Adv Syst Sci Appl 2023; 04:1–7 Published online at https://ijassa.ipu.ru. An Exact Solution of the Hunter–Saxton–Calogero Equation by Contact Linearization Method Svetlana Mukhina* V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia Abstract: In this paper we consider a class of generalized nonlinear hyperbolic partial differential equations of the Hunter–Saxton–Calogero type, which arise in the theory of control of liquid crystals and in the control of unsteady gas flows. We found such conditions that the original equation can be reduced to linear one by contact transformations. The general exact multivalued solutions of the Hunter–Saxton–Calogero equation are found. The obtained solutions are visualized. Keywords: contact transformations, Cartan form, nematic crystals, exact solution, nonlinear partial differential equation, differential forms. 1. INTRODUCTION Let us consider the generalized nonlinear second-order Hunter–Saxton–Calogero partial differential equation utx = uuxx +G(ux), (1.1) where u(t, x) is an unknown function, t and x are the time and the spatial coordinates, respectively. Such equations withG(ux) = κu2x and k = 1 2 arise in the theory of nematic liquid crystals. If, initially, all molecules of a liquid crystal are aligned, then some of them will shift slightly and disorientation will spread throughout the crystal. In this case, the function u(t, x) describes the propagation of weak linear orientation waves in the nematic liquid crystal [1]. The equation with κ ̸= 1 2 is used in hydrodynamics [2], in the geometry of Einstein–Weyl spaces [3]. The contact equivalence of equation (1.1) and the Euler–Poisson equation was established for G(ux) = κu2x in [4]. Calogero [5], while studying waves in shallow water, found a complex solution of equation (1.1). In this article we present conditions, under which nonlinear equation (1.1) is equivalent to a linear equation with respect to a pseudo-group of contact transformations. This allows us to construct its exact multivalued solutions. These solutions can be used to control the propagation of orientation waves in a nematic crystal. This paper continues the series of articles [6–9] on the application of geometric theory of nonlinear differential equations to constructing their exact solutions. We use the methods developed in [10–12]. ∗Corresponding author: ssmukhina@edu.hse.ru 2 S. MUKHINA 2. GEOMETRY OF THE GENERALIZED HUNTER–SAXTON–CALOGERO EQUATION Let J1 be the 1-jet space of functions on R2 with two independent variables t, x and let t, x, u, p1, p2 be the canonical coordinates on this space. The Cartan form κ = du− p1dt− p2dx defines a contact structure on J1 (the so called Cartan distribution) C : J1 ∋ θ 7→ C(θ) = kerκθ ⊂ TθJ 1. The Cartan distribution C is generated by the vector fields ∂ ∂t + p1 ∂ ∂u , ∂ ∂x + p2 ∂ ∂u , ∂ ∂p1 , ∂ ∂p2 . (2.2) A two-dimensional surface Γ1 v = { u = v(t, x), p1 = ∂v ∂t , p2 = ∂v ∂x } ⊂ J1 is called a 1-graph of a function v(t, x). Let Ω2(R2) be the module of differential 2-forms on R2. For an arbitrary differential 2- form ω on J1, we can construct the Lychagin differential operator ∆ω, which acts by the following rule (see [13]): ∆ω : C∞(R2) → Ω2(R2), ∆ω(v) = ω|Γ1 v . Here ω|Γ1 u is a restriction of ω to Γ1 v. The equation ∆ω(v) = 0 (2.3) is a second-order differential equation of the Monge–Ampere class. The restriction of ω to the surface Γ1 v vanishes if and only if the function v is a solution of equation (2.3). A surface L ⊂ J1R2 is called a multivalued solution of equation (2.3) if ω|L = 0 and κ|L = 0. Equation (1.1) belongs to the class