The Constant Astigmatism Equation. New Exact Solution
aa r X i v : . [ n li n . S I] N ov The Constant Astigmatism Equation.New Exact Solution.
Natale Manganaro , Maxim V. Pavlov , , Department of Mathematics and Informatics,University of Messina,Viale Ferdinando Stango D’Alcontres 31, 98166 Messina, Italy Sector of Mathematical Physics,Lebedev Physical Institute of Russian Academy of Sciences,Leninskij Prospekt 53, 119991 Moscow, Russia Laboratory of Geometric Methods in Mathematical Physics,Lomonosov Moscow State University,Leninskie Gory 1, 119991, Moscow, Russia Mathematical institute in Opava,Silesian University in Opava,Na Rybn´ıˇcku 1, 746 01 Opava, Czech Republic
Abstract
In this paper we present a new solution for the Constant Astigma-tism equation. This solution is parameterized by an arbitrary functionof a single variable.
Contents eferences 10 keywords : Constant Astigmatism equation, differential constraints, overde-termined systems, characteristic method, integrable systems, reductions.MSC: 35L40, 70H06;PACS: 02.30.Ik. The theory of integrable systems usually associates with the method of theInverse Scattering Transform, which allows to construct multi-soliton, multi-gap and similarity solutions, i.e. parameterized by an arbitrary number ofconstants. Nevertheless, some integrable nonlinear systems in partial deriva-tives possess solutions parameterized by arbitrary functions of a single vari-able. The most known example is the three-wave interaction, which weredeeply investigated in [18].In this paper we deal with the Constant Astigmatism equation u tt + (cid:18) u (cid:19) xx + 2 = 0 , (1)which was considered in a set of papers (see detail in [6], [7], [8], [9], [10],[17]). This equation is connected with the remarkable Bonnet (also knownas the Sine-Gordon) equation by a reciprocal transformation. However, weshow that just the Constant Astigmatism equation has the particular solutionparameterized by an arbitrary function of a single variable.To illustrate this phenomenon, we just mention another remarkable inte-grable nonlinear system known as the Kaup–Boussinesq system u t + (cid:18) u − η (cid:19) x = 0 , η t + (cid:18) uη − u xx (cid:19) x = 0 . (2)Under the potential substitution u = z x and η = z x / z t , the secondequation becomes z tt + 2 z x z xt + (cid:18) z t + 32 z x (cid:19) z xx − z xxxx = 0 . This equation possesses the reduction z t + 12 ( z x − z xx ) = 0 , u t + uu x − u xx = 0 , (3)which is linearizable to the heat equation v t = 12 v xx , by the Cole–Hopf substitution u = − v x /v (indeed, one can see that the firstequation in (2) reduces to (3) if η = u x /
2; then the second equation in (2) au-tomatically satisfies). Since, the heat equation has a solution parameterizedby an arbitrary function of a single variable v ( x ), then the Kaup–Boussinesqsystem also admits a particular solution u = z x and η = z x / z t , where: u = − ∂ x ln √ πt ∞ Z −∞ exp − ( x − y ) t − y Z u ( ξ, dξ dy ,η = − ∂ x ln √ πt ∞ Z −∞ exp − ( x − y ) t − x ′ Z u ( ξ, dξ dy , which depends on an arbitrary function u ( x ) = u ( x, t ) | t =0 . The aim of this section is to develop a reduction procedure for (1) within theframework of the theory of differential constraints. The method is based uponappending a set of PDEs to a given governing system of field equations andit was first applied by Janenko [11] to the gas dynamics model. The auxiliaryequations play the role of differential constraints because they select classesof solutions of the system under interest. The method is very general and,in fact, it includes many of the reduction approaches known in a literature.Unfortunately often the generality of the method limits according diversityof applications. In each particular case, the method of differential constraintsutilizes specific features of a corresponding nonlinear system (see Refs. [13],[15], [12], [4], [5], [1], [2], [3]).First, for further convenience we change the sign u → − u in the ConstantAstigmatism equation (1) u tt = 2 − (cid:18) u (cid:19) xx . (4)3ext, we reduce the governing equation (4) to the pair of equations a t − b x = 0 b t − a + t ) a x = 0 (5)where a ( x, t ) = u ( x, t ) − t , (6)and b ( x, t ) is an auxiliary function. System (5) is strictly hyperbolic andthe eigenvalues of the matrix coefficients as well as the corresponding lefteigenvectors are, respectively, λ = ∓ f ( a, t ); l ( ∓ ) = ( − λ ; 1) , (7)where f ( a, t ) = 1 a + t . (8)Therefore, according to [14], owing to (7), the more general first order differ-ential constraint admitted by (5) takes the form l ( ∓ ) · U x = p ( x, t, a, b ) , with U = (cid:18) ab (cid:19) , which in our case