Supersymmetric Ito Equation: Bosonization and Exact Solutions Bo Ren, Ji Lin, and Jun Yu

Supersymmetric Ito equation: Bosonization and exact solutions Bo Ren, Ji Lin, and Jun Yu

Citation: AIP Advances 3, 042129 (2013); doi: 10.1063/1.4802969 View online: http://dx.doi.org/10.1063/1.4802969 View Table of Contents: http://aipadvances.aip.org/resource/1/AAIDBI/v3/i4 Published by the American Institute of Physics.

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Supersymmetric Ito equation: Bosonization and exact solutions Bo Ren,1 Ji Lin,2 and Jun Yu1,a 1Institute of Nonlinear Science, Shaoxing University, Shaoxing 312000, China 2Institute of Nonlinear Physics, ZheJiang Normal University, Jinhua, 321004, China (Received 13 September 2012; accepted 10 April 2013; published online 19 April 2013)

Based on the bosonization approach, the N = 1 supersymmetric Ito (sIto) system is changed to a system of coupled bosonic equations. The approach can effectively avoid difﬁculties caused by intractable fermionic ﬁelds which are anticommuting. By solving the coupled bosonic equations, the traveling wave solutions of the sIto system are obtained with the mapping and deformation method. Some novel types of exact solutions for the supersymmetric system are constructed with the solutions and symmetries of the usual Ito equation. In the meanwhile, the similarity reduction solutions of the model are also studied with the Lie point symmetry theory. Copyright 2013 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4802969]

I. INTRODUCTION The study of supersymmetric integrable systems has assumed great importance in the past few years. It has played a more and more important role in physics and mathematics, especially, in modern particle physics and quantum mechanics.1, 2 Over the last three decades, a number of important soliton systems, such as Korteweg-de Vries, Sine-Gorden, KP hierarchy and nonlinear Schordinger¨ equation have been embedded into their supersymmetric counterparts.3–8 Though the supersymmetric integrable systems have been studied by many authors in the both quantum and classical levels, many important problems are still open.9 How to find a proper bosonization procedure is an important problem for quantum and classical supersymmetric integrable models.10 Recently, a simple bosonization approach to treat the super integrable systems has been proposed.11, 12 One essential advantage of the method is that it can effectively avoid difficulties caused by intractable fermionic fields. In this letter, we shall use the bosonization approach on the N = 1 supersymmetric Ito (sIto) system. And then many kinds of solutions for the sIto system are obtained through the coupled bosonic equations. The well known Ito equation is

utt + 6uxxut + 6ux uxt + uxxxt = 0, (1) which was ﬁrst proposed by Ito, and its bilinear Backlund¨ transformation, Lax pair and multi-soliton solutions were obtained.13 Since Ito equation possesses typical properties of a soliton equation, a great deal of research works on the Ito equation have been carried out. The integrable properties of this equation such as the nonlinear superposition formula, Kac-Moody algebra, bi-Hamiltonian structure have been further found.14–16 Recently, the supersymmetric version of the Ito equation and its one, two and three soliton solutions were obtained.17 The N = 1 sIto system reads as17

Dt t + 6(x (Dt ))x + Dt xxx = 0, (2)

aElectronic mail: [email protected].

2158-3226/2013/3(4)/042129/12 3, 042129-1 C Author(s) 2013

where Dt = ∂θ + θ∂t is the covariant derivative. It is established with usual spatial variable x and super temporal variables (t, θ), where θ is a Grassmann variable, and the ﬁeld u leads to a fermionic superﬁeld (θ, x, t) = ξ(x, t) + θu(x, t). The component version of (2) reads as

utt + 6uxxut + 6ux uxt + uxxxt + 6(ξx ξ)x = 0, (3a)

ξt + 6ξx ux + 6uxxξ + ξxxx = 0, (3b) where u and ξ are bosonic and fermionic component ﬁelds, respectively. When the fermionic sectors vanish, the supersymmetric system degenerates to the known classical one. The structure of this paper is organized as follows. In section II, we shall use the bosonization approach on the N = 1 sIto system by expanding the superﬁelds with introducing two fermionic parameters. And then the exact solutions of the model are found using the mapping and deformation method and Lie point symmetry. In Sections III and IV, we extend the bosonization approach for the sIto system to the case of three fermionic parameters and N fermionic parameters, respectively. The exact solutions are obtained with the same methods. The last section is a simple summary and discussion.

