Bosonization, Painleve property, exact solutions for N=1 supersymmetric mKdV equation
aa r X i v : . [ n li n . S I] M a y Bosonization, Painlev´e property, exact solutions for N = 1 supersymmetricmKdV equation Bo Ren ∗ , Jian-Rong Yang , Ping Liu , Xi-Zhong Liu Institute of Nonlinear Science, Shaoxing University, Shaoxing 312000, China Department of Physics and Electronics, Shangrao Normal University, Shangrao 334001, China College of Electron and Information Engineering,University of Electronic Science and Technology of China Zhongshan Institute, Zhongshan 528402, China (Dated: July 25, 2018)The N = 1 supersymmetric modified Korteweg-de Vries (SmKdV) system is transformedto a system of coupled bosonic equations with the bosonization approach. The bosonizedSmKdV (BSmKdV) passes the Painlev´e test and allows a set of B¨acklund transformation(BT) by truncating the series expansions of the solutions about the singularity manifold.The traveling wave solutions of the BSmKdV system are obtained using the mapping anddeformation method. Some special types of exact solutions for the BSmKdV system arefound with the solutions and symmetries of the usual mKdV equation. In the meanwhile, thesimilarity reduction solutions of the system are investigated by using the Lie point symmetrytheory. The generalized tanh function expansion method for the BSmKdV system leads toa nonauto-BT theorem. Using the nonauto-BT theorem, the novel exact explicit solutionsof the BSmKdV system can be obtained. All these solutions obtained via the bosonizationprocedure are different from those obtained via other methods. I. INTRODUCTION
The mathematical formulation of supersymmetry is based on the introduction of Grassmannvariables. It exists an extensive literature devoted to construction the symmsymmetric inte-grable models, such as Korteweg-de Vries [1], modified Korteweg-de Vries [2–5], Sine-Gorden [6],Kadomtsev-Petviashvili hierarchy [7] and nonlinear Sch¨ordinger equation [8]. It has shown thatthese supersymmetric integrable systems possess the Painlev´e property, the Lax representation,an infinite number of conservation laws, the B¨acklund and the Darboux transformations, bilinearforms and multi-soliton solutions [9–16]. However, to treat the integrable systems with fermionssuch as the supersymmetric integrable systems and pure integrable fermionic systems is muchmore complicated than to study the integrable pure bosonic systems [17]. It is significant if onecan establish a proper bosonization procedure to deal with the supersymmetric systems. Recently,a simple bosonization approach to treat the super integrable systems has been proposed [18, 19].The method can effectively avoid difficulties caused by intractable fermionic fields which are anti-commuting [18–21].In this letter, we shall use the bosonization approach to the SmKdV system [2–5]. It readsΦ t + D Φ − D Φ D Φ − D Φ) D Φ = 0 , (1)where D = ∂ θ + θ∂ x is the covariant derivative. It is established with the usual independent variable { x, t } and a Grassmann variable θ , and expansion Φ in terms of θ yields Φ( θ, x, t ) = ξ ( x, t )+ θu ( x, t ) . (1) is related to the N = 1 supersymmetric KdV through a Miura type of transformation [4, 5]and has a bilinear B¨aklund transformation [10]. It shares the common conserved quantities with ∗ Electronic mail: [email protected]. the supersymetric Sine-Gordon equation [13]. The quasi-periodic wave solutions are constructedwith the Hirota bilinear method and the Riemann theta function recently [22].The paper is organized as follows. In section 2, based on the bosonization approach, the N = 1SmKdV system is changed to a system of coupled bosonic equations. The Painlev´e property andthe BT of the coupled bosonic equations are studied by the standard singularity analysis. Insections 3, some special types of exact solutions can be explicitly found by means of the mappingand deformation method. In sections 4, the reduction solutions for the usual Painlev´e II are foundusing the Lie point symmetry. Section 5 is devoted to the generalized tanh function expansionapproach for the coupled bosonic equations. The explicit novel exact solution of the BSmKdV isinvestigated. The last section is a simple summary and discussion. II. BOSIONIZATION OF THE SMKDV EQUATION AND PAINLEV´E ANALYSISA. Bosonization approach with two fermionic parameters
