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ON A CERTAIN CLASS OF HYPERBOLIC EQUATIONS WITH SECOND-ORDER INTEGRALS UDC 517.9

A. V. Zhiber and A. M. Yur’eva

Abstract. In this paper, we examine a special class of nonlinear hyperbolic equations possessing a second-order y-integral. We clarify the structure of x-integrals and prove that they are x-integrals of a hyperbolic equation with a first-order y-integral. We also prove that this class contains the well-known Laine equation. Keywords and phrases: Liouville-type equations, differential substitutions, x- and y-integrals. AMS Subject Classification: 35Q51, 37K60

1.

Introduction. We consider equations of the following form: p − ϕu q √ uxy = ux + ux , ϕuy ϕuy

(1)

where p and q are functions of the variables x, y, and u and ϕ is a function of the variables x, y, u, and uy . In [5] it was shown that if Eq. (1) has a second-order y-integral   ¯ =W ¯ x, y, u, uy , uyy , D W ¯ = 0, (2) W then the function ϕ is independent of the variable x. Here D is the operator of complete differentiation with respect to the variable x. Note that Eqs. (1) have not been examined in [4]. We assume that (3) v = ϕ(y, u, uy ). Then from Eq. (3) it follows that and Eq. (1) can be written as follows:

uy = Φ(y, u, v),

(4)

√ Dv = p · ux + q ux .

(5)

In [5], necessary and sufficient conditions under which Eq. (1) has a second-order y-integral were obtained. Taking into account (4), we can write the integral (2) as follows: ¯ = 0, W ¯ =W ¯ (x, y, u, v, vy ). (6) DW Then Eq. (6) is equivalent to the system of equations ¯ = 0, L2 W ¯ = 0, L1 W

¯ = 0, L3 W

(7)

where the operators ∂ 1 ∂ + q 2 Φv · , ∂x 2 ∂¯ v1   ∂ ∂ ∂ +p + py + pu Φ + pΦu + p2 Φv , L2 = ∂u ∂v ∂¯ v1   qy ∂ 3 qu 1 ∂ + pΦv + + Φ + Φu , v¯1 = vy , L3 = ∂v 2 q q 2 ∂¯ v1 L1 =

(8)

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 152, Mathematical Physics, 2018.

168

c 2021 Springer Science+Business Media, LLC 1072–3374/21/2522–0168 

and the conditions of the existence of a y-integral (6) for Eq. (1) have the following form: px 2 2 Φv + 2 (ln q)xu · Φ + 2 (ln q)xy , q2 q q  qu qu 1 1 1 pu + p Φv + pΦvu + (ln q)yu + Φu + (ln q)uu · Φ + Φuu + p2 Φvv = 0, 2 q q 2 2   px q y px q u 1 1 1 1 + pux − Φ + px Φ u + ppx − qqu Φv − q 2 Φuv − pq 2 Φvv = 0. pxy − q q 2 2 2 2 Φvv = 3

(9) (10) (11)

In [5], a partial analysis of the conditions (9)–(10) was performed in the case where the solution of Eq. (9) is determined by the formula Φ=− where S1,2

√ A± Δ = , 2

C + D(y, u)eS1 v + R(y, u)eS2 v , B Δ = A2 + 4B = 0,

B = 0,

(12)

R · D = 0

(13)

and A=3

px , q2

2 (ln q)xu , q2

B=

C=

2 (ln q)xy , q2

(14)

under the assumption Ax = 0.

(15)

In this case (see [5, Theorem 4.1]), the following relations are possible: S1 = 2,

S2 = 1,

A = 3,

B = −2,

D = 1,

R = −S(x, y)

qx H(x, y) + , q2 q

p=−

qu . q

(16)

In this paper, we describe a class of Eqs. (1) with second-order y-integrals satisfying the conditions (12)–(16). It is shown that one of examples of such equations is the following