II. TWO FERMIONIC PARAMETERS BOSIONIZATION AND ITS SOLUTIONS A. Bosonization approach with two fermionic parameters The supersymmetric extension of the Ito equation is a system of coupled equations for a commut- ing and an anticommuting field. In order to avoid the difficulties in dealing with the anticommutative fermionic field of the supersymmetric equations, we expand the component fields ξ and u with the two fermionic parameters11, 12

ξ(x, t) = p1ζ1 + p2ζ2, (4a)

u(x, t) = u0 + u12ζ1ζ2, (4b)

where ζ 1 and ζ 2 are two Grassmann parameters, while the coefﬁcients p1 = p1(x, t), p2 = p2(x, t), u0 = u0(x, t) and u12 = u12(x, t) are four usual real or complex functions with respect to the spacetime variables x and t. Then, substituting (4) into the Ito system (3), we get the equations

u0,tt + u0,xxxt + 6u0,xxu0,t + 6u0,x u0,xt = 0, (5a)

p1,t + p1,xxx + 6p1,x u0,x + 6p1u0,xx = 0, (5b)

p2,t + p2,xxx + 6p2,x u0,x + 6p2u0,xx = 0, (5c)

u12,tt + u12,xxxt + 6u12,xxu0,t + 6u0,xxu12,t + 6(u0,x u12,x )t = 6(p1 p2,xx − p2 p1,xx). (5d) The above way is just the bosonic procedure for the sIto system (3) with two fermionic parameters. Equation (5a) is exactly the usual Ito equation which has been widely studied.14–16 Equations (5b) and (5c) are linear homogeneous in p1 and p2, respectively. Equation (5d) is linear nonhomogeneous in u12. These equations which are usual pure bosonic systems can be easily solved in principle. This is just one of the advantages of the bosonization approach.

B. Traveling wave solutions with mapping and deformation method Now let us consider the traveling wave solutions of the bosonic (5). Introducing the traveling wave variable X = kx + ωt + c0 with constants k, ω and c0, (5) is transformed to the ordinary

differential equations (ODEs)

3 2 k u0,XXXX + (ω + 12k u0,X )u0,XX = 0, (6a)

3 2 2 k p1,XXX + ωp1,X + 6k p1,X u0,X + 6k p1u0,XX = 0, (6b)

3 2 2 k p2,XXX + ωp2,X + 6k p2,X u0,X + 6k p2u0,XX = 0, (6c)

3 2 2 2 k ωu12,XXXX + ω u12,XX + 12k ω(u0,X u12,X )X = 6k p1 p2,XX − p2 p1,XX . (6d) As the well know exact solutions of (6a), we try to build the mapping and deformation relationship between the traveling wave solutions of the usual Ito equation and the sIto equation, then to get the exact solutions of the sIto equation with the known solutions of Ito equation. At ﬁrst, we solve out u0, X from (6a) 2 ω u0 u , =− − . (7) 0 X k 4k2 To simplify (6d), we integrate the linear inhomogeneous equation once

3 2 2 2 k ωu12,XXX + (ω + 12k ωu0,X )u12,X = 6k (p1 p2,X − p2 p1,X ) + c1, (8)

where c1 is the integral constant. In order to get the mapping relationship between p1, p2, u12 and u0, we introduce the variable transformations as follows

p1(X) = P1(u0(X)), p2(X) = P2(u0(X)), u12(X) = U12(u0(X)). (9)

Using the transformation (9) and vanishing u0, X via (7), the linear ODEs (6b-6c) as well as (8) become d3 P d2 P 2 4 + ω 2 + ω2 1 + 2 3 + ω 1 − 2 = , 16k u0 8k u0 3 96k u0 24k u0 2 192k u0 P1 0 (10a) du0 du0

d3 P d2 P 2 4 + ω 2 + ω2 2 + 2 3 + ω 2 − 2 = , 16k u0 8k u0 3 96k u0 24k u0 2 192k u0 P2 0 (10b) du0 du0