In terms of the component fields, (1) is equivalent to u t + u xxx − u u x − ξ ( uξ x ) x = 0 , (2a) ξ t + ξ xxx − uu x ξ − u ξ x = 0 . (2b)It is obvious that (2) includes a commuting u and an anticommuting ξ field. It will degeneratesto the usual classical system with vanishing the fermionic sector. In order to avoid the difficultiesin dealing with the anticommutative fermionic field ξ , we expand the component fields ξ and u byintroducing the two fermionic parameters [18–21] u ( x, t ) = v + wζ ζ , (3a) ξ ( x, t ) = pζ + qζ , (3b)where ζ and ζ are two Grassmann parameters, while the coefficients v , p , q and w are four usualreal or complex functions with respect to the spacetime variable { x, t } . Substituting (3) into theSmKdV system (2), we obtain v t + v xxx − v v x = 0 , (4a) p t + p xxx − v p x − pvv x = 0 , (4b) q t + q xxx − v q x − qvv x = 0 , (4c) w t + w xxx − v w ) x − v x ( pq ) x − vpq xx − vqp xx = 0 . (4d)The above way is just the bosonic procedure for the SmKdV system (2) with two fermionic pa-rameters (BSmKdV-2). (4a) is exactly the usual mKdV equation which has been widely studied[23–26]. (4b)-(4d) are linear homogeneous in p , q and w , respectively. These pure bosonic systemscan be easily solved theoretically. This is just one of the advantages of the bosonization approach. B. Painlev´e analysis and B¨acklund transformations for the BSmKdV-2 system
In this part, we will study the Painlev´e property and the BT of the BSmKdV-2 system. If allthe movable singularities of its solutions are only poles, the model is called Painlev´e integrable. Inorder to perform the Painlev´e analysis, the bosonic fields v, p, q, w expand about the singularitymanifold φ ( x, t ) = 0 as v = ∞ X j =0 v j φ j − α , p = ∞ X j =0 p j φ j − α , q = ∞ X j =0 q j φ j − α , w = ∞ X j =0 w j φ j − α , (5)with { v j , p j , q j , w j } being arbitrary functions of { x, t } . From the leading order analysis result, theall constants α , α , α and α are positive integers, i.e., 1 , , , v j , p j , q j and w j can be obtained, the resonancevalues of j are given j = − , , , , , , , , , , , . (6)After the detailed calculations, the resonance conditions are satisfied identically because the func-tions v j , p j , q j and w j are all determined by twelve arbitrary functions φ , p , q , w , p , q , v , v , p , q , w and w . From the above considerations we deduce that the BSmKdV-2 is really Painlev´eintegrable.Using the standard truncated Painlev´e expansion, the BT is v = v φ + v , p = p φ + p , q = q φ + q , w = w φ + w φ + w , (7)where v = ± φ x , w = ( φ x w ,x − φ xx w ) φ − x and { v , p , q , w } satisfy BSmKdV-2 system. Be-sides, we find the fields φ , p , q and w are the solutions of the following Schwarzian BSmKdV-2system φ t + φ xxx − φ xx φ x = 0 , (8a) p ,t + p ,xxx + p ,x φ xx φ x + p φ xx φ xxx φ x − p ,x φ xxx φ x − p φ xx φ x − p ,xx φ xx φ x = 0 , (8b) q ,t + q ,xxx + q ,x φ xx φ x + q φ xx φ xxx φ x − q ,x φ xxx φ x − q φ xx φ x − q ,xx φ xx φ x = 0 , (8c) w ,t + w ,xxx + w ,x φ xx φ x + 6 w φ xx φ xxx φ x − w φ xx φ x − w ,x φ xxx φ x − w ,xx φ xx φ x + 3 q p ,xx φ xx φ x (8d) − p q ,xx φ xx φ x + p q ,x φ xxx φ x − q p ,x φ xxx φ x + p q ,x φ xx φ x − q p ,x φ xx φ x + 3 p ,x q ,xx − q ,x p ,xx = 0 , with the solutions φ , p , q and w are related by v = − φ xx φ x , p = p φ xx − p ,x φ x φ x , q = q φ xx − q ,x φ x φ x ,w = 112 φ x (12 q φ xx p ,x φ x − p φ xx q ,x φ x + 6 p q ,xx φ x − q p ,xx φ x − w φ t φ x − p φ x q + 12 q φ x w + 6 w ,xx φ x − w ,x φ x φ xx + 21 w φ xx ) . It is obvious that an auto-BT (7) and a nonauto-BT (8) are obtained with the singularity analysis.