specializes to b x − λa x = p ( x, t, a, b ) , (9)where the function p ( x, t, a, b ) must be determined during the process. Inthe following without loss of generality we consider the case λ = + f ( a, t ).The consistency requirement between (5) and (9) leads to a linear expressionwith respect to a x whose coefficients depend only on ( x, t, a, b ). Thus bothcoefficients must vanish independently: p a + f p b = − f t + pf a f p t + f p x = f t + pf a f p. (10)Both these equations can be integrated by the method of characteristics, i.e.we have dbda = f, dpda = − f t + pf a f , dxdt = f, dpdt = f t + pf a f p. a ) and two other ordinary differentialequations (with respect to t ). However, the solution of (10) p = − t ± √ a + t (11)follows directly from the compatibility condition ddt (cid:18) dpda (cid:19) = dda (cid:18) dpdt (cid:19) . Therefore, taking (9) and (11) into account, equations (5) can be written inthe form a t − f a x = pb t − f b x = − f p. (12)Then the first equation a t − a + t a x = − t ± √ a + t (13)again can be integrated by the method of characteristic, while the function b ( x, t ) satisfying the second equation can be found in quadratures (see (5)) db = a t dx + a x ( a + t ) dt ≡ ( u t − t ) dx + u x u dt, because the compatibility condition ( b t ) x = ( b x ) t is fulfilled by virtue of (4).Moreover, since b t = f a x and b x = a t , one can see that the second equationin (12) is equivalent to the first one in (12).Of course a similar analysis holds in the case λ = − f ( a, t ).In order to obtain the required reduction of the governing equation (4),by writing equations (9) and (12) in terms of the original variable u we get(see (6) and, for instance, (13) for the case λ = + f ( a, t )) u t = ± u u x ± √ u. (14)So that exact solutions of (4) are determined by integrating the first or-der equation (14) using the characteristic method. Therefore we proved thefollowing: Theorem 1 : Constant Astigmatism equation (4) possesses four naturalreductions (14).
Thus, Constant Astigmatism equation (4) has four particu-lar solutions parameterized by an arbitrary function of a single variable. New Particular Solution
Let us first prove the following
Theorem 2 : The first order equation u t = F ( x, t, u, u x ) . (15) is a reduction of Constant Astigmatism equation (4) if it specializes to (14). Proof : Substitution of (15) into (4) leads to F t + F F u + F y ( F x + F u u x + F y u xx ) − u u xx + 2 u u x = 2 , (16)where we set y = u x . By requiring (16) is satisfied ∀ u xx we get F = ± yu + G ( x, t, u ) , (17)where the function G ( x, t, u ) needs to be determined. Substitution of (17) in(16) yields G t + (cid:16) G ± yu (cid:17) (cid:16) G u ∓ yu (cid:17) ± u (cid:16) G x + (cid:16) G u ∓ yu (cid:17) y (cid:17) + 2 u y = 2 . (18)Since relation (18) must be satisfied ∀ y , we obtain G = ± √ u . Thus, invirtue of (17), the Theorem is proved.Next we solve equation (14) by a slightly modified version of the charac-teristic method.Under the point transformation u = v (14) reduces to the form v t = ± v v x ± . This equation can be written in the conservative form w t ± (cid:18) w ± t (cid:19) x = 0 , where v = w ± t . Then the potential function z can be introduced such that dz = wdx ∓ dtw ± t . In the first case d [ z + ln( w + t )] = wdx + dww + t . (cid:0) w + t (cid:1) x = 1. Thus w + t = x + h ( w ),where h ( w ) is an arbitrary function. So, the first particular solution of theConstant Astigmatism equation is u (1) = ( w + t ) , (19)where w ( x, t ) is a solution of the algebraic equation( x + h ( w ))( t + w ) = 1 . (20)Analogously, in the second case: u (2) = ( w − t ) , where w ( x, t ) is a solution of the algebraic equation( x + h ( w ))( t − w ) = 1;In the third case: u (3) = ( w + t ) , where w ( x, t ) is a solution of the algebraic equation( h ( w ) − x )( t + w ) = 1;In the fourth case: u (4) = ( w − t ) , where w ( x, t ) is a solution of the algebraic equation( h ( w ) + x )( w − t ) = 1 . Remark 1 : The Constant Astigmatism equation preserves itself underthe transformation x ↔ t and u → /u . Also its reduction (14) preservesitself under the same transformation. Remark 2 : A relationship between the Sine-Gordon and the ConstantAstigmatism equations was presented in [10]. Corresponding reciprocal trans-formation (see formulas (29), (30) in this cited paper) contains two distinctexpressions (cid:16) u x u ± u t (cid:17) − u, which vanishes if the function u ( x, t ) satisfies (14). Thus, the above fourparticular solutions parameterized by an arbitrary function of a single vari-able cannot be transformed to corresponding solutions of the Sine-Gordonequation. 