Discrete Symmetries and Nonlocal Reductions
aa r X i v : . [ n li n . S I] J un Discrete Symmetries and Nonlocal Reductions
Metin G¨urses ∗ Department of Mathematics, Faculty of SciencesBilkent University, 06800 Ankara, Turkey
Aslı Pekcan † Department of Mathematics , Hacettepe University, 06800 Ankara, Turkey
Konstyantyn Zheltukhin ‡ Department of Mathematics , Middle East Technical University, 06800 Ankara, Turkey
Abstract
We show that nonlocal reductions of systems of integrable nonlinear partial differential equations arethe special discrete symmetry transformations.
Keywords.
Integrable systems, Scale symmetries, Discrete symmetries, Nonlocal reductions.
Nonlocal reductions of systems of integrable nonlinear partial differential equations which were inventedfirst by Ablowitz and Musslimani [1]-[3], attracted many researchers in the field. Ablowitz and Musslimanihave first constructed nonlocal reduction for nonlinear Schr¨odinger (NLS) system of equations and obtainednonlocal nonlinear Schr¨odinger (nNLS) equation [1], [2]. They showed that nNLS equation is integrable, i.e.,it admits a Lax pair, and found soliton solutions by the use of the inverse scattering method. Ablowitz andMusslimani have later extended their nonlocal reductions, corresponding to space reflection, time reflection,and space-time reflection to modified Korteweg-de Vries (mKdV) system, sine-Gordon (SG) system, Davey-Stewartson (DS) system and so on. After Ablowitz and Musslimani’s works there is a huge interest inobtaining nonlocal reductions of systems of integrable equations and finding interesting wave solutions ofthese systems. Specific examples are nonlocal NLS equation [1]-[14], nonlocal mKdV equation [2]-[4], [13], [15]-[18], nonlocal SG equation [2]-[4], [19], nonlocal DS equation [3], [20]-[24], nonlocal Fordy-Kulish equations[13], [25], nonlocal N -wave systems [3], [26], nonlocal vector NLS equations [27]-[30], nonlocal (2 + 1)-dimensional negative AKNS systems [31], nonlocal coupled Hirota-Iwao mKdV systems [32]. See [33] forthe discussion of superposition of nonlocal integrable equations, and [34] for the nonlocal reductions of theintegrable equations of hydrodynamic type. The connection between local and nonlocal reductions is given in[35], [36]. In all these works the soliton solutions and their properties were investigated by using the inversescattering method, by the Hirota bilinear method, and by Darboux transformations. ∗ email: [email protected] † email: [email protected] ‡ email: [email protected]
1n the last decade we observe that even as the number of systems of integrable nonlinear differentialequations possessing nonlocal reductions is increasing, there is no one so far explaining how or where suchnonlocal reductions come from. The origin of nonlocal reductions was mysterious. In this work we addressto this problem. We show that those systems possessing nonlocal reductions admit discrete symmetry trans-formations which leave the systems invariant. A special case of discrete symmetry transformation turns outto be the nonlocal reductions of the same systems. We show this fact for NLS, mKdV, SG, DS, coupledNLS-derivative NLS, loop soliton systems, hydrodynamic type systems, and Fordy-Kulish equations, andderive all possible nonlocal reductions from the discrete symmetry transformations of these systems.