III. TRAVELING WAVE SOLUTIONS WITH MAPPING AND DEFORMATIONMETHOD
Now the traveling wave solutions of the bosonic (4) will be studied. Introducing the travelingwave variable X = kx + ωt + c with constants k , ω and c , (4) is transformed to the ordinarydifferential equations (ODEs) k v XXX + ωv X − kv v X = 0 , (9a) k p XXX + ωp X − kpvv X − kv p X = 0 , (9b) k q XXX + ωq X − kqvv X − kv q X = 0 , (9c) k w XXX + ωw X − k ( v w ) X + 3 k q ( vp X ) X − k p ( vq X ) X = 0 . (9d)As the well known exact solutions of (9a), we try to build the mapping and deformation relationshipbetween the traveling wave solutions v and { p, q, w } , then the exact solutions of the BSmKdV-2equation can be obtained with the known solutions of mKdV equation.At first, we get v X from (9a) v X = a p k ( kv − ωv − c v + c k ) k , (10)where c and c are the integral constants and a = 1. In order to get the mapping relationshipbetween v and { p, q, w } , we introduce the variable transformations p ( X ) = P ( v ( X )) , q ( X ) = Q ( v ( X )) , w ( X ) = W ( v ( X )) . (11)Using the transformation (11) and vanishing v X via (10), the linear ODEs (9b)-(9d) become (cid:0) kv − ωv − c v + c k (cid:1) d P d v + (cid:0) kv − ωv − c (cid:1) d P d v + 3 kv d P d v − kvP = 0 , (12a) (cid:0) kv − ωv − c v + c k (cid:1) d Q d v + (cid:0) kv − ωv − c (cid:1) d Q d v + 3 kv d Q d v − kvQ = 0 , (12b) (cid:0) kv − ωv − c v + c k (cid:1) d W d v + (cid:0) kv − ωv − c (cid:1) d W d v + ( ω − kv ) W − F ( v ) = 0 , (12c)where F ( v ) = 3 a p k ( kv − ωv − c v + c k ) (cid:16) P d Q d v − Q d P d v (cid:17) + c . The mapping and deformation relations are constructed via (12) P = A v + ( A + A ) v cos( R ( v )) − ( A − A ) v sin( R ( v )) , (13a) Q = A v + ( A + A ) v cos( R ( v )) − ( A − A ) v sin( R ( v )) , (13b) W = (cid:18) A + Z v A + yF ( y )( ky − ωy − c y + c k ) / dy (cid:19)p kv − ωv − c v + c k , (13c)where A k ( k = 1 , , · · · ,
8) are arbitrary constants, and R ( v ) = Z u ik √ c ky p ky − ωy − c y + c k dy. If we know the solution of v , the traveling wave solution of BSmKdV-2 system will be given withconsidering (13) and (11). For a special case, A = c = 0, A = A and A = A , the travelingwave solution W is an ordinary type of the symmetries of the traveling wave equation (9a). Infact, for any given a solution v of the usual mKdV equation, a special type solutions of the bosonicequation (4) can be constructed p = P = A v, q = Q = A v, w = W = A σ ( v ) , (14)where σ ( v ) is the symmetry of the usual mKdV equation (9a). The field w exactly satisfy thesymmetry equation of the usual mKdV system. The solution v is not restricted to the travelingwave solutions. We can construct not only traveling wave solutions but also some novel types ofsolutions of the BSmKdV-2 system by using the solutions and symmetries of the mKdV equation. IV. SIMILARITY REDUCTION SOLUTIONS WITH LIE POINT SYMMETRYTHEORY
It is well known that the Lie point symmetry play an important role in the investigation ofnonlinear partial differential equations (PDEs) in modern mathematical physics. The approach iseffective methods to obtain the explicit exact solutions [27–30]. Our aim is to apply the techniquesof Lie group theory to the coupled bosonic equation in order to obtain particular exact solutionsand to study their properties. First, we assume the corresponding Lie point symmetry has thevector form V = X ∂∂x + T ∂∂t + V ∂∂v + P ∂∂p + Q ∂∂q + W ∂∂w , (15)where X , T , V , P , Q and W are functions with respect to x , t , v , p , q and w . The symmetrysupposes σ = Xv x + T v t − V, σ = Xp x + T p t − P, σ = Xq x + T q t − Q, σ = Xw x + T w t − W. (16)The symmetry σ k ( k = 0 , , ,