7 emark 3 : A Cauchy problem for a nonlinear equation in partial deriva-tives of a second order is based on u | t =0 = u ( x ) and u t | t =0 = u ( x ). Since,we can consider a nonlinear equation in partial derivatives of a first order,we investigate a Cauchy problem restricted on a narrow class of solutions.Indeed, for instance, in the first above case, a Cauchy problem has a solution u = ( W ( x, t ) + t ) , (21)where W ( x, t ) is a solution of an algebraic equation (cid:18) x + 1 W − X ( W ) (cid:19) ( t + W ) = 1 , and the function X ( W ) is determined by the equation X ( W ( x )) = x , where W ( x ) = p u ( x ) (see equation (21) for t = 0). Then u ( x ) cannot be anarbitrary function: u ( x ) = 2 1 + X ′ ( W ) W X ′ ( W ) W | W = √ u ( x ) . According to [8], particular solutions of the Constant Astigmatism equationcan be replicated infinitely many times starting from any initial solution bythe formulas x (1) = xux u − , t (1) = η, u (1) = ( x u − u ; (22) x ( − = ξ, t ( − = tt − u , u ( − = u ( t − u ) , (23)where − dη = xu t dx + (cid:18) x u x u + 1 u + x (cid:19) dt, dξ = ( tu t − u − t ) dx + t u x u dt. (24)Combination of these two transformations gives an infinite series of par-ticular solutions in a general case. In this Section we consider replicationof particular solutions starting from the solution constructed in the previousSection. Without loss of generality, we consider the first such a particularsolution determined by (19), (20), i.e. u = ( w + t ) , w ( x, t ) is a solution of the algebraic equation x = 1 t + w − h ( w ) . (25)Then the functions η, ξ (see (24)) can be found in quadratures, i.e. t (1) ≡ η = 2 h ( w ) − wh ( w ) + Z h ( w ) dw − h ( w ) tx ( − ≡ ξ = − w t + w + 2 w + Z w h ′ ( w ) dw, while (22) leads to a first iteration u (1) = ( t (1) + w (1) ) , where w (1) ( x (1) , t (1) ) is a solution of the algebraic equation (cf. (25)) x (1) = 1 t (1) + w (1) − h (1) ( w (1) ) . (26)In this case t ≡ η (1) = 2 h (1) ( w (1) ) − w (1) [ h (1) ( w (1) )] + Z [ h (1) ( w (1) )] dw (1) − [ h (1) ( w (1) )] t (1) , where w (1) = − Z h ( w ) dw, h (1) ( w (1) ) = 1 h ( w ) , w = − Z [ h (1) ( w (1) )] dw (1) . Meanwhile, transformation (23) leads to a first “negative” iteration u ( − = ( w ( − + t ( − ) , where w ( − ( x ( − , t ( − ) is a solution of the algebraic equation (cf. (25) and(26)) x ( − = 1 t ( − + w ( − − h ( − ( w ( − ) . In this case t ( − = − tw + 2 wt , t = − t ( − ( w ( − ) + 2 w ( − t ( − , where h ( − ( w ( − ) = − Z w h ′ ( w ) dw, w ( − = w − . Thus, we see that both transformations ( x, t, u ) → ( x (1) , t (1) , u (1) ) and( x, t, u ) → ( x ( − , t ( − , u ( − ) preserve the class of solutions (see Remark 2 inthe previous Section) which cannot be associated with any solutions of theSine-Gordon equation, i.e. all such iterated solutions are solutions of reducedequation (14). The above formulas just allow to connect an infinite set ofsolutions determined by different expressions h ( w ).9 Conclusion
It is well known that integrable systems conditionally can be split on twowide classes: S and C integrable systems. Usually, we understand that C integrable systems are linearizable systems by appropriate transformations.General solutions of C integrable systems can be expressed explicitly, andthese solutions are parameterized by arbitrary functions. S integrable sys-tems possess infinitely many particular solutions which are parameterized bysufficiently many arbitrary constants in general. Also, we know that some S integrable systems can be degenerated (for instance, the Sinh-Gordon equa-tion u xt = c e u + c e − u in a particular case becomes the Liouville equation u xt = e u , whose general solution is parameterized by two arbitrary functionsof a single variable; see other detail in [16]). In this paper we considered Con-stant Astigmatism equation (4), which is integrable by the Inverse ScatteringTransform (see again detail in [7], [8], [9], [10], [17]) but also admits particu-lar solutions parameterized by an arbitrary function of a single variable evenin a nondegenerate case. We hope that particular solutions parameterized byarbitrary functions of a single variable for some other important integrablesystems will be found soon.In the previous Section we consider transformations for the ConstantAstigmatism equation, which replicate solutions of its reduced version (14)only. We hope that a more general transformation connecting solutions of(14) and (4) can be found. Acknowledgement
We thank Michal Marvan for fruitful discussions. MVP was supportedby GA ˇCR under the project P201/11/0356, by the RF Government grant
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