Let the dynamical variables q i ( t, x ) and r i ( t, x ) ( i = 1 , , · · · , N ), in (1 + 1)-dimensions, satisfy the followingsystem of integrable evolution equations q it = F i ( q j , r j , q jx , r jx , q jxx , r jxx , · · · ) , i, j = 1 , , · · · , N, (2.1) r it = G i ( q j , r j , q jx , r jx , q jxx , r jxx , · · · ) , i, j = 1 , , · · · , N, (2.2)where F i and G i ( i = 1 , , · · · , N ) are functions of the dynamical variables q i ( t, x ), r i ( t, x ), and their partialderivatives with respect to x . The above system of equations is integrable, so it has a Lax pair and arecursion operator R . Some of these equations admit local and nonlocal reductions. Let us assume that theabove system of equations (2.1) and (2.2) admits the following reductions.(a) Local reductions:The local reductions are given by r i ( t, x ) = κ q i ( t, x ) , i = 1 , , · · · , N, (2.3)and r i ( t, x ) = κ ¯ q i ( t, x ) , i = 1 , , · · · , N, (2.4)where κ and κ are real constants. Throughout this paper a bar over a letter is defined as1) for a complex number q = α + iβ , ¯ q = α − iβ , i = − q = α + iβ , ¯ q = α − iβ , i = 1.If a reduction is consistent the system of equations (2.1) and (2.2) is reduced to a system for q i ’s q it = ˜ F i ( q j , q jx , q jxx , · · · ) , i, j = 1 , , · · · , N (2.5)for the reduction (2.3) and q it = ˜ F i ( q j , ¯ q j , q jx , ¯ q jx , q jxx , ¯ q jxx , · · · ) , i, j = 1 , , · · · , N (2.6)2or the reduction (2.4), where ˜ F = F | r i = κ q i or ˜ F = F | r i = κ ¯ q i , respectively. (b) Nonlocal reductions:Recently, Ablowitz and Musslimani introduced new type of reductions [1]-[3] r i ( t, x ) = τ q i ( ε t, ε x ) = τ q iε , (2.7)and r i ( t, x ) = τ ¯ q i ( ε t, ε x ) = τ ¯ q iε , (2.8)for i = 1 , , · · · , N . Here τ and τ are real constants and ε = ε = 1.When ( ε , ε ) = ( − , , (1 , − , ( − , −
1) the above constraints reduce the system (2.1) and (2.2) to nonlocalspace reflection symmetric (S-symmetric), time reflection symmetric (T-symmetric), or space-time reflectionsymmetric (ST-symmetric) differential equations.Since the reductions are done consistently the reduced systems of equations are also integrable. This meansthat the reduced systems admit recursion operators and Lax pairs. We can obtain N -soliton solutions ofthe reduced systems by the inverse scattering method [1]-[3], [10], [11], [14], [17], [19], [27], by the Darbouxtransformation [9], [16], [18], [22], [23], and by the Hirota bilinear method [7], [13], [15], [21], [31]-[33]. In this section we will show that nonlocal reductions arise from scaling symmetries of integrable system ofequations. A scaling symmetry of a system of differential equations is the scale transformation which leavesthese equations invariant. Scaling symmetries group is a subgroup of the symmetry groups of differentialequations [37] and discrete symmetries are special cases of the scaling symmetries [38]. (a) NLS System : This system is given by aq t = − q xx + q r, (3.1) ar t = 12 r xx − q r , (3.2)where a is any constant. This constant is the imaginary unit for the original NLS system but we change itby redefining the t variable. We search for a symmetry transformations such that the NLS system is leftinvariant. In general we choose the symmetry transformation as T : ( q ( t, x ) , r ( t, x )) → ( q ′ ( t ′ , x ′ ) , r ′ ( t ′ , x ′ ))where primed system satisfies also the NLS system, i.e., aq ′ t ′ = − q ′ x ′ x ′ + ( q ′ ) r ′ , (3.3) ar ′ t ′ = 12 r ′ x ′ x ′ − q ′ ( r ′ ) . (3.4)3e shall consider the real and complex dynamical systems separately. For the real case the symmetrytransformation that we are interested in is the scale transformations t ′ = β t, x ′ = α x, (3.5) q ′ = γ q + δ r, (3.6) r ′ = γ r + δ q, (3.7)where α, β, γ , γ , δ , and δ are real constants. We have two possible cases:(a) First type of real scale symmetry transformation is t ′ = − α t, x ′ = α x, (3.8) q ′ = δ r, (3.9) r ′ = 1 δ α q, (3.10)where α and δ are arbitrary constants.(b) Second type of real scale symmetry transformation is t ′ = α t, x ′ = α x, (3.11) q ′ = γ q, (3.12) r ′ = 1 γ α r, (3.13)where α and γ are arbitrary constants. These two parameter transformations map solutions to solutions ofthe NLS system.From the above scale symmetry transformation we can obtain discrete symmetry transformations by letting α = ǫ = ±