d3U d2U dU 2 + ω 12 + 12 − 12 = , (4ku0 ) 3 24ku0 2 24k F(u0) (10c) du0 du0 du0 where 96k3 dP dP 64k3c F(u ) = 2 P − 1 P + 1 . 0 2 2 1 2 2 4ku ω + ω du0 du0 ω 2 + ω 0 4ku0 The mapping and deformation relations are constructed via (10) √ u0 c2 2 2 1 2 P = + 64c k ω kωu , dy + 4c k u , , (11a) 1 2 3 0 X 2 3 4 0 X 4k u0,X (4ky + ω) √ u0 c5 2 2 1 2 P = + 64c k ω kωu , dy + 4c k u , , (11b) 2 2 6 0 X 2 3 7 0 X 4k u0,X (4ky + ω) √ u0 1 120ku3 + 26ωu U = c 6 kω 20kωu2 + ω dy + 0 0 + c 20ku2 + ω , (11c) 12 8 0 2 + ω 2 + ω 9 0 4ky 4ku0

where ci (i = 2,3,...,9)arearbitraryconstants. The solution U12 in (11c) can be obtained with F(u0) = 0. If we know the solution of u0, the traveling wave solution of two fermionic parameters

sIto system will be given with considering (11), (9) and (4). Here, we list one solution as the example. Thesolutionofu0 can be expressed as the follow form using (7) √ 1 ω kω(X + c ) u =− tan 10 . (12) 0 2 k 2k2 Substituting (12) into (11) and combining (9) and (4), we can get the solution of sIto system. Besides, for any given a solution u0 of the usual Ito equation, a certain type solutions of the bosonic equation (5) can be constructed

p1 = c11u0,t , (13a)

p2 = c12u0,t , (13b)

u12 = σ (u0), (13c)

where σ(u0) represents the symmetry of the usual Ito equation (5a). The solution of sIto system will be expressed

u = u0 + σ(u0)ζ1ζ2,ξ= (c11ζ1 + c12ζ2)u0,t . (14)

While p1 and p2 describe as the form of (13), u0 can be chosen as any solution of the Ito equation. Obviously, the first three equations of the bosonic-looking equations (5) will be satisfied automat- ically. The right hand side of the nonhomogeneous equation (5d) equals to zero with considering (13a) and (13b). Then, u12 of (5d) exactly satisfies the symmetry equation of the usual Ito system. It means that we have much freedom to choose u0 so as to construct solutions of the sIto equations. The solution u0 is not restricted to the traveling wave solutions. All in all, we can construct not only traveling wave solutions but also some novel types of solutions of the sIto system using the solutions and symmetries of the Ito equation. As illustrative example, the N-soliton solution of the Ito equation reads13 N k = + η , u Ito 2 log 1 Aim in exp i (15) > xx k=1 i1>i2>···>ik m n i=1

(k −k )(k3−k3) = i j i j where Aij + 3+ 3 which is the phase shift from the interaction of the soliton “i” with the soli- (ki k j )(ki k j ) η = − 3 + η0 ,η0,η, = , , ··· , ton “j”, i ki x ki t i and arbitrary constants (ki i i i 1 2 N). Correspondingly, a special type of multiple soliton solutions of sIto system reads

u = u Ito + c13(xuIto,x + 3tuIto,t + u Ito)ζ1ζ2, (16a)

ξ = (c11ζ1 + c12ζ2)u Ito,t , (16b)

with the arbitrary constant c13. The multi-soliton solutions obtained here have not yet been obtained by using other methods.

C. Similarity reduction solutions with symmetry reduction approach The Lie group technique provides a very important method in mathematics and physics. Usually, the classical Lie group approach18 and the non-classical Lie group approach19 are powerful and effective methods to obtain the explicit exact solutions.20–22 Now, we shall use the Lie point symmetry approach to study the usual pure bosonic systems. First, we assume the corresponding Lie point symmetry has the vector form ∂ ∂ ∂ ∂ ∂ ∂ V = X + T + U0 + U12 + P1 + P2 , (17) ∂x ∂t ∂u0 ∂u12 ∂p1 ∂p2

where X, T, U0, U12, P1 and P2 are functions with respect to x, t, u0, u12, p1 and p2. It means the system of (5) is invariant under the following transformations

{x, t, u0, u12, p1, p2}→{x + X, t + T, u0 + U0, u12 + U12, p1 + P1, p2 + P2}, (18) with an inﬁnitesimal parameter . The symmetry can be supposed to have the following form

σ0 = Xu0,x + Tu0,t − U0,σ12 = Xu12,x + Tu12,t − U12,σi = Xpi,x + Tpi,t − Pi (i = 1, 2). (19)

Considering the notation (19), σ k is the solution of the linearized equations of (5)

σ0,tt + σ0,xxxt + 6u0,xxσ0,t + 6σ0,xxu0,t + 6(σ0,x u0,x )t = 0, (20a)

