An Extended Modified Kadomtsov-Petviashvili Equation: Ermakov-Painlevé II Symmetry Reduction with Moving Boundary Application

An Extended Mo dified Kadom tso v-P etviash vili Equation: Ermak o v-P ainlev ´ e I I Symmetry Reduction with Mo ving Boundary Application Colin Rogers 1 and P ablo Amster 2 1 Sc ho ol of Mathematics and Statistics, Univ ersity of New South W ales, Sydney , Australia. Email: c.rogers@unsw.edu.au 2 Depto. de Matem´ atica, F acultad de Ciencias Exactas y Naturales, Univ ersidad de Buenos Aires & IMAS-CONICET. Ciudad Univ ersitaria. Pabell´ on I (1428), Buenos Aires, Argentina Email: pamster@dm.uba.ar Abstract Here, a nov el 2+1-dimensional nonlinear evolution equation with tem- p oral mo dulation is introduced whic h admits in tegrable Ermak o v-Painlev ´ e I I symmetry reduction. Application is made to obtain exact solution to a class of Stefan-t yp e moving b oundary problems for this 2+1-dimensional nonlinear ev olution equation. In volutory transformations with origin in autonomisation of certain Ermako v-type coupled systems are extended to 2+1-dimensions and applied to derive a wide 2+1-dimensional class with temp oral modulation and whic h inherits the property of admittance of suc h h ybrid Ermak o v-Painlev ´ e I I symmetry reduction applicable to cer- tain moving b oundary problems. 1 In tro duction Hybrid Ermak ov-P ainlev´ e I I systems w ere originally derived in [1] via wa ve pac ket representations admitted b y multi-dimensional coupled nonlinear Schr¨ o- dinger systems incorp orating de Broglie-Bohm p oten tial terms. Therein, in particular, the canonical single comp onen t Ermako v-P ainlev´ e II equation w as sho wn to o ccur notably in the analysis of transv erse wa ve propagation in a gen- eralised Mooney-Rivlin h yp erelastic material. It has subsequently b een deriv ed in such diverse physical applications as cold plasma physics [2], Kortew eg capil- larit y theory [3] and in the analysis of Dirichlet b oundary v alue problems which arise out of the classical Nernst-Planc k electrolytic system [4]. Underlying in- tegrable structure in m ulti-comp onen t Ermak o v-Painlev ´ e I I systems has b een delimited in [5] wherein an admitted Ermako v system in v arian t w as applied 1 to construct an algorithmic solution pro cedure inv olving a Painlev ´ e I I connec- tion. A B¨ ac klund transformation admitted b y Painlev ´ e II was applied iteratively therein and the imp ortan t link b etw een the Ermako v-Painlev ´ e II and integrable P ainlev´ e XXXIV equation set down. The application of the in verse scattering transform and its dev elopments to solv e initial/b oundary problems in mo dern soliton theory has an extensiv e lit- erature (qv [6, 7, 8] and work cited therein). Ho wev er, the analysis of moving b oundary problems of Stefan-type for solitonic equations has but a recent ori- gin with motiv ation in the classical Saffman-T aylor mo del of [9]. The latter was concerned with description of the percolation of a liquid into a porous medium or Hele-Shaw cell. In a remark able later dev elopment in [10], the canonical Dym equation of soliton theory [11] w as derived in a related analysis