1. In particular the first type produces a discrete symmetry transformation if α = ǫ and δ = k then q ( t, x ) = k r ′ ( − t, ǫx ) , (3.14) r ( t, x ) = k q ′ ( − t, ǫx ) , (3.15)where ǫ = k = 1. A special discrete symmetry transformation is obtained when we take q ′ = q and r ′ = r .This special discrete symmetry is the well-known nonlocal reductions r ( t, x ) = kq ( − t, x ) and r ( t, x ) = kq ( − t, − x ) [3], [4], [6], [10], [13], [14].For the complex dynamical systems the scale symmetry transformation T : (¯ q ( t, x ) , ¯ r ( t, x )) → ( q ′ ( t ′ , x ′ ) , r ′ ( t ′ , x ′ ))takes the following form t ′ = β t, x ′ = α x, (3.16) q ′ = γ ¯ q + δ ¯ r, (3.17) r ′ = γ ¯ r + δ ¯ q. (3.18)4here α, β, γ , γ , δ and δ are real constants. We have two possible cases:(a) First type of complex scale symmetry transformation is t ′ = β t, x ′ = α x, (3.19) q ′ = δ ¯ r, (3.20) r ′ = δ ¯ q, (3.21)with ¯ a β = − aα , δ δ α = 1 . (3.22)(b) Second type of complex scale symmetry transformation is t ′ = β t, x ′ = α x, (3.23) q ′ = γ ¯ q, (3.24) r ′ = γ ¯ r, (3.25)with ¯ a β = aα , γ γ α = 1 . (3.26)These two parameter transformations map also solutions to solutions of the NLS system. From these scalesymmetry transformations we obtain discrete symmetry transformation by letting α = ǫ = ± β = ǫ = ± γ = γ = k = ±
1. In particular the first type produces a discrete symmetry transformation of the form q ( t, x ) = k ¯ r ′ ( ǫ t, ǫ x ) , (3.27) r ( t, x ) = k ¯ q ′ ( ǫ t, ǫ x ) , (3.28)where ǫ = ǫ = k = 1 and ¯ aǫ = − a which follows from (3.22). A special discrete symmetry transformationis obtained when we take q ′ = q and r ′ = r . This special symmetry is the well-known nonlocal reductions r ( t, x ) = k ¯ q ( − t, x ) with ¯ a = − a , r ( t, x ) = k ¯ q ( t, − x ) with ¯ a = a , and r ( t, x ) = k ¯ q ( − t, − x ) with ¯ a = − a [1],[2], [4]-[9], [11]-[14].The examples that we consider in the rest of the paper share similar real and complex scale symmetrytransformations and the associated discrete symmetry transformations. Since we are interested in nonlocalreductions of the integrable systems of equations we will present only the first type real and complex discretetransformations and the corresponding nonlocal reductions. (b) MKdV System : This system is given by aq t = − q xxx + 32 q r q x , (3.29) ar t = − r xxx + 32 q r r x . (3.30)We will write the discrete symmetry transformations directly. We have two different cases: Let ( q, r ) and( q ′ , r ′ ) satisfy the mKdV system of equations (3.29) and (3.30).5or the real case we have q ( t, x ) = kr ′ ( ǫ t, ǫ x ) , r ( t, x ) = kq ′ ( ǫ t, ǫ x ) , (3.31)where k = 1 and ǫ ǫ = 1. When we take q ′ = q and r ′ = r we obtain the nonlocal reduction r ( t, x ) = k q ( − t, − x ) [2]-[4], [13], [15]-[17].For the complex case we have q ( t, x ) = k ¯ r ′ ( ǫ t, ǫ x ) , r ( t, x ) = k ¯ q ′ ( ǫ t, ǫ x ) , (3.32)where ¯ a ǫ ǫ = a and k = 1. These special discrete transformations produce different nonlocal reductionswhen q ′ = q and r ′ = r with different values of ǫ = ± ǫ = ± r ( t, x ) = k ¯ q ( − t, x ) with ¯ a = − a , r ( t, x ) = k ¯ q ( t, − x ) with ¯ a = − a , and r ( t, x ) = k ¯ q ( − t, − x ) with ¯ a = a [2]-[4], [13], [15], [18]. (c) SG System : This system is given by q xt + 2 s