On Quasi-integrable Deformation Scheme of The KdV System
OOn Quasi-integrable Deformation Scheme of The KdV
System
Kumar Abhinav ∗ and Partha Guha † The Institute for Fundamental Study (IF), Naresuan University,Phitsanulok 65000, Thailand Khalifa University of Science and Technology,PO Box 127788, Abu Dhabi, UAEFebruary 22, 2021
Abstract
We put forward a general approach to quasi-deform the KdV equation by deforming the cor-responding Hamiltonian. Following the standard Abelianization process based on the inherent sl (2) loop algebra, an infinite number of anomalous conservation laws are obtained, which yieldconserved charges if the deformed solution has definite space-time parity. Judicious choice ofthe deformed Hamiltonian leads to an integrable system with scaled parameters as well as to ahierarchy of deformed systems, some of which possibly being quasi-integrable. As a particularcase, one such deformed KdV system maps to the known quasi-NLS soliton in the alreadyknown weak-coupling limit, whereas a generic scaling of the KdV amplitude u → u (cid:15) alsogoes to possible quasi-integrability under an order-by-order expansion. Following a genericparity analysis of the deformed system, these deformed KdV solutions need to be parity-evenfor quasi-conservation which may be the case here following our analytical approach. From theestablished quasi-integrability of RLW and mRLW systems [Nucl. Phys. B (2019) 49–94],which are particular cases of the present approach, exact solitons of the quasi-KdV systemcould be obtained numerically. Mathematics Subject Classifications (2010) : 37K10, 37K55. 37K30.
Keywords and Keyphrases : Quasi-integrable deformation, KdV equation, NLS equation. ∗ E-mail: [email protected] † E-mail: [email protected] a r X i v : . [ n li n . S I] F e b Introduction
The -dimensional Korteweg-de Vries (KdV) equation [1] is applicable to many real-lifephenomena such as flow of a shallow fluid. Being third order in space derivative, thoughthis non-linear partial differential equation (PDE) has a dispersion odd in momentum powers,it is completely integrable [2] and supports localized soliton solutions [3] that can representdifferent observable physical objects in fluid dynamics like tidal waves. Solitonic structures arewell-known in other nonlinear systems in -dimensions such as the nonlinear Schrödinger(NLS) equation which is also integrable and may seem to be more closely related to physicalphenomena as the NLS solitons have been observed in Bose-Einstein condensates, cold atomsand optics [4]. The NLS with quadratic dispersion is more suitable for a ‘physical visualization’is fundamentally different from the KdV system. The KdV is geometrically connected todiffeomorphism group [5] whereas NLS is tagged with loop algebra [6]. Moreover the usual
Laxrepresentation [7] of KdV system involves second and third order monic differential operators(
L, A ), whereas that of the NLS system is given by × matrices [2]. However, it is knownthat in a suitable weak-coupling limit, their respective solutions map into each-other [8, 9],including their soliton solutions.The NLS soliton dynamics has been well-studied including its various deformations [4] invarious physical systems and same can be said about the exact KdV system [10]. However,detailed studies of localized structures for deformations of the KdV system are sparse whichcould be due to it being third-order in space-derivative that effects its solvability more severely.Deformations of such continuous systems may not be integrable in general as the delicatebalance between dispersion and nonlinearity is crucial for the infinitely many conserved quan-tities (charges) hallmarking their integrability. Due to such high sensitivity, most of thesedeformations do not posses any conserved charges. Subsequently, it