of the motion of the interface betw een a viscous and non-viscous liquid. In [12], a P ainlev´ e I I symmetry reduction w as applied to derive exact solutions to a class of mo ving boundary problems of generalised Stefan-type for the Dym equa- tion and its reciprocal asso ciates. This type of reduction had its origin in an analysis of the evolution of the interface in a Hele-Shaw cell [13]. In [14], an extended S-in tegrable version of the Dym equation w as deriv ed in the geometric con text of binormal motion of inextensible curves. In te rms of physical applica- tion, this generalisation o ccurs in the h ydro dynamics of unidirectional disp ersive shallo w w ater propagation with nov el peaked soliton phenomena (Camassa and Holm [15]). Iterated action of a classical B¨ ac klund transformation admitted by P ainlev´ e I I was applied to this extended Dym equation in [16] to generate exact solution to a no vel class of Stefan-type moving boundary problems in terms of Y ablonski-V orob’ev p olynomials. In mo dern soliton theory , recipro cal transformations asso ciated with ad- mitted conserv ation laws were in tro duced in [17] and subsequently applied in [18] in the link age of the canonical AKNS and WKI inv erse scattering schemes of [19] and [20] resp ectiv ely . Recipro cal transformations likewise, imp ortan tly , connect certain classes of 1+1-dimensional solitonic hierarchies [21]. In 2+1- dimensional systems, recipro cal-type transformations were constructed in [22] and subsequently applied in soliton theory in the link age of the triad of canonical Kadom tsev-Petviash vili, Dym and mo dified Kadom tsev-Petviash vili hierarchies [23]. Mo ving boundary problems in 1+1-dimensional soliton theory of Stefan-type and their reciprocal associates which are amenable to exact solution via Painlev ´ e I I symmetry reduction hav e b een detailed in [24, 25, 26, 27]. In [28], a recipro- cal transformation allied with a M¨ obius-type mapping w as applied to a class of Stefan-t yp e moving boundary problems for the solitonic Dym equation to gen- erate exact parametric solution to a class for a base member of the WKI inv erse scattering scheme. It is remarked that conjugation of recipro cal transforma- tions and in volutory-t yp e transformations with origin in the autonomisation of Ermak ov systems [29] has application in soliton theory . A nov el v arian t with temporal mo dulation of the mo dified Kadomtsev– P etviashvili equation of 2+1-dimensional soliton theory is presented here which admits Ermako v-P ainlev´ e I I symmetry reduction. The latter is applied to obtain 2 exact solution to a class of nonlinear moving b oundary problems of Stefan-type asso ciated. A 2+1-dimensional generalisation of in volutory-t yp e transforma- tions with origin in autonomisation of the canonical Ermak ov-Ra y-Reid system in [29] is applied to embed the extended mKP equation in a wide asso ciated class with temp oral modulation which inherits admittance of Ermak o v-Painlev ´ e I I sysmmetry reduction. 