of Monge–Ampere equations and, therefore, it can be associated with the differential 2-form ω = −2G(p2)dt ∧ dx+ dt ∧ dp1 − dx ∧ dp2 − 2udt ∧ dp2. (2.4) Let us introduce a “non-holonomic symplectic structure” Ω ∈ Ω2(C): Ω = dκ|C Since the Cartan distribution is not completely integrable, this 2-form is defined on vector fields that belong to C only. Differential form (2.4) is effective, i.e., ∂u⌋ω = 0 and ω ∧ Ω = 0. Moreover, it is hyperbolic: ω ∧ ω + Ω ∧ Ω = 0. (2.5) Define the linear operator Aω : D(C) → D(C) as follows: AωX⌋Ω = X ⌋ω, Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) AN EXACT SOLUTION OF THE HUNTER–SAXTON–CALOGERO EQUATION 3 where D(C) is a module of vector fields that belong to the Cartan distribution C. The operator Aω has the following matrix representation in basis (2.2): Aω =  1 0 0 0 −2u −1 0 0 0 −2G(p2) 1 2u 2G(p2) 0 0 −1  . Its square is scalar: A2 ω = 1, therefore, its eigenvalues are ±1. The eigenvectors define two 2-dimensional characteristic distributions C+ = { X+ = ∂ ∂t − u ∂ ∂x + (−p2u+ p1) ∂ ∂u +G(p2) ∂ ∂p2 , Y+ = G(p2) ∂ ∂p1 } , C− = { X− = −u ∂ ∂x − up2 ∂ ∂u +G(p2) ∂ ∂p2 , Y− = ∂ ∂x + p2 ∂ ∂u +G(p2) ∂ ∂p1 } . The vector fields X±, Y± form a basis of the module D(C±). The first derivatives of distributions C (1) ± = {X±, Y±, [X±, Y±]} are 3-dimensional. Therefore, in the 5-dimensional space J1, they intersect along a 1- dimensional distribution l = C (1) + ∩ C(1) − , which is generated by the vector field Z = G (p2) ∂ ∂u +G(p2) (G ′′ (p2)− p2) ∂ ∂p1 . At any point a ∈ J1, the tangent space TaJ1 can be decomposed into a direct sum TaJ 1 = C+(a)⊕ l(a)⊕ C−(a). Denote the distributions C+, l, and C− as P1, P2, and P3, respectively. Let Dj be the module of vector fields from the distribution Pj and let Pj : D(J1) → Dj be projectors. Define the tensors qsj,k ∈ Ω2(J1)⊗D(J1) (see [12]): qsj,k(X, Y ) := −Ps[PjX,PkY ], where j, k, s = 1, 2, 3; s ̸= j, k, and skew contraction of two decomposable tensors α⊗ X, β ⊗ Y ∈ Ω2(J1)⊗D(J1): ⟨α⊗X, β ⊗ Y ⟩ = (Y ⌋α) ∧ (X⌋β) . This definition is extended to the remaining tensors by linearity. Tensor invariants of equation (1.1) have the form: q12,3 = (p2dt ∧ dx+ dt ∧ du)⊗ ( G(p2) 2 ∂ ∂p1 −G (p2) ∂ ∂x +G (p2) p2 ∂ ∂u ) , q31,2 = (p2dt ∧ dx− dt ∧ du+ p1dt ∧ dp2 + p2dx ∧ dp2 − du ∧ dp2) ⊗ ( − (G′′ (p2)− 2) (G (p2)) 2 ∂ ∂p1 ) , q21,1 = (G (p2) dt ∧ dx+ udt ∧ dp2 + dx ∧ dp2)⊗ ( −G (p2) ∂ ∂u + (G′ (p2)− p2)G(p2) ∂ ∂p1 ) , q23,3 = (( G′ (p2) p2 − p22 −G (p2) ) dt ∧ dx+ (−G′ (p2) + p2) dt ∧ du+ dt ∧ dp1 − udt ∧ dp2 ) ⊗ ( G (p2) ∂ ∂u − (G′ (p2)− p2)G (p2) ∂ ∂p1 ) . Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 4 S. MUKHINA The invariant Laplace forms for equation (1.1) are λ+ = 〈 q21,1, q 1 2,3 〉 = −dt ∧ dp2, λ− = 〈 q23,3, q 3 1,2 〉 = −(G′′(p2)− 2)dt ∧ dp2. Equation (1.1) satisfies the conditions of contact linearization λ− = 0, λ+ ∧ λ+ = 0, dλ+ = 0 if and only if the function G(p2) has the form G(p2) = p22 + 2k1p2 + k0, where k0, k1 are arbitrary constants. Then equation (1.1) has the form utx − uuxx − 2k1ux − u2x − k0 = 0. (2.6) Let us construct a linearizing contact transformation. Equation (2.6) corresponds to the differential 2-form ω = −2(u2x + 2k1ux + k0)dt ∧ dx+ dt ∧ dut − 2udt ∧ dux − dx ∧ dux. (2.7) We apply the partial Legendre transform to this 2-form: Φ: (t, x, u, p1, p2) 7→ (t, −p2,−xp2 + u, p1, x) . Applying this transformation to differential form (2.7), we obtain a new form ω1 = Φ∗(ω) = (2xp2 − 2u)dt ∧ dx+ dt ∧ dp1 + ( 2x2 + 4k1x+ 2k0 ) dt ∧ dp2 − dx ∧ dp2, which corresponds to the linear equation utx + (x2 + k1x+ k0)uxx + xux − u = 0. (2.8) Equation (2.8) can be