q = 0 , (3.33) r xt + 2 s r = 0 , (3.34) s x + ( q r ) t = 0 , (3.35)where q = q ( t, x ), r = r ( t, x ), and s = s ( t, x ). We have the following two discrete symmetry transformations.For the real case, q ( t, x ) = kr ′ ( ǫ t, ǫ x ) , r ( t, x ) = kq ′ ( ǫ t, ǫ x ) , s ( t, x ) = s ′ ( ǫ t, ǫ x ) , (3.36)where ǫ = ǫ = ± k = 1. If we take q ′ = q and r ′ = r these special discrete transformations producethe nonlocal reductions: r ( t, x ) = kq ( − t, x ), r ( t, x ) = kq ( t, − x ), and r ( t, x ) = kq ( − t, − x ) [2]-[4], [19].For the complex case, q ( t, x ) = k ¯ r ′ ( ǫ t, ǫ x ) , r ( t, x ) = k ¯ q ′ ( ǫ t, ǫ x ) , s ( t, x ) = ¯ s ′ ( ǫ t, ǫ x ) , (3.37)where ǫ = ǫ = ± k = 1. When q ′ = q and r ′ = r these special discrete transformations produce thenonlocal reductions: r ( t, x ) = k ¯ q ( − t, x ), r ( t, x ) = k ¯ q ( t, − x ), and r ( t, x ) = k ¯ q ( − t, − x ) [4]. (d) DS System : This system is given by aq t + 12 [ γ q xx + q yy ] + q r = φq, (3.38) − ar t + 12 [ γ r xx + r yy ] + r q = φr, (3.39) φ xx − γ φ yy = 2( qr ) xx , (3.40)where q = q ( t, x, y ), r = r ( t, x, y ), φ = φ ( t, x, y ), γ = ±
1, and a is a constant. We have the following discretesymmetry transformations. For the real case, q ( t, x, y ) = kr ′ ( ǫ t, ǫ x, ǫ y ) , (3.41) r ( t, x, y ) = kq ′ ( ǫ t, ǫ x, ǫ y ) , (3.42) φ ( t, x, y ) = φ ′ ( ǫ t, ǫ x, ǫ y ) , (3.43)6here ǫ = − k = 1. These special discrete transformations produce the nonlocal reductions when q ′ = q and r ′ = r with different values of ǫ = − ǫ = ± ǫ = ± r ( t, x, y ) = k q ( − t, x, y ), r ( t, x, y ) = k q ( − t, − x, y ), r ( t, x, y ) = k q ( − t, x, − y ), and r ( t, x, y ) = k q ( − t, − x, − y ) [3].For the complex case, q ( t, x, y ) = k ¯ r ′ ( ǫ t, ǫ x, ǫ y ) , (3.44) r ( t, x, y ) = k ¯ q ′ ( ǫ t, ǫ x, ǫ y ) , (3.45) φ ( t, x, y ) = ¯ φ ′ ( ǫ t, ǫ x, ǫ y ) , (3.46)where k = 1 , ǫ = ǫ = ǫ = 1, and ¯ aǫ = − a . We observe that these discrete transformations producemany different nonlocal reductions when q ′ = q , r ′ = r , and φ ′ = φ with different values of ǫ = ± ǫ = ± ǫ = ± r ( t, x, y ) = k ¯ q ( − t, x, y ), r ( t, x, y ) = k ¯ q ( − t, − x, y ), r ( t, x, y ) = k ¯ q ( − t, x, − y ), r ( t, x, y ) = k ¯ q ( − t, − x, − y ) with ¯ a = a ; r ( t, x, y ) = k ¯ q ( t, − x, y ), r ( t, x, y ) = k ¯ q ( t, x, − y ), r ( t, x, y ) = k ¯ q ( t, − x, − y ) with¯ a = − a [3], [20]-[24]. (e) Coupled NLS-derivative NLS System : This system [39] is given by aq t = iq xx + α ( rq ) x + iβrq , (3.47) ar t = − ir xx + α ( rq ) x − iβr q, (3.48)where α, β ∈ R , and a is any constant. We have the following discrete symmetry transformations. For thereal case, q ( t, x ) = kr ′ ( ǫ t, ǫ x ) , r ( t, x ) = kq ′ ( ǫ t, ǫ x ) , (3.49)where ǫ = ǫ = − k = 1. When q ′ = q and r ′ = r , these discrete transformations produce the nonlocalreduction r ( t, x ) = kq ( − t, − x ) [3].For the complex case, q ( t, x ) = k ¯ r ′ ( ǫ t, ǫ x ) , r ( t, x ) = k ¯ q ′ ( ǫ t, ǫ x ) , (3.50)where ǫ = 1, ¯ aǫ = a , and k = 1. From these discrete transformations we have different nonlocal reductionswhen q ′ = q and r ′ = r with different values of ǫ = ± ǫ = ± r ( t, x ) = k ¯ q ( − t, x ) with ¯ a = a , r ( t, x ) = k ¯ q ( t, − x ) with ¯ a = − a , and r ( t, x ) = k ¯ q ( − t, − x ) with ¯ a = − a . (f ) Loop-soliton System : This system [39], [40] is given by aq t + ∂ ∂x h q x (1 − rq ) / i = 0 , (3.51) ar t + ∂ ∂x h r x (1 − rq ) / i = 0 . (3.52)We have the following discrete symmetry transformations.For the real case, q ( t, x ) = kr ′ ( ǫ t, ǫ x ) , r ( t, x ) = kq ′ ( ǫ t, ǫ x ) , (3.53)where ǫ = ǫ = − k = 1. When q ′ = q and r ′ = r , these discrete transformations produce the nonlocalreduction r ( t, x ) = kq ( − t, − x ) [3]. 