σ1,t + σ1,xxx + 6u0,xxσ1 + 6σ0,xxu1,x + 6u1,x σ0,x + 6σ1,x u0,x = 0, (20b)

σ2,t + σ2,xxx + 6u0,xxσ2 + 6σ0,xxu2,x + 6u2,x σ0,x + 6σ2,x u0,x = 0, (20c)

σ12,tt + σ12,xxxt + 6(u0,x σ12,x )t + 6(σ0,x u12,x )t + 6σ0,xxu12,t + 6σ12,xxu0,t + (20d)

6u12,xxσ0,t + 6u0,xxσ12,t = 6(u1σ2,xx − σ2u1,xx + σ1u2,xx − u2σ1,xx),

Substituting (19) into the symmetry equations (20) with u0, u12, p1 and p2 satisfying (5), we obtain the determining equations by identifying all coefﬁcients of derivatives of u0, u12, p1 and p2.The solutions of the functions X, T, U0, U12, P1 and P2 can be concluded using the determining equations C x C T = C t + C , X = 1 + C , U =− 1 u , P = C p + C p , 1 2 3 3 0 3 0 1 7 1 8 2 (−4C − 3C + 3C ) P = 1 7 5 p + C p , U = C u + C , (21) 2 3 2 9 1 12 5 12 6

where Ci (i = 1,2,...,9)arearbitraryconstants. Then, one can solve the characteristic equations to obtain similarity solutions dx dt du dt dp dt dp dt du dt = , 0 = , 1 = , 2 = , 12 = , (22) X T U0 T P1 T P2 T U12 T

where X, T, U0, U12, P1 and P2 are given by (21). Two subcases are distinguished concerning the solutions of (5) in the following.

= → 1 → = Case I. When C1 0, we make the transformations C3 3 C1C3, C2 C1C2 and C6 C8 = C9 = 0, the similarity solutions for the u0, u12, p1 and p2 express as − 1 u0 = U0(ξ)(t + C2) 3 , (23a)

C7 C p1 = P1(ξ)(t + C2) 1 , (23b)

− C5 C7 − 4 C 3 p2 = P2(ξ)(t + C2) 1 , (23c)

C5 C u12 = U12(ξ)(t + C2) 1 . (23d)

− 1 where the similarity variable is ξ = (x + C3)(t + C2) 3 . Substituting (23) into (5), the reduction equations lead to ξ + − ξ 2 − ξ − + 2 + ξ + = , 3 U0,ξξξξ 12U0,ξξξ U0,ξξ 6 U0,ξ 4U0 36U0,ξ 36 U0,ξξ U0,ξ 18U0U0,ξξ 0 (24a)

3C7 3P1,ξξξ − ξ P1,ξ + P1 + 18(U0,ξ P1)ξ = 0, (24b) C1

3C5 − 3C7 − 4 3P2,ξξξ − ξ P2,ξ + P2 + 18(U0,ξ P2)ξ = 0, (24c) C1 9C5 2 54C5 6C5 3ξU12,ξξξξ + 9 − U12,ξξξ + (18U0 − ξ )U12,ξξ + 54U0,ξ − U0,ξ + ξ − 4ξ U12,ξ C1 C1 C1 2 9C5 9C5 54C5 + − − ,ξξ + ξ ,ξ ,ξ ξ − ,ξξ + ,ξξ = . 2 U0 U12 36 (U0 U12 ) 54(P2 P1 P1 P2 ) 0 (24d) C1 C1 C1 These reduction equations are linear ODEs while the previous functions are known, we can theoretically solve U0, P1, P2 and U12 one after another.

Case II. When C1 = 0, we make C4 = C5 = C6 = C7 = 0, we can ﬁnd the similarity solutions after solving out the characteristic equations

u0 = U0(ξ), u12 = U12(ξ), p1 = P1(ξ), p2 = P2(ξ), (25)

with the similarity variable ξ = t − (C2/C3)x. We redeﬁne the similarity variable as ξ = x + ct with c an arbitrary velocity constant. Substituting (25) into (5), the invariant functions U0, P1, P2, U12 satisfy the following reduction systems

U0,ξξξξ + cU0,ξξ + 12U0,ξξ U0,ξ = 0, (26a)

P1,ξξξ + cP1,ξ + 6(P1U0,ξ )ξ = 0, (26b)

P2,ξξξ + cP2,ξ + 6(P2U0,ξ )ξ = 0, (26c)

2 cU12,ξξξξ + c U12,ξξ + 12(U12,ξ U0,ξ )ξ + 6P2 P1,ξξ − 6P1 P2,ξξ = 0. (26d) In the same procedures, (26) can theoretically be solved one after another.

