ) , h ′′ = 0 , (2.35)where ( f + 2 u x ) ( u x − f ) = 1 , ( g + 2 u y ) ( u y − g ) = 1 , ( S ′ − S ) ( S ′ + S ) = c , ω ′ = 4 ω + c . We are planning to consider the Tzitzeica type equations separately. One of the problemshere is that the list of integrable fifth order evolution equations from [3] possibly is notcomplete.The second question is: why we restrict ourselves by symmetries of the form u t = u xxx + F ( u, u x , u xx ) instead of general symmetries of the form u t = Φ( u, u x , u xx , u xxx ) ? (2.36)The main reason is the following statement (see [12]): suppose equation (2.36) is a symmetryfor equation (1.19). Then ddy (cid:16) ∂ Φ( u, u x , u xx , u xxx ) ∂u xxx (cid:17) = 0 . Therefore, if we assume that (1.19) has no nontrivial integrals, then ∂ Φ( u, u x , u xx , u xxx ) ∂u xxx = const. . Acknowledgments.
Authors thank A.V. Zhiber and S.J. Statsev for fruitful discussions.V.S. was partially supported by the RFBR grants 08-01-00440 and NS 3472.2008.2. A.M.was supported by Russian Federal Agency for Higher Education. Project 1.5.07.
Appendix 1. Symmetries, integrals and differentialsubstitutions
Here an information of the integrable equations from the list (2.1)–(2.21) is presented. Onlythe simplest equation from the equivalence class is shown. The existence of x -integrals J ( u, u y , u yy , . . . ) and y -integrals I ( u, u x , u xx , . . . ) was checked till the seventh order.10inearizing substitutions from Liouville type equations to v xy = 0 have the form I = v x or J = v y . More complicated substitutions I = f ( v x , v xx , . . . ) or J = g ( v y , v yy , . . . ) arepresented explicitly. Equation (2.1).
The symmetries have the following form: u t = u xxx − c u x − f ( u ) u x , u t = u yyy − u y u yy u y + 1) − c u y . There are two the sin -Gordon type equations: (2.1a). u xy = u p u x + 1 ; (2.1b). u xy = sin u p u x + 1and two Liouville type equations: (2.1c). u xy = p u x + 1 ; the integrals are: I = u xx p u x + 1 , J = u yy − u ;the linearizing substitution u x = sinh( y + v x ) gives rise to the general solution: u = Z sinh( y + f ( x )) dx + g ( y ); (2.1d). u xy = e u p u x + 1; the integrals are: I = u xx p u x + 1 − p u x + 1 , J = u yy − u y − e u , The general solution is given by u ( x, y ) = ln − ϕ ( x ) g ′ ( y ) (cid:0) g ( y ) + h ( x ) (cid:1)(cid:0) ϕ ( x ) + f ( x ) (cid:0) g ( y ) + h ( x ) (cid:1)(cid:1) ! ,ϕ ( x ) = exp (cid:18)Z f ( x )4 f ′ ( x ) dx (cid:19) , h ( x ) = Z f ′ ( x ) ϕ ( x ) f ( x ) dx. Equation (2.2).
Both x - and y -symmetries have the form (1.5), where c = c = 0. If ab = 0, then we have the sin -Gordon equation. (2.2a). u xy = e u is the Liouville equation. Its symmetries have the same form as for the sin -Gordon equation. The integrals were shown in the Introduction. The general solution u ( x, y ) = log (cid:18) f ′ ( x ) g ′ ( y )( f ( x ) + g ( y )) (cid:19) ;was found by Liouville in 1853. Equation (2.3).
The x -symmetry has the form (1.10), where c = 0 , c = − /
4; the y -symmetry is of the form (1.6), where c = c = 0. It is an S-integrable equation.11 quation (2.4). Both x - and y -symmetries have the form (1.6), where c = 0 , c = − /
2. It is an S-integrable equation.
Equation (2.5).