becomes very difficult toobtain localized solutions for these deformed systems since the integrable solitons derive theirrobustness of existence from infinitely many conservation laws [11].On the other hand, real physical systems are characterized by finite number of degrees offreedom, prohibiting integrability of the corresponding field-theoretical models in principle.Yet, they are physically known to posses solitonic states, very similar in structure to the in-tegrable ones. Some examples include particular deformations of sine-Gordon (SG) [12]. Thismotivates the study of continuous systems as slightly deformed integrable models. In a recentwork [13, 14], the SG model was shown to be deformable into an approximate system that sup-ports the conservation of only a subset of the charges while the others behaving anomalously,corresponding to an anomalous zero-curvature condition. Such behavior was seen in the super-symmetric extension to the SG system [15] and for certain deformations of the NLS [16] and AB[17] systems also. Moreover, the anomalous charges are seen to regain conservation for local-ized solutions which are far apart. In particular cases single and multi-solitonic solutions werenumerically obtained for these deformations [12, 13, 14, 16]. Expectantly the correspondingcharges were anomalous when these solitonic structures interacted locally, but when the latterare well-separated these charges return to being conserved. This was interpreted as asymptoticintegrability and the corresponding systems are deemed as quasi-integrable (QI). or the particular case of NLS equation, Ferreira et. al. [16] modified NLS potential , theterm in the Hamiltonian that leads to the nonlinearity in the NLS equation as V ( ϕ ) = | ϕ | → (cid:0) | ϕ | (cid:1) (cid:15) . Here ϕ satisfies the NLS equation and (cid:15) is the deformation parameter. It was foundthat this model possesses an infinite number of quasi-conserved (anomalous) charges that areconserved asymptotically corresponding to numerically obtained solitonic structures. It wasfurther found that the anomaly functions corresponding to the deformed curvature and thatfor the anomalous charge evolution have definite parity properties essential for the asymptoticintegrability of the system. Owing to the closeness with the NLS system [8, 9], the KdV systemis also expected to display such quasi-integrability. However this third order equation in spaceeludes a direct dynamical interpretation in the usual classical sense and there are no ‘potential’analogue here. More importantly the Lax formalism is essential to the quasi-integrabilitymechanism wherein the inherent SU (2) structure leading to an sl (2) loop algebra is utilized[12, 13, 14, 16]. The usual KdV Lax pair is made of monic differential operators devoid ofsuch algebra. SU (2) representations of the KdV Lax pair, however, exists [18] with propergrade structure [19] for the Abelianization procedure [12, 13] required for quasi-integrability.Subsequently, a general structure for the quasi-integrable deformation of the KdV system isyet to be obtained.In a recent and important work [20] particular deformation of the KdV equations, whichcan be identified with non-integrable systems such as the regularized long-wave (RLW) [21,22] and the modified regularized long-wave (mRLW) [23] equations, were shown to be quasi-integrable. Detailed analytical results, supported by numerical evaluation of single and multi-soliton structures, ensured asymptotic integrability of such systems for certain ranges of thedeformation parameters. However, a general way to quasi-deform the original KdV system„in the likeness of sine-Gordon [13, 14] or NLS [16, 24], has not been proposed yet. In thepresent work we propose a general framework of deformation of the KdV equation that leadsto