solved by cascade integration method: u(t, x) = ek1t (∫ t t0 F1(τ)e −k1τ cosh ( (τ − t) √ k21 − k0 − arctanh ( x+ k1√ k21 − k0 )) dτ+ +F2 −t− arctanh ( x+k1√ k21−k0 ) √ k21 − k0   √ 2k1x+ x2 + k0 k0 − k21 , (2.9) where F1, F2 are arbitrary functions. Note that the Legendre transformation maps the multivalued solutions of equation (2.6) to the solutions of equation (2.8). But the inverse Legendre transformation maps classical solutions (2.9) to multivalued ones. Apply the inverse transformation Φ−1 : (t, x, u, p1, p2) 7→ (t, p2,−xp2 + u, p1,−x) to (2.9). Let us choose t and p2 as parameters β, α, respectively. Then we get general multivalued solution of equation (2.6): Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) AN EXACT SOLUTION OF THE HUNTER–SAXTON–CALOGERO EQUATION 5 L :  t =β, x =− 1√ −(α2 + 2αk1 + k0)γ ( ek1β ( (α + k1) (∫ β β0 F1(τ)e −k1τ cosh(ψ)dτ ) +(α + k1)F2(η) + γ (∫ β β0 F1(τ)e −k1τ sinh (ψ) dτ + F ′ 2(η) ))) , u =− 1√ −(α2 + 2αk1 + k0)γ (( (αk1 + k0) (∫ β β0 F1(τ)e −k1τ cosh (ψ) dτ ) +(αk1 + k0)F2(η)− α ( γ (∫ β β0 F1(τ)e −k1τ sinh (ψ) dτ ) + F ′ 2(η) )) ek1β ) , ut =e k1β √ α2 + 2αk1 + k0 −γ ( −γ (∫ β β0 F1(τ)e −k1τ sinh (ψ) dτ ) +k1 (∫ β β0 F1(τ)e −k1τ cosh (ψ) dτ ) − F ′ 2(η) + k1F2 ) + F1(β), ux =α, where F1, F2 are arbitrary functions, α, β are parameters, γ = √ k21 − k0, η = − βγ + artanh ( α + k1 γ ) γ  , ψ = ( −arctanh ( α + k1 γ ) + (τ − β)γ ) . To show that L is indeed a multivalued solution, it is enough to check that the restriction of the 2-form ω to it vanishes. 3. VISUALIZATION Let us consider an example of visualization for constructed solution. Let k0 = 2, k1 = 0. Choose the functions F1(τ) = −τ, F2(η) = −η. Then we have t =β, x = β + (arctan (α)α + 1− βα) √ α2 + 1 + α2β α2 + 1 , u = (β + α− arctan (α)) √ α2 + 1− α2 − 1 α2 + 1 . The graph of this solution is shown in Fig. 3.1. Solution graphs for other F1 and F2 are presented in Fig. 3.2 and Fig. 3.3. Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) 6 S. MUKHINA Fig. 3.1. Solution of equation (2.6) with F1(τ) = −τ, F2(η) = −η. Fig. 3.2. Solution with F1(τ) = τ2, F2(η) = η2 Fig. 3.3. Solution with F1(τ) = τ3, F2(η) = η3 Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) AN EXACT SOLUTION OF THE HUNTER–SAXTON–CALOGERO EQUATION 7 ACKNOWLEDGEMENTS This work is supported by the Russian Science Foundation (grant 23-21-00390). REFERENCES 1. Hunter, J. K. & Saxton, R. (1991). Dynamics of director fields, SIAM J. Appl. Math., 51, 1498–1521. 2. Golovin, S. V. (2004). Group foliation of Euler equations in nonstationary rotationally symmetrical case, Proc. Inst. Math. NAS of Ukraine, 50, 110–117. 3. 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Evolutionary systems and flows on solutions spaces of finite type equations, Lobachevskii Journal of Mathematics, 44(9), 3945–3951. 10. Krasilshchik, I. S., Lychagin, V. V. & Vinogradov, A. M. (1986). Geometry of jet spaces and nonlinear partial differential equations. New York, NY: Gordon and Breach. 11. Kushner, A. G., Lychagin, V. V. & Rubtsov, V. N. (2007). Contact geometry and nonlinear differential equations. Encyclopedia of Mathematics and Its Applications. Cambrigde, UK: Cambridge University Press. 12. Kushner, A. G. (2008). A contact linearization problem for Monge–Ampere equations and Laplace invariants, Acta Appl. Math, 101, 177–189. 13. Lychagin, V. V. (1979). Contact geometry and non-linear second-order differential equations, Russian Math. Surveys, 34(1), 149–180. Copyright © 2023 ASSA. Adv Syst Sci Appl (2023) Introduction Geometry of the generalized Hunter–Saxton–Calogero equation Visualization