The x -symmetry is of the form (1.7), the form of the y -symmetry isanalogous: u t = u yyy − u y u yy u y + a ) − ℘ ( u ) u y ( u y + a ) . If a = 0 , then this symmetry is equivalent to (1.9).In the general case the equation can be rewritten using the Jacobi function sn as: u xy = 1sn( u, k ) q u x + 1 q u y + a. (2.37)This is an S-integrable equation except for the degenerate cases considered below. Noticethat the formulas p ℘ ( u, g , g ) − µ = cn( u, k )sn( u, k ) , p ℘ ( u, g , g ) − µ = cn( u, k )dn( u, k ) . lead to another forms of this equation. They can be reduced to (2.37) by the substitution( u, k ) → ( λu, f ( k )) (see [13], Sec. 13.22).There are two degenerate cases for the Weierstrass function. In the first case when ℘ ( u ) = u − we have µ = 0 and √ ℘ − µ = u − . In the second case ℘ ( u ) = sin − u − , µ = − and √ ℘ − µ = sin − u . (2.5a). Equation u xy = u − p u x + 1 p u y + a is C-integrable, the integrals are: I = u xx p u x + 1 + 1 u p u x + 1 , J = u yy p u y + a + 1 u q u y + a. The general solution is given by: u ( x, y ) = p f ( x ) + g ( y ) (cid:18) − Z dxf ′ ( x ) − a Z dxg ′ ( y ) (cid:19) / . (2.5b). Equation u xy = (sin u ) − p u x + 1 p u y + a is C-integrable, the integrals are: I = u xx p u x + 1 + cot u p u x + 1 , J = u yy p u y + a + cot u q u y + a. If a = 0 , then the general solution is u ( x, y ) = 2 arccos (cid:18) f ( x ) + h ( x ) + g ( y )2 f ( x ) (cid:19) / , h ( x ) = Z p f ′ − f dx. If a = 0 , then the general solution is u ( x, y ) = arccos Ψ( x, y ) , Ψ( x, y ) = 12 w ( x ) (cid:2) e g ( ξ + h ) − e − g (cid:3) (2 w ′ + f w ) + ( ξ + h ) e g , g = g ( y ) ,h ( y ) = Z e − g p g ′ − a dy, f ′ ( x ) = 12 (1 + f ) − w ′′ w , ξ ( x ) = Z dxw ( x ) . a = 0 , u xy = f ( u ) u y p u x + 1. There exists the following y -integral I = ( u x + p u x + 1) exp( − ξ ( u )) , ξ ( u ) = Z f ( u ) du for all f ( u ). This leads to the first order ODE: u x = 12 (cid:16) h ( x ) e ξ − (cid:0) h ( x ) e ξ (cid:1) − (cid:17) . All remaining equations are C-integrable. Some of them have two integrals and can bereduced to the D’Alembert equation. Others have no integrals and can be reduced to theKlein-Gordon equation.
Equation (2.6).
The x -symmetry has the form (1.9) and the y -symmetry is the mKdVequation u t = u yyy − u u y . The integrals are: I = u xxx u x − u xx u x , J = u y − u . The general solution is given by u ( x, y ) = g ′′ ( y )2 g ′ ( y ) − g ′ ( y ) f ( x ) + g ( y ) . Equation (2.7).
The x -symmetry has the form (1.9) and the y -symmetry is (1.10),where c = 0 , c = −
3. The integrals are: I = u xxx u x − u xx u x , J = √ u y − u. The general solution is given by u ( x, y ) = − g ′ ( y ) f ( x ) + g ( y ) + Z ( g ′′ ) g ′ dy. Equation (2.8).
The y -symmetry takes the form (1.6), where c = 0 , c = − / x -symmetry is u t = u xxx − u xx u x − u x . This symmetry can be reduced to (1.9) by u → ln u . The integrals are: I = u xxx u x − u xx u x − u x , J = ( u y + q u y + 1 ) e − u . The general solution is given by u ( x, y ) = ln (cid:20) g ( y ) f ( x ) + h ( y ) (cid:21) + Z g − p g ′ − g dy, h = − g − Z p g ′ − g dy. quation (2.9). (The Goursat equation.) Both x - and y -symmetries have the form (1.11)with arbitrary constant c .The equation is reduced to the Klein-Gordon equation v xy = v by any of the followingtwo differential substitutions:(1) u x = 4 v x , u y = v ; (2) u x = v , u y = 4 v y . Equation (2.10).
The x -symmetry has the form (1.11), where c = 0 and the y -symmetrycan be obtained from (1.5) by the substitution c = 0 , u → − ln u . Moreover, there existsthe following second order y -symmetry u t = u yy − u − ( u y + au y ).The integrals and the general solution are: I = u xx u x , J = u y + au ; u ( x, y ) = f ( x ) − ag ( y ) g ′ ( y ) . Equation (2.11).
The x -symmetry has the form (2.27), where q = − and the y -symmetry is given by (1.5), where c = − a , c = − b . The integrals are: I = u xxx u x − u xx u x − u x , J = u y − ae u + be − u . In the case a = 0 the general solution is given by u ( x, y ) = ln g ( y ) + ln (cid:20) h ( y ) f ( x ) − aϕ ( y ) (cid:21) , ln h = Z ( ag + bg − ) dy, ϕ = Z gh dy ;if a = 0 then u ( x, y ) = ln f ( x ) − bg ( y ) g ′ ( y ) . Equation (2.12).