quasi-integrability. We obtain the sl (2) loop algebraic Abelianization [14, 16] of the KdVsystem and obtain the anomalous charges. We further provide the generic parity analysis ofthis deformation based on the said loop algebra to obtain definite parity structure of the quasi-deformation anomalies, crucial for the known cases. Finally a few of the deformed solutions andcorresponding anomalies are analyzed approximately that conforms to the quasi-integrabilitystructure.As mentioned before, being a third order differential system, the KdV equation does notaccommodate a dynamical deformation of the Lax pair. Instead, a more general off-shell deformation scheme, that of the KdV Hamiltonian has been achieved, allowing for suitablydeformed Lax component, that further allows for a hierarchy of higher-derivative extensions ofKdV. At the simplest level, such deformations lead to scaling of the KdV parameters and thusretaining integrability. In the perturbative domain, as the KdV and NLS systems are relatedthrough a weak-coupling map [8, 9] between the solutions, we obtain a map between thisquasi-KdV and the known quasi-NLS results [16]. We further infer about the connection of thequasi-KdV system to its non-holonomic (NH) deformation [18], the latter retaining integrability.Since the connection between quasi- and non-holonomic NLS system had been compared [24]and are found to mutually correspond asymptotically, we expect a similar property for therespective deformations of the KdV system. n the following, section 2 provides a detailed loop algebraic structure of the general quasi-deformed KdV system. In section 3 we obtain a general analysis of the deformation algebraicstructure of the deformed system ensuring quasi-conservation of the charges for localized so-lutions. We obtain some detailed results in the perturbative limit in section 4 with particularexamples. Finally we discuss and conclude in section 5 highlighting remaining issues and furtherpossibilities. It is well known that a systematic procedure of obtaining most finite dimensional completelyintegrable systems is Adler, Kostant and Symes ( AKS) theorem [25, 26] applying to someLie algebra g equipped with an ad-invariant non-degenerate bi-linear form. When this schemeis applied to loop algebra and the Fordy-Kulish decomposition scheme is invoked then theNLS [27] and KdV [28] equations can be formulated from there. This mechanism can also beapplied for the construction of hierarchies too [29]. In case of the KdV system, the most generalconstruction that can be derived from this AKS procedure is a pair of coupled complex KdVequations [18], through construction of the Lax pair: A = Q and B = T + [ S, Q ] , (1)where, T = − Q xx + (cid:2) Q + , (cid:2) Q − , Q + (cid:3)(cid:3) − (cid:2) Q − , (cid:2) Q − , Q + (cid:3)(cid:3) and S = Q + x + Q − x + 4 c (cid:0) Q + + Q − (cid:1) , c ∈ R . (2)With the definitions, Q = (cid:18) q − q (cid:19) ; Q + = (cid:18) q (cid:19) , Q − = (cid:18) − q (cid:19) , (3) Q = Q + + Q − ≡ ¯ qσ + − qσ − . where, the Pauli matrices satisfy the SU (2) algebra: [ σ + , σ − ] = σ and [ σ , σ ± ] = ± σ ± , (4)and q, ¯ q are mutually conjugate amplitudes, this leads to the coupled KdV equations. It iseasy to see that an sl (2) -loop algebra can be constructed on the SU (2) basis, which in turnenables a complete gauge-group interpretation of this system.Incorporating the generic representation of Eq.s 3, the Lax pair takes the form, A = ¯ qσ + − qσ − and B = (¯ qq x − q ¯ q x ) σ − (cid:0) ¯ q xx + 2 q ¯ q (cid:1) σ + + (cid:0) q xx + 2¯ qq (cid:1) σ − . (5) he corresponding curvature then can be evaluated as, F tx = (¯ q t + ¯ q xxx + 6 q ¯ q ¯ q x ) σ + − ( q t + q xxx + 6 q ¯ qq x ) σ − , (6)yielding two coupled KdV-like equations, ¯ q t + ¯ q xxx + 6 q ¯ q ¯ q x = 0 and q t + q xxx + 6 q ¯ qq x = 0 , (7)under zero-curvature condition, by considering each linearly independent component of thecurvature matrix. Although the above equations posses higher order non-linearity than theusual KdV system, a straight-forward choice of variables, ¯ q ( q ) = 1 , and q (¯ q ) = u ; u ∈ R , (8)immediately leads to the non-coupled (usual) KdV equation, u t + 6 uu x + u xxx = 0 . (9)The other possibility: ¯ q ( q ) = − u, q (¯ q ) = 1 leads to a KdV equation with a negative sign tothe non-linear term, which can be transformed to the ‘usual’ one through the transformation u → − u . The Lax pair corresponding to the choice in Eq. 8 is, A = σ + − uσ − and B = u x σ − uσ + + (cid:0) u xx + 2 u (cid:1) σ − , (10)leading to the curvature, F tx := A t − B x + [ A, B ] ≡ − ( u t + 6 uu x + u xxx ) σ − , (11)which vanishes on-shell subjected to the KdV equation. This algebraic structure stemmingfrom the SU (2) representation allows construction of the sl (2) loop algebra necessary for theAbelianization procedure of quasi-integrability [13, 14, 16]. This would not have been possiblewith the more common monic Lax pair, A = ∂ x + ∂ x + u, and B = − ∂ x − u∂ x − u x , (12)for the KdV equation. In the following we explicate the Abelianization procedure in detail. Since the KdV equation has derivatives higher than two, a dynamic interpretation of the sameis not possible at the level of the equation itself. In order to employ the quasi-integrabilitymechanism of Ref.s [13, 14, 16, 15], the notion of potential is essential, that emerges from suchan interpretation of the equation. In case of KdV system, however, a well-known Hamiltonianformulation [2] exists. In fact, the KdV equation 9 can be shown to emerge from two differentequivalent Hamiltonians. Subjected to the order of non-linearity appearing in the Lax pair ofEq. 10, we opt for the following Hamiltonian, H [ u ] = (cid:90) ∞−∞ dx (cid:18) u x − u (cid:19) with δH [ u ] δu ( x ) = − u − u xx . (13) his enables us to re-express the temporal Lax component ( B ) of the Lax pair as, B ≡ u x σ − uσ + + (cid:20) u xx − (cid:18) δH [ u ] δu ( x ) + u xx (cid:19)(cid:21) σ − . (14)The above is a general expression to accommodate any possible deformation at the Hamiltonianlevel. We propose that the deformation of the system is implemented in the non-linear part ofHamiltonian for the KdV system to impart quasi-integrability, the explicit form of which willbe discussed below. The corresponding curvature takes the form: F tx ≡ (cid:20) u t + u xxx − ∂ x (cid:18) δH [ u ] δu ( x ) + u xx (cid:19) + 2 uu x (cid:21) σ + + X σ , (15)with the supposed anomaly term, X = 2 u + 23 (cid:18) δH [ u ] δu ( x ) + u xx (cid:19) , (16)that vanishes for undeformed system . In the presence of this anomaly, implementation of the deformed ‘equation of motion’ (EOM) ( i. e. , the KdV equation), u t + u xxx − ∂ x (cid:18) δH [ u ] δu ( x ) + u xx (cid:19) + 2 uu x = 0 , or , u t + 6 uu x + u xxx = X x , (17)leaves the curvature non-zero.Starting from the deformed Lax pair, one can construct an infinite number of quasi-conservedcharges through the Abelianization procedure applied in Ref.s [13, 14, 16], through gauge-transforming the Lax components: ( A, B ) → U ( A, B ) U − + U ( x,t ) U − = ⇒ F tx → U F tx U − , (18)In doing so, the anomaly X prevents rotation of both of them into the same infinite dimensionalAbelian subalgebra of the characteristic sl (2) loop algebra, eventually leading to an infinite setof quasi-conservation laws characterized by X . The sl (2) loop algebra: The SU (2) algebraic structure for the KdV system [18] enables theconstruction of an sl (2) loop algebra: (cid:2) F m , F m ± (cid:3) = 2 F m + n ∓ , (cid:2) F m − , F n + (cid:3) = F m + n +1 , (19)consistent with the definitions, One can very well work with the second