2 Ermak o v-P ainlev ´ e I I Symmetry Reduction The canonical solitonic modified Kadomtsev-P etviash vili (mKP) equation is giv en by [7] V t + V xxx − 3 V 2 V x − 3 V x ∂ − 1 x V y + 3 σ 2 ∂ − 1 x V y y = 0 , σ 2 = ± 1 (1) and, in particular, is amenable to the ¯ ∂ -dressing pro cedure of soliton theory [30]. Here, a nov el real v arian t of the preceding is introduced whic h admits Ermak ov-P ainlev´ e I I symmetry reduction, namely U t + U xxx − 3 U 2 U x − 3 U x ∂ − 1 x U y + δ ∗ U ∂ − 1 x U y y + λ ( t + a ) µ U − 4 U x = 0 . (2) Here, an ansatz similarit y reduction with U = ( t + a ) m Ψ  x + α ∗ y ( t + a ) n  (3) is p ostulated. Under the latter ∂ − 1 x U y = α ∗ ( t + a ) m Ψ = α ∗ U , ∂ − 1 x U y y = α ∗ 2 ( t + a ) m − n Ψ ′ = α ∗ U y = α ∗ 2 U x (4) whence with δ ∗ = 3 /α ∗ , (2) reduces to U t + U xxx − 3 U 2 U x + λ ( t + a ) µ U − 4 U x = 0 . (5) The latter constitutes an extension of the solitonic mKdV equation in U ( x, y , t ) wherein y o ccurs implicitly . Insertion of the similarit y representation (3) in to (5) yields m ( t + a ) m − 1 Ψ + ( t + a ) m − 1 ( − nξ )Ψ ′ + ( t + a ) m − 3 n Ψ ′′′ − 3( t + a ) 3 m − n Ψ 2 Ψ ′ + λ ( t + a ) µ − 3 m − n Ψ − 4 Ψ ′ = 0 , (6) wherein Ψ = Ψ( ξ ), ξ = ( x + α ∗ y ) / ( t + a ) n . Th us, m Ψ − ( nξ Ψ) ′ + n Ψ + ( t + a ) 1 − 3 n Ψ ′′′ − 3( t + a ) 2 m +1 − n Ψ 2 Ψ ′ + λ ( t + a ) µ − 4 m − n +1 Ψ − 4 Ψ ′ = 0 (7) 3 whence, with m = − 1 / 3, n = 1 / 3 and µ = − 2 there results Ψ ′′′ − 3Ψ 2 Ψ ′ − (1 / 3)( ξ Ψ) ′ + λ Ψ − 4 Ψ ′ = 0 . (8) On in tegration, the latter yields Ψ ′′ − Ψ 3 − (1 / 3) ξ Ψ − ( λ/ 3)Ψ − 3 = ζ , ζ ϵ R (9) whic h, on appropriate scalings Ψ = γ w ∗ , and ξ = ϵz and with ζ = 0 reduces to the canonical Ermak ov-P ainlev´ e I I equation [1] w ∗ = 2 w ∗ 3 + 2 w ∗ + χw ∗− 3 (10) with χ = ( λ/ 3) γ − 4 ϵ 2 . 3 A Class of Stefan-Type Moving Boundary Prob- lems Here, moving b oundary problems associated with U ( x, y, t ) are in tro duced gov- erned b y the nonlinear system U t + U xxx − 3 U 2 U x + λ ( t + a ) − 2 U − 4 U x = 0 U xx − U 3 − ( λ/ 3)( t + a ) − 2 U − 3 = L m S i ˙ S U = P m S j ) on x + α ∗ y = S ( t ) , t > 0 ( U xx − U 3 − ( λ/ 3)( t + a ) − 2 U − 3 ) | x + α ∗ y =0 = H 0 ( t + a ) k , t > 0 . (11) wherein S ( t ) = γ ( t + a ) 1 / 3 . Boundary Conditions I. U xx − U 3 − ( λ/ 3)( t + a ) − 2 U − 3 = L m S i ˙ S on x + α ∗ y = S ( t ) = γ ( t + a ) 1 / 3 , t > 0. Insertion of the similarit y represen tation (3) with m = − 1 / 3, n = 1 / 3 in to the preceding yields Ψ ′′ ( γ ) − Ψ 3 ( γ ) − ( λ/ 3)Ψ − 3 ( γ ) = L m ( t + a ) S i ˙ S . (12) On application of the Ermako v-Painlev ´ e I I reduction corresp onding to ζ = 0 the relation L m = γ Ψ( γ ) (13) results together with i = − 1. I I. U = P m S j on x + α ∗ y = S ( t ) = γ ( t + a ) 1 / 3 , t > 0. 4 Here, the relation P m = γ Ψ( γ ) (14) is deriv ed along with j = − 1. I II. ( U xx − U 3 − ( λ/ 3)( t + a ) − 2 U − 3 ) | x + α ∗ y =0 = H 0 ( t + a ) k . This yields Ψ ′′ (0) − Ψ 3 (0) − ( λ/ 3)Ψ − 3 (0) = H 0 ( t + a ) k +1 (15) whence, by virtue of the Ermako v-Painlev ´ e I I reduction in Ψ( ξ ) at ξ = 0 it is required that H 0 = 0. 