III. THREE FERMIONIC PARAMETERS BOSONIZATION Similar the processes of the last section, we use the bosonization approach with three fermionic parameters. Then, the general traveling wave solutions of the model are found with the mapping and deformation method. The similarity solutions will also study with the Lie point symmetry theory.

A. Bosonization approach with three fermionic parameters

In the case of three Grassmann parameters ζ 1, ζ 2 and ζ 3, the component ﬁelds ξ and u expand

ξ(x, t) = p1ζ1 + p2ζ2 + p3ζ3 + p123ζ1ζ2ζ3 (27a)

u(x, t) = u0 + u12ζ1ζ2 + u23ζ2ζ3 + u13ζ1ζ3, (27b)

where ζ i (i = 1, 2, 3) are Grassmann parameters, while the coefﬁcients u0, uij (ij = 12, 23, 13), pi = pi(x, t)(i = 1, 2, 3) and p123 are eight usual real or complex functions, the sIto system (3) is changed to

u0,tt + u0,xxxt + 6u0,x u0,xt + 6u0,xxu0,t = 0, (28a)

pi,t + pi,xxx + 6pi,x u0,x + 6pi u0,xx = 0, (28b)

uij,tt + uij,xxxt + 6uij,xxu0,t + 6u0,xxuij,t + 6(u0,x uij,x )t = 6(pi p j,xx − p j pi,xx), (28c)

p123,t + p123,xxx + 6(p123u0,x )x =−6(p1u23,x + p2u13,x + p3u12,x )x . (28d) Equation (28) is just the bosonic looking of the sIto system (3) in three fermionic parameter case.

B. Traveling wave solutions with mapping and deformation method Now we want to get the traveling wave solutions of the bosonic equations (28). Introducing the traveling wave variable X = kx + ωt + c0, (28) is transformed to the ODEs 3 2 k u0,XXXX + (ω + 12k u0,X )u0,XX = 0, (29a)

3 2 2 k pi,XXX + ωpi,X + 6k pi,X u0,X + 6k pi u0,XX = 0, (29b)

3 2 2 2 k ωuij,XXXX + ω uij,XX + 12k ω(u0,X uij,X )X = 6k pi p j,XX − p j pi,XX , (29c)

3 2 2 k p123,XXX + ωp123,X + 6k (p123u0,X )X + 6k (p1u23,X + p2u13,X + p3u12,X )X = 0. (29d) The approach adopted in the previous section, we ﬁrst integrate the inhomogeneous ODEs (30c)-(30d) once and get 3 2 2 2 k ωuij,XXX + (ω + 12k ωu0,X )uij,X = 6k (pi p j,X − p j pi,X ) − ci , (30a)

3 2 2 k p123,XX + (ω + 6k u0,X )p123 =−6k (p1u23,X + p2u13,X + p3u12,X ) − c4, (30b)

where cj (j = 1, 2, 3, 4) are arbitrary constants. We consider the variable transformations as follows

pi (X) = Pi (u0(X)), p123(X) = P123(u0(X)), uij(X) = Uij(u0(X)). (31)

With the above transformation and eliminating u0, X via (7), the linear ODEs (30b) as well as (30) are changed to d3 P d2 P 2 4 + ω 2 + ω2 i + 2 3 + ω i − 2 = , 16k u0 8k u0 3 96k u0 24k u0 2 192k u0 Pi 0 (32a) du0 du0

d3U d2U dU 2 + ω ij + ij − ij = , = , , , 4ku0 3 24ku0 2 24k Fk (u0) (k 3 1 2) (32b) du0 du0 du0

d2U dU 2 4 + ω 2 + ω2 123 + 2 3 + ω 123 − 2 + ω = , 16k u0 8k u0 2 32k u0 8k u0 96k u0 8k U123 F4(u0) du0 du0 (32c) where 96k3 dP dP 64k3c F (u ) = j P − i P + i k 0 2 2 i j 2 4kωu + ω du0 du0 ω 2 + ω 0 4ku0 dU dU dU = 2 + ω 23 + 13 + 12 + F4(u0) 24k 4ku0 P1 P2 P3 c4 du0 du0 du0 The general three fermionic parameters traveling wave solution can be obtained using (32) √ u0 c4+i 2 2 1 2 P = + 64c + k ω kωu , dy + 4c + k u , , (33a) i 2 5 i 0 X 2 3 6 i 0 X 4k u0,X (4ky + ω)