The x -symmetry has the form (1.13). There are the following y -symmetries: u t = u yyy −
32 (3 + coth u ) u y u yy + 14 (3 coth u + 6 coth u + 7) u y ,u t = u yy −
12 (3 + coth u ) u y . The integrals are: I = e − u η − η + e u sinh u , J = u yy u y − u y (coth u + 3) . The general solution is given by: u ( x, y ) = −
12 ln(1 + ψ ) , ψ = f ( x )( g ( y ) + h ( x )) , f ′ = f − f h ′ . quation (2.13). The x -symmetry has the form (1.13). There are the following y -symmetries: u t = u yyy − u y u yy coth u + 2(3 coth u − u y , u t = u yy − u y coth u. The integrals are: I = η − e u η − e − u , J = u yy u y − u y coth u. The general solution is: u ( x, y ) = 12 ln (cid:12)(cid:12)(cid:12)(cid:12) ψ + 1 ψ − (cid:12)(cid:12)(cid:12)(cid:12) , ψ = f ( x )( g ( y ) + h ( x )) , h ′ = − f ′ + 4 f f . Equation (2.14).
Both x - and y -symmetries have the form (1.13). The equation isreduced to the Klein-Gordon one v xy = v by the following differential substitution: u x = (cid:0) v − v x sinh u + cosh u (cid:1) − , u y = (cid:0) v − v y sinh u + cosh u (cid:1) − . Equation (2.15).
There are x -symmetry of the form (1.12) and the following y -symmetry: u t = u yyy − u y u yy u + 3 u y u − c uu yy + 2 u y − cu u y ) . The equation can be reduced to the Klein-Gordon equation v xy = cv by the following differ-ential substitution: u = v /z, z x = − v x , z y = − cv . If c = 0 then the Klein-Gordon equation is reduced to the D’Alembert equation and thefollowing two integrals appear: I = ( η − u , J = u yy u y − u y u . The general solution is: u ( x, y ) = ( f ( x ) + g ( y )) z ( x ) , z ( x ) = − Z f ′ ( x ) dx. Notice that if c = 0 the equation admits a second order symmetry. Equation (2.16).
There are x -symmetry of the form (1.12) and two the following y -symmetries: u t = u yyy − u − u y u yy + 6 u − u y , u t = u yy − u − u y . The integrals and the general solution are given by: I = η − u , J = u yy u y − u y u ; u ( x, y ) = f ( x ) h ( x ) + g ( y ) , h ( x ) = − Z f ′ ( x ) dx. quation (2.17). Both x - and y -symmetries have the form (1.12). The integrals are ofthe form: I = u xx η ( η − − u η ( η − , J = u yy ξ ( ξ − − u ξ ( ξ − . The general solution is given by: u ( x, y ) = ( f ( x ) + g ( y )) z ( x, y ) , z ( x, y ) = − Z f ′ ( x ) dx − Z g ′ ( y ) dy. Equation (2.18).
There are x -symmetry of the form (1.14) and two the following y -symmetries: u t = u yyy − u − u y u yy + 12 u − u y , u t = u yy − u − u y . The integrals are of the form: I = u x u + u , J = u yy u y − u y u . The general solution is: u ( x, y ) = f ′ ( x )2 (cid:0) f ( x ) + g ( y ) (cid:1) ! / . Equation (2.19).
There are x -symmetry of the form (1.15) and two the following y -symmetries: u t = u yyy − u − (2 u y + a ) u yy + 3 au − u y (3 u y + a ) + 6 u − u y , u t = u yy − u − u y ( u y + a ) . When a = 0 the y -symmetry (1.9) is also admitted. The equation can be reduced to theKlein-Gordon one v xy = − av by the following substitution: u x = (cid:16) v x v − u (cid:17) ( u − λ ) , u y = 1 λ (cid:16) u v y v + a (cid:17) ( u − λ ) , where λ is arbitrary parameter. If a = 0 , then the Klein-Gordon equation is reduced to theD’Alembert equation and two integrals appear: I = u x u + u, J = u yy u y − u y u , In this case the general solution is in the form u ( x, y ) = f ′ ( x )( f ( x ) + g ( y )) − . Equation (2.20).
The x -symmetry is u t = u xxx − a u xx and the y -symmetry has theform (1.11), where c = 0 and x → y . The integrals are of the form: I = u xxx − a u xx + a u x , J = u yy a u y + √ u y . u ( x, y ) = f ( x ) + e ax Z (cid:18) g ( y ) + 1 − e − ax/ a (cid:19) dy. The limit a → Equation (2.21).
There are infinitely many symmetries of the form u t = P ( ∂ x , ∂ y ) u, where P is an arbitrary polynomial with constant coefficients. In particular, there exist x -and y -symmetries of the form u t = P ( ∂ x ) u and u t = P ( ∂ y ) u . If c = 0 integrals do notexist otherwise the simplest integrals are: I = u x , J = u y . References [1] Sokolov V.V., Shabat A.B., Classification of Integrable Evolution Equations,