4 An Airy Reduction The Ermako v-Painlev ´ e I I equation (10) with w ∗ = ρ 1 / 2 , ρ > 0 on appropriate scaling yields ρ z z = ( ρ z ) 2 / 2 ρ + 2 ρ 2 + z ρ + 2 δ /ρ , δ ϵ R (16) whic h is equiv alent to the classical in tegrable Painlev ´ e XXXIV equation (qv [3], [5]). In particular, it admits a particular solution ρ ( z ) = w z + w 2 + (1 / 2) z (17) wherein w ( z ) is gov erned b y the canonical P ainlev´ e I I equation. Th us, the Ermak ov-P ainlev´ e I I equation in turn admits a class of exact solutions Ψ = γ w ∗ = γ [ w z + w 2 + (1 / 2) z ] 1 / 2 . (18) P ainlev´ e I I admits an imp ortant sub class of exact solutions when the PI I parameter α = 1 / 2, namely , that with w = − ϕ ′ ( z ) /ϕ ( z ) (19) with ϕ ( z ) go verned b y the classical Airy equation ϕ ′′ + (1 / 2) z ϕ = 0 . (20) Accordingly (18) shows that the Ermako v-P ainlev´ e I I in Ψ admits a particular class of solutions Ψ = γ [ 2( ϕ ′ /ϕ ) 2 + z ] 1 / 2 , z = ξ /ϵ (21) wherein ϕ = aAi (2 − 1 / 3 z ) + bB i (2 − 1 / 3 z ) , a, b ϵ R (22) Ermak ov-P ainlev´ e II solutions of the preceding Airy-type hav e physical appli- cation in [3] to the classical Korteweg capillarit y system. Therein, the linked P ainlev´ e XXXIV equation arises as a symmetry reduction of a Bernoulli integral of motion. It is remark ed that in [31], the Airy-type solution (19) of Painlev ´ e I I 5 with α = 1 / 2 and the subsequen t class of exact solutions as generated via the iterated action of an admitted B¨ acklund transformation has b een applied in the analysis of certain b oundary v alue problems for the Nernst-Planc k electrolytic system. In the present con text of the class of moving b oundary problems determined b y the nonlinear system (11) with the preceding class of exact solutions of Airy- t yp e for the Ermako v-P ainlev´ e I I equation, the parameters L m and P m in the mo ving b oundary conditions are determined b y L m = P m = γ Ψ( γ ) (23) with Ψ( γ ) = γ [ 2( ϕ ′ /ϕ ) 2 + z ] 1 / 2 | z = γ /ϵ (24) and ϕ determined in terms of Ai ( − z − 1 / 3 z ), B i ( − 2 − 1 / 3 z ) b y (22). It is remarked that the iterativ e action of the B¨ ac klund transformation ad- mitted by Painlev ´ e I I as applied in the analysis of b oundary problems for the Nernst-Planc k system in [31] ma y b e used to generate with (21) as seed solu- tion a wide class of asso ciated solutions Ψ of the Ermako v-Painlev ´ e I I equation and thereb y of moving b oundary problems of whic h the latter constitutes a symmetry reduction. 5 T emp oral Mo dulation via a Class of In v olu- tory T ransformations Under the action on (5) of the class of transformations dt ∗ = ρ − 2 ( t ) dt , dx ∗ = dx , dy ∗ = dy U ∗ = U /ρ ( t ) , ρ ∗ = 1 /ρ ( t ) ( I ∗ ) there results a diverse range of asso ciated nonlinear ev olution equations which incorp orate temporal mo dulation, namely ρ ∗ 2 ∂ /∂ t ∗ ( ρ ∗− 1 U ∗ ) + ρ ∗− 1 U ∗ x ∗ x ∗ x ∗ − 3 ρ ∗− 3 U ∗ 2 U ∗ x ∗ + λ ( t + a ) µ ρ ∗ 3 U ∗− 4 U ∗ x ∗ = 0 (25) wherein dt = ρ ∗− 2 dt ∗ . Under I ∗ , dt ∗∗ = ρ ∗− 2 dt ∗ = dt , dx ∗∗ = dx , dy ∗∗ = dy , U ∗∗ = U ∗ /ρ ∗ = U , ρ ∗∗ = 1 /ρ ∗ = ρ (26) so that the inv olutory prop erty I ∗∗ = I holds. Thereby , application of I ∗ to the sub class of U determined by (3) which admits Ermako v-Painlev ´ e I I symmetry reduction generates a wide asso ciated class with temp oral mo dulation which inherits this prop ert y . Application of the inv olutory