Downloaded 20 Apr 2013 to 117.147.163.195. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/ 042129-8 Ren, Lin, and Yu AIP Advances 3, 042129 (2013) √ u0 1 120ku3 + 26ωu U = a 6 kω 20kωu2 + ω dy + 0 0 + b 20ku2 + ω , ij ij 0 2 + ω 2 + ω ij 0 4ky 4ku0 (33b) y u0 + 2 c11 F4(y1)dy1 P = 4k u , c + dy , (33c) 123 0 X 10 (4ky2 + ω2)3

with aij, bij, ci (i = 5, 6, ···, 11) are arbitrary constants. Similar to the two fermionic parameters case, we can write down solutions of (28)

pi = d j u0,t , p123 = d4u0,t , uij = σij(u0), (34)

where dj (j = 1, 2, 3, 4) are arbitrary constants, σ ij(u0) are arbitrary symmetries of the usual Ito equation. Therefore, the sIto system (3) possesses the following special solution

u = u0 + σ12(u0)ζ1ζ2 + σ23(u0)ζ2ζ3 + σ13(u0)ζ1ζ3, (35a)

ξ = (d1ζ1 + d2ζ2 + d3ζ3 + d4ζ1ζ2ζ3)u0,t . (35b)

While one of the Grassmann numbers ζ i (i = 1, 2, 3) tends to zero, the solution (25) turns back to the last section for two fermionic parameters. Actually, applying the similar procedure for any numbers of the fermionic parameters, one can obtain various exact solutions such as the general traveling wave solution and the special solutions like (35).

C. Similarity reduction solutions with symmetry reduction approach Due to the same way, we only write the two cases of similarity solutions and reduction equations for the model (28).

= → 1 → Case I. When C1 0, for simplicity, we make the transformations C3 2 C1C3 and C2 C1C2, and retain the terms with same index of the invariant functions and the bosonic systems’s functions, the similarity solutions become

− 1 u0 = U0(ξ)(t + C2) 3 , (36a)

+ − C4 C5 C6 − 2 2C 3 p1 = P1(ξ)(t + C2) 1 , (36b)

+ − C6 C4 C5 − 2 2C 3 p2 = P2(ξ)(t + C2) 1 , (36c)

+ − C5 C6 C4 − 2 2C 3 p3 = P3(ξ)(t + C2) 1 , (36d)

C7−k C uij = Uij(ξ)(t + C2) 1 , (36e)

+ + C4 C5 C6 − 1 2C 3 p123 = P123(ξ)(t + C2) 1 , (36f)

− 1 where ξ = (x + C3)(t + C2) 3 is group invariant. The invariant functions U0, Uij, Pi and P123 satisfy the following reduction equations ξ + − ξ 2 − ξ − + 2 + ξ + = , 3 U0,ξξξξ 12U0,ξξξ U0,ξξ 6 U0,ξ 4U0 36U0,ξ 36 U0,ξξ U0,ξ 18U0U0,ξξ 0 (37a)

3C4 + 3C5 − 3C6 − 4C1 6P1,ξξξ − 2ξ P1,ξ + P1 + 36U0,ξ P1,ξ + 36U0,ξξ P1 = 0, (37b) C1

3C6 + 3C4 − 3C5 − 4C1 6P2,ξξξ − 2ξ P2,ξ + P2 + 36U0,ξ P2,ξ + 36U0,ξξ P2 = 0, (37c) C1

3C5 + 3C6 − 3C4 − 4C1 6P3,ξξξ − 2ξ P3,ξ + P3 + 36U0,ξ P3,ξ + 36U0,ξξ P3 = 0, (37d) C1 9C7−k 2 3ξUij,ξξξξ + 9 − Uij,ξξξ + (18U0 − ξ )Uij,ξξ C1 54C7−k 6C7−k + 54U0,ξ − U0,ξ + ξ − 4ξ Uij,ξ C1 C1 2 9C7−k 9C7−k 54C7−k + − − ,ξξ + ξ ,ξ ,ξ ξ 2 U0 Uij 36 (U0 Uij ) C1 C1 C1

