On a Certain Class of Hyperbolic Equations with Second-Order Integrals
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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.
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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
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