transformations I ∗ to the 6 mo ving b oundary problems determined b y the system (11) em b eds them in a wide class whic h admits exact solution via Ermako v-Painlev ´ e I I symmetry reduction. In volutory transformations I ∗ of the type applied here hav e their genesis in a geometric analysis of coupled t wo-component Ermako v-Ray-Reid systems. The latter hav e application notably in nonlinear optics [31, 32] and 2+1-dimensional shallo w w ater hydrodynamics [33]. They ma y b e set in the con text of generalised nonlinear coupled Ermak ov systems which hav e extensive applications in b oth ph ysics and contin uum mechanics [34]. Ermak ov-t yp e mo dulation of physical systems has b een detailed in [35, 36, 37, 38] notably in connection with nonlinear Schr¨ odinger mo dels and Kepler triads. In [39], spatially mo dulated coupled systems of sine-Gordon, Demoulin and Manako v NLS type were reduced to their unmodulated solitonic coun ter- parts via inv olutory transformations. Thereby canonical solitonic prop erties w ere inherited by the modulated systems. In the present context, a wide range of temp oral mo dulation of (5) is gen- erated if ρ ∗ ( t ∗ ) in the class of in volutory transformations I ∗ is go verned b y the classical Ermak ov equation ρ ∗ t ∗ t ∗ + ω ( t ∗ ) ρ ∗ = E /ρ ∗ 3 , E ϵ R . (27) Application ma y then b e made of its nonlinear sup erp osition principle [34]. The latter has physical application, in particular, in the analysis of initial-b oundary v alue problems descriptive of the large amplitude radial oscillations of thin shells comp osed of hyperelastic materials of Mo oney-Rivlin t yp e and sub ject to a range of boundary loadings [40]. The nonlinear sup erposition principle may b e derived via a Lie group pro cedure as in [41]. Lie theoretical generalisation and dis- cretisation of Ermako v-type equations which preserve admittance of nonlinear sup erposition principles hav e b een detailed in [42]. 6 Conclusion Here, moving b oundary problems of Stefan-type ha ve b een sho wn to be amenable to exact solution via Ermako v-Painlev ´ e I I symmetry reduction of a no vel class of 2+1-dimensional nonlinear evolution equations. It is remarked that the re- duction pro cedure and application of in volutory transformations to incorp orate temp oral mo dulation may likewise b e adapted to certain v arian ts of the 2+1- dimensional Bogoy a vlensky-Konop elc henko equation with its div erse physical applications. References [1] C. Rogers, A nov el Ermako v-P ainlev´ e I I system: N+1-dimensional coupled NLS and elasto dynamic reductions, Stud. Appl. Math. 133 , 214–231 (2014). 7 [2] C. Rogers and P .A. Clarkson, Ermako v-Painlev ´ e II reduction in cold plasma ph ysics. Application of a B¨ acklund transformation, J. Nonline ar Mathemat- ic al Physics 25 , 247–261 (2018). [3] C. Rogers and P .A. Clarkson, Ermak ov-P ainlev´ e I I symmetry reduction in a Korteweg capillarit y system, Symmetry, Inte gr ability and Ge ometry: Metho ds and Applic ations 13 , 018 (2017). [4] P . Amster and C. Rogers, On a Ermak ov-P ainlev´ e I I reduction in three-ion electro diffusion. 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