− 54(Pj Pi,ξξ + Pi Pj,ξξ ) = 0, (37e)

3C4 + 3C5 + 3C6 − 2C1 6U123,ξξξ + (36U0,ξ − 2ξ)U123,ξ + 36U0,ξξ U123 + U123 C1

+ 36(U1,ξ U23,ξ + U2,ξ U13,ξ + U3,ξ U12,ξ + U1U23,ξξ + U2U13,ξξ + U3U12,ξξ ) = 0. (37f)

Case II. When C1 = 0, for simplicity, we make the invariant functions equal to zero, and then we can ﬁnd the general similarity solutions

u0 = U0(ξ), uij = Uij(ξ), pi = Pi (ξ), p123 = P123(ξ). (38) The reduction equations lead to

U0,ξξξξ + cU0,ξξ + 12U0,ξξ U0,ξ = 0, (39a)

Pi,ξξξ + cPi,ξ + 6(Pi U0,ξ )ξ = 0, (39b)

2 cUij,ξξξξ + c Uij,ξξ + 12(Uij,ξ U0,ξ )ξ + 6Pj Pi,ξξ − 6Pi Pj,ξξ = 0, (39c)

P123,ξξξ + cP123,ξ + 6(P123U0,ξ )ξ + 6(P1U23,ξ + P2U13,ξ + P3U12,ξ )ξ = 0, (39d) where the similarity variable ξ = x + ct. Similar to the two fermionic parameter section, (39) can theoretically be solved one after another.

IV. N FERMIONIC PARAMETERS BOSONIZATION We can also introduce N fermionic parameters to expand the sIto system. For the N ≥ 2 fermionic parameters ζ i (i = 1, 2, . . . , N) instance, the component ﬁelds u and ξ can be expanded as

[ N+1 ] 2 ξ , = ζ ζ ···ζ , (x t) pi1i2···i2n−1 i1 i2 i2n−1 (40a)

n=1 1≤i1<···

[ N+1 ] 2 , = + ζ ζ ···ζ , u(x t) u0 ui1i2···i2n i1 i2 i2n (40b)

n=1 1≤i1<···

, ≤ < ···< ≤ ≤ < ···< where the coefﬁcients u0 ui1i2···i2n (1 i1 i2n N) and pi1i2···i2n−1 (1 i1 i2n−1 ≤ N)are2N real or complex bosonic functions of classical spacetime variable x, t. Substituting (41) into the sIto model (3), we obtain the following bosonic system of 2N equations

u0,tt + u0,xxxt + 6u0,x u0,xt + 6u0,xxu0,t = 0, (41a)

⎧ ⎨ 0, n = 1; ˆ = Lo pi1i2···i2n−1 ⎩ τ( j , j ,..., j − ) N+1 −6 (−1) 1 2 2n 1 (p ··· u ··· , ) , n = 2, 3,...,[ ], W1 i j1 i j2 i j2l i j2l+1 i j2l+2 i j2n−1 x x 2 (41b) ⎧ τ( j , j ) ⎪ 6 (−1) 1 2 (pi pi ,x )x , n = 1; ⎪ W2 j1 j2 ⎨ τ , ,..., ˆ = − ( j1 j2 j2n ) Leui i ···i 6 ( 1) (pi j i j ···i j pi j i j ···i j ,x )x (41c) 1 2 2n ⎪ W2 1 2 2l−1 2l 2l+1 2n ⎪ ⎩ τ( j , j ,..., j ) N −6 (−1) 1 2 2n (u ··· , u ··· , ) , n = 2, 3,...,[ ], W3 i j1 i j2 i j2l x i j2l+1 i j2l+2 i j2n t x 2 where

Lˆ o = ∂tt + ∂xxxt + 6u0,x ∂x + 6u0,xx,

Lˆ e = ∂tt + ∂xxxt + 6u0,t ∂xx + 6u0,xx∂t + 6u0,xt∂x + 6u0,x ∂xt,

0, j1, j2,..., jN is even permutation; τ( j1, j2,..., jN ) = 1, j1, j2,..., jN is odd permutation,

W1 ={( j1, j2,..., j2n−1)|1 ≤ j1 < j2 < ···< j2l ≤ 2n − 1, 1 ≤ j2l+1 < j2l+2 < ···< j2n−1 ≤ − , ≤ ≤ − , = = }, 2n 1 1 l n 1 jh1 jh2 (h1 h2)