7or the complex case, q ( t, x ) = k ¯ r ′ ( ǫ t, ǫ x ) , r ( t, x ) = k ¯ q ′ ( ǫ t, ǫ x ) , (3.54)where ¯ aǫ ǫ = a , and k = 1. These discrete transformations produce different nonlocal reductions when q ′ = q and r ′ = r with different values of ǫ = ± ǫ = ± r ( t, x ) = k ¯ q ( − t, x ) with ¯ a = − a , r ( t, x ) = k ¯ q ( t, − x ) with ¯ a = − a , and r ( t, x ) = k ¯ q ( − t, − x ) with ¯ a = a . (g) Hydrodynamic type of systems: Shallow water waves Recently we studied the reductions in equations of hydrodynamic type [34] and obtained several examplesof nonlocal version of these equations. An example of equations of hydrodynamic type is the shallow waterwaves system [41] aq t = ( q + r ) q x + q r x , (3.55) ar t = ( q + r ) r x + r q x . (3.56)Here a is a nonzero constant. The discrete transformations which leave this system invariant are following.For the real case, r ( t, x ) = k q ′ ( ǫ t, ǫ x ) , (3.57) q ( t, x ) = k r ′ ( ǫ t, ǫ x ) , (3.58)where k = ǫ ǫ . For the complex case r ( t, x ) = k ¯ q ′ ( ǫ t, ǫ x ) , (3.59) q ( t, x ) = k, ¯ r ′ ( ǫ t, ǫ x ) , (3.60)where ¯ a k ǫ ǫ = a . In both cases k = ǫ = ǫ = 1 [34].If we let q ′ = q and r ′ = r we get the special discrete symmetry transformations which lead to the localand nonlocal reductions. When q and r are real variables we have r ( t, x ) = k q ( ǫ t, ǫ x ) then the reducedequation is aq t ( t, x ) = ( q ( t, x ) + kq ( ǫ t, ǫ x )) q x ( t, x ) + kq ( t, x ) q x ( ǫ t, ǫ x ) , (3.61)provided that k = ǫ ǫ and a is real.When q and r are complex variables we have r ( t, x ) = k ¯ q ( ǫ t.ǫ x ) then the reduced equation is aq t ( t, x ) = ( q ( t, x ) + k ¯ q ( ǫ t, ǫ x )) q x ( t, x ) + kq ( t, x ) ¯ q x ( ǫ t, ǫ x ) , (3.62)provided that ¯ ak ǫ ǫ = a [34]. (h) Fordy-Kulish Equations Let q α ( t, x ) and r α ( t, x ) be the complex dynamical variables where α = 1 , , · · · , N , then the Fordy-Kulish(FK) integrable system is given by [42] 8 q αt = q αxx + R α βγ − δ q β q γ r δ , (3.63) − ar αt = r αxx + R − α − βγδ r β r γ q δ , (3.64)where R α βγ − δ , R − α − β − γδ are the curvature tensors of a Hermitian symmetric space with( R α βγ − δ ) ⋆ = R − α − β − γδ , (3.65)and a is a complex number. Here we use the summation convention, i.e., the repeated indices are summedup from 1 to N . These equations are known as the FK system which is integrable in the sense that they areobtained from the zero curvature condition of a connection defined on a Hermitian symmetric space. TheFK equations (3.63) and (3.64) are invariant under the discrete transformations r α ( t, x ) = k ¯ q ′ α ( ǫ t, ǫ x ) , (3.66) q α ( t, x ) = k ¯ r ′ α ( ǫ t, ǫ x ) , (3.67)where k = ǫ = ǫ = 1 and ¯ aǫ = − a . If we let r ′ α = r α and q ′ α = q α we obtain the special discretesymmetry transformations and hence the nonlocal reductions r α ( t, x ) = k ¯ q α ( ǫ t, ǫ x ) [25]. Then the reducednonlocal FK equations are aq αt ( t, x ) = q αxx ( t, x ) + k R α βγ − δ q β ( t, x ) q γ ( t, x ) ¯ q δ ( ǫ t, ǫ x ) . (3.68) In this work we showed that the discrete symmetries of systems of integrable equations are important infinding the nonlocal reductions. For this reason we started first with the scale symmetry transformationsof real and complex dynamical systems. Discrete symmetry transformations are special cases of the scaletransformations. There are two different types of discrete symmetry transformations both for real and com-plex dynamical variables. Using this fact we can find all discrete symmetry transformations of the systemof equations. Among these discrete symmetry transformations the first types are the origins of the nonlocalreductions of these systems. We showed that a special discrete symmetry transformation of the first typeproduces all the well known nonlocal reductions.
This work is partially supported by the Scientific and Technological Research Council of Turkey (T ¨UB˙ITAK).