3) is the solution of the linearized equations of (4) σ ,t + σ ,xxx − v σ ) x = 0 , (17a) σ ,t + σ ,xxx − vv x σ − v σ ,x − p ( σ v ) x − vσ p x = 0 , (17b) σ ,t + σ ,xxx − vv x σ − v σ ,x − q ( σ v ) x − vσ q x = 0 , (17c) σ ,t + σ ,xxx − σ v ) x − σ vw ) x + 3 σ ( vp x ) x − σ ( vq x ) x − p ( σ q x + σ ,x v ) x + 3 q ( σ p x + σ ,x v ) x = 0 . (17d)Substituting (16) into the symmetry equations (17) and eliminating v , p , q and w in terms of (4),the solutions X , T , V , P , Q and W can be concluded using the determining equations T = C t + C , X = C x C , V = − C v, (18) P = C p + C q, Q = C q + C p, W = w ( C + C ) , where C i ( i = 1 , , ...,
7) are arbitrary constants. Then, one can solve the characteristic equations d xX = d tT , d vV = d tT , d pP = d tT , d qQ = d tT , d wW = d tT , (19)where X , T , V , P , Q , and W are given by (18). One case about the solution (4) will discuss inthe following.When C = C = C = 0, we can find the similarity solutions after solving out the characteristicequations v = V ( ξ ) , p = P ( ξ ) e C xC , q = Q ( ξ ) e C xC , w = W ( ξ ) e ( C C xC , (20)with the similarity variable ξ = t − (cid:0) C /C (cid:1) x . Substituting (20) into (4), the invariant functions V , P , Q and W satisfy the reduction systems V ξξξ − C C V ξ − C C V V ξ = 0 , (21a) P ξξξ − C C P ξξ − C C P ξ V − C C P V V ξ + 3 C C − C C P ξ V − C C P + 3 C C C P V = 0 , (21b) Q ξξξ − C C Q ξξ − C C Q ξ V − C C QV V ξ + 3 C C − C C Q ξ V − C C Q + 3 C C C QV = 0 (21c) W ξξξ + C − C C − C C C W ξξ − C C ( V W ) ξ − C + C + 3 C C + 3 C C C W + 6 C ( C + C ) C V W + 3 C C + 3 C C + 6 C C C − C C W ξ + 3 C ( C − C ) C V P Q (21d) − C C Q ( V P ξ ) ξ + 3 C C P ( V Q ξ ) ξ + 3 C ( C − C ) C V ξ P Q + 6 C C V ( C P ξ Q − C P Q ξ ) = 0 . which (21a) satisfies the Painlev´e II equation. These reduction equations are linear ODEs whilethe previous functions are known, we can solve V , P , Q , and W one after another in principle.The group invariant solution of an interaction solution among a Painlev´e II wave and a soliton isobtained with the Lie point symmetry theory. V. GENERALIZED TANH FUNCTION EXPANSION METHOD OF BSMKDV-2SYSTEM
The truncated Painlev´e expansion approach and the generalized tanh function expansionmethod are established to find interactions among different nonlinear excitations [31]. The methodsare valid for all integrable systems and or even nonintegrable models because both the truncatedPainlev´e analysis and the tanh expansion method can be used to find exact solutions of the partiallysolvable nonlinear models [21, 31].According to the usual tanh function expansion method, the generalized expansion solution hasthe form v = v + v tanh( f ) , (22a) p = p + p tanh( f ) , (22b) q = q + q tanh( f ) , (22c) w = w + w tanh( f ) + w tanh ( f ) . (22d)where v , v , p , p , q , q , w , w , w and f are functions of { x, t } and should be determinedlater. After some detail calculations by substituting (22) into the BSmKdV-2 system (4), we canprove the following nonauto-BT theorem. Theorem (Nonauto-BT theorem). If { f, g, h, n } is a solution of f t + f xxx − f x − f xx f x = 0 , (23a) g t + g xxx − g x f x − g xx f xx f x + g x f t f x + g x f xx f x − g f t f xx f x = 0 , (23b) h t + h xxx − h x f x − h xx f xx f x + h x f t f x + h x f xx f x − h f t f xx f x = 0 , (23c) n t + n xxx − n xx f xx f x + n x (cid:0) f xx f x + 3 f t f x − f x (cid:1) − n (cid:0) f t f xx f x + 3 f xx f x − f x f xx (cid:1) + 6 f x ( gh x − hg x )+ 3 f t f x ( hg x − gh x ) + 9 f xx f x ( hg x − gh x ) + 3 f xx f x ( hg xx − gh xx ) + 3 g x h xx − h x g xx = 0 . (23d)then { v, p, q, w } with v = − f xx f x + f x tanh( f ) , (24a) p = f xx f x g − g x f x + g tanh( f ) , (24b) q = f xx f x h − h x f x + h tanh( f ) , (24c) w = n xx f x − n x f xx f x + n ( f t f x + 3 f xx f x −
2) + 12 f x ( gh xx − hg xx ) + f xx f x ( hg x − gh x )+ ( f xx nf x − n x f x ) tanh( f ) + n tanh ( f ) . (24d)is a solution of the BSmKdV-2 system (4).We can find some nontrivial solutions of the BSmKdV-2 from some quite trivial solutions of(23). Here we list an interesting examples. A quite trivial straight-line solution of (23) has theform f = k x + ω t + l , g = k x + ω t + l , h = k x + ω t + l , n = k x + ω t + l ,ω = 2 k , ω = 3 k k , ω = 3 k k , ω = 3 k (2 k + k l − k l ) , (25)where k , k , k , k , l , l , l and l are all the free constants. Substituting the line solution (25)into the nonauto-BT theorem yields the following soliton solution of BSmKdV-2 system v = k tanh( f ) , (26a) p = g tanh( f ) − k k , (26b) q = h tanh( f ) − k k , (26c) w = n tanh ( f ) − k k tanh( f ) − n. (26d)Though the soliton solution (26) is a traveling wave in the space time { x, t } for the boson field v , it is not a traveling wave for other boson fields p , q and w , then the superfiled Φ of SmKdVis not a traveling wave except for the case of g , h and n being constants, i.e., k = k = k = 0.This example reveals that an straightening the single soliton to a straight-line solution for theBSmKdV-2 is given by the nonauto-BT theorem. VI. CONCLUSIONS
In summary, the bosonization procedure has been successfully applied to the SmKdV equation.The SmKdV equation is simplified to the mKdV equation together with three linear differentialequations. The BSmKdV-2 system is proved to possess Painlev´e property and to be completelyintegrable. The auto-BT and nonauto-BT are constructed by truncating the standard Painlev´eexpansion.The traveling wave solutions are studied by using the mapping and deformation method. Somespecial types of exact solutions can be given straightforwardly through the exact solutions of themKdV equation and its symmetries. In addition, the group invariant solutions of the system arederived with the Lie point symmetry method. The generalized tanh function expansion method isdeveloped to find interaction solutions among different nonlinear excitations. Straightening a singlesoliton to a straight-line solution for the BSmKdV-2 system is constructed with the generalizedtanh function expansion method. Using the nonauto-BT theorem, various exact explicit solutionsof the BSmKdV-2 system can be obtained. All these solutions obtained via the bosonizationprocedure are different from those obtained via other methods such as the Hirota bilinear methodand the Riemann theta function [10, 22].In this paper, the properties and exact solutions of the BSmKdV-2 system are investigated, wecan also introduce N fermionic parameters to expand the SmKdV system (BSmKdV-N). For the N ≥ ζ i ( i = 1 , , ..., N ) instance, the component fields u and ξ expand ξ ( x, t ) = [ N +12 ] X n =1 X ≤ i < ···
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