W2 ={( j1, j2,..., j2n)|1 ≤ j1 < j2 < ···< j2l−1 ≤ 2n, 1 ≤ j2l < j2l+1 < ···< j2n ≤ 2n, ≤ ≤ , = = }, 1 l n jh1 jh2 (h1 h2)

W3 ={( j1, j2,..., j2n)|1 ≤ j1 < j2 < ···< j2l ≤ 2n, 1 ≤ j2l+1 < j2l+2 < ···< j2n ≤ 2n, ≤ ≤ − , = = }. 1 l n 1 jh1 jh2 (h1 h2) The general traveling wave solution of the sIto equation with N fermionic parameters can be written [ N+1 ] [ N+1 ] 2 2 = ζ ζ ···ζ + θ + ζ ζ ···ζ , pi1i2···i2n−1 i1 i2 i2n−1 u0 ui1i2···i2n i1 i2 i2n

n=1 1≤i1<···

with τ , ,··· , = 2 + ω − ( j1 j2 j2n−1) + , F ··· − (u ) 24k 4ku ( 1) P ··· U ··· , e ··· − i1i2 i2n 1 0 0 i j1 i j2 i2l i j2l+1 i j2l+2 i j2n−1 u0 i1i2 i2n 1 W1

where u0 represent the solution of the usual Ito equation, ai1i2 , bi1i2 , ci1i2 , ai1i2···i2n−1 , bi1i2···i2n−1 , ci1i2···i2n ,

di1i2···i2n and ei1i2···i2n−1 are arbitrary constants. The reduction equations of the sIto equation with the N fermionic parameters can be also obtained. Here we just list case II as the example

U0,ξξξξ + (12U0,ξ + c)U0,ξξ = 0, (43a)

0, n = 1; ˆ = L1 Pi i ···i − 1 2 2n 1 τ( j , j ,..., j − ) N+1 −6 (−1) 1 2 2n 1 P ··· U ··· ,ξ , n = 2, 3,...,[ ], W1 i j1 i j2 i j2l i j2l+1 i j2l+2 i j2n−1 ξ 2 (43b) ⎧ τ( j , j ) ⎪ − 1 2 ,ξ , = ⎪ 6 W ( 1) Pi j Pi j ξ n 1; ⎨ 2 1 2 τ( j , j ,..., j ) ˆ = − 1 2 2n ··· ··· ,ξ L2Ui i ···i 6 W ( 1) Pi j i j i j − Pi j i j + i j ξ 1 2 2n ⎪ 2 1 2 2l 1 2l 2l 1 2n ⎩ τ( j , j ,..., j ) N −12c (−1) 1 2 2n U ··· ,ξ U ··· ,ξ , n = 2, 3,...,[ ], W3 i j1 i j2 i j2l i j2l+1 i j2l+2 i j2n ξ 2 (43c) where two operators are

Lˆ 1 = ∂ξξξ + c∂ξ + 6U0,ξξ + 6U0,ξ ∂ξ 2 Lˆ 2 = c∂ξξξξ + c ∂ξξ + 12U0,ξξ ∂ξ + 12U0,ξ ∂ξξ In principle, we can solve (44) one after another.

V. CONCLUSIONS In summary, by means of the bosonization approach, the N = 1 supersymmetry Ito system is changed to a system of coupled bosonic equations. The coupled bosonic systems are obtained with introducing two, three and N fermionic parameters. The systems are just the usual Ito equation together with several linear differential equations. Therefore, the approach can avoid the difﬁculties caused by intractable anticommuting fermionic ﬁelds. Using the the mapping and deformation method, the traveling wave solutions of the bosonized systems are obtained. Besides, some special types of exact solutions can be given straightforwardly through the exact solutions of the Ito equation and its symmetries. In addition, the similarity reduction solutions of the model are derived using the Lie point symmetry theory. The solutions obtained via the bosonization procedure are completely different from those obtained via other methods such as the bilinear approach.23 The bosonization procedure should be effectively applicable to not only the supersymmetric integrable systems but also other not integrable supersymmetric systems. The method is powerful tool for solving these problems and future work on these aspects is worthy of studying.

ACKNOWLEDGMENT We would like to thank S. Y. Lou for useful discussions. This work was partially supported by the National Natural Science Foundation of China Nos. 11175158 and 11275129 and the Starting Research Funds from the Shaoxing University under Grant No. 20125035. The authors are indebted to the referees’ useful comments.

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