Hamiltonian form for the KdV system [2]: H [ u ] = − (cid:82) ∞−∞ dx u ( x ) ,with the alternate fundamental bracket defined as { u ( x ) , u ( y ) } = (cid:2) ∂ x + 2 ( u x + u∂ x ) (cid:3) δ ( x − y ) . Then, the timecomponent of the Lax pair will take the form: B ≡ − u x σ − (cid:104) u xx − δH [ u ] δx (cid:105) σ + + 2 uσ − . Rest will follow throughthe replacement: (cid:16) δH [ u ] δu ( x ) + u xx (cid:17) → δH [ u ] δx . n = λ n σ , F n − = λ n √ σ + − λσ − ) and F n + = λ n √ σ + + λσ − ) , (20)with λ being the spectral parameter. Such a structure is essentially same as that in Ref.[16] for quasi-integrable (QI) NLS systems. This serves as a strong connection between thequasi-deformations of the two systems, which we will address soon. The Gauge Transformation:
The Lax pair of Eq. 10 with B deformed according to Eq. 14,however, is not suitable for the Abelianization (a version of the Drinfeld-Sokolov reduction)as the the spatial component A does not contain a constant semi-simple element of the sl (2) algebra that split the algebra into the correspondingKernel and Image subspaces. In otherwords, this mandates the presence of the spectral parameter ( λ ) in the Lax pair in a particularway. A Lax pair that fulfills this algebraic requirement and also leads to the quasi-KdV equationis [19], ¯ A = σ + − ( u − λ ) σ − and¯ B = u x σ − (2 u + 4 λ ) σ + + (cid:26) u xx − (cid:18) δH [ u ] δu + u xx (cid:19) + 2 λu − λ (cid:27) σ − , (21)with B being suitably quasi-deformed which we will adopt for the remaining analysis. It is clearthat the spatial Lax operator is free from the quasi-deformation, a crucial property exploitedby the Abelianization procedure to obtain the general form of the quasi-conserved charges.The undeformed version of the above Lax pair can be obtained from the previous one througha gauge transformation corresponding to a unitary operator , G = exp ( a + σ + + a − σ − ) ; a + = λ √ λ − u (cid:90) x √ λ − u , a − = λ √ λ − u (cid:90) x √ λ − u . (22)In terms of the sl (2) generators the new Lax operators take the forms, ¯ A = √ F − u √ (cid:0) F − − F − − (cid:1) and¯ B = − √ F + u x F − √ uF + f ( u ) √ (cid:0) F − − F − − (cid:1) (23) where f ( u ) = (cid:26) u xx − (cid:18) δH [ u ] δu (cid:19) + u xx (cid:27) , which are essentially similar to those given in Ref. [20] subjected to a particular interpre-tation of the deformed Hamiltonian H [ u ] . The above structure incorporates all possibilitiesof anomalous deformation of the KdV equations and therefore should correspond to multiplequasi-KdV systems. It might not be the most general gauge transformation that leads to the desired Lax pair. Further the time-dependence of a ± may include some non-trivial extensions. As the particular form in ¯ A is needed, the correspondingundeformed ¯ B may suitably constructed through term-by-term compensation starting with B . ollowing the general approach of in the Ref.s [13, 14, 16] for Abelianization by gauge-rotating the spatial Lax operator exclusively to the image of sl (2) , we undertake the gaugetransformation defined by, g = e G , G = −∞ (cid:88) n = − α n F n − + β n F n . (24)Here the coefficients α − n , β − n are to be chosen such that the transformed spatial component ˜ A = g ¯ Ag − + g x g − depends only on F n + s: ˜ A ≡ −∞ (cid:88) n =0 γ n F n + . (25)On employing the BCH formula e X Y e − X = Y + [ X, Y ] + [ X, [ X, Y ]] + [ X, [ X, [ X, Y ]]] + · · · the new spatial component has the general form, ˜ A = ¯ A + [ G, ¯ A ] + 12! [ G, [ G, ¯ A ]] + 13! [ G, [ G, [ G, ¯ A ]]] + · · · + G x + 12! [ G, G x ] + 13! [ G, [ G, G x ]] + · · · , (26)The first few of the individual commutators are, [ G, ¯ A ] = −∞ (cid:88) n = − (cid:104) √ α n (cid:16) F n +1 − u F n (cid:17) + √ β n (cid:0) F n − − uβ n F n − − + uβ n F n − (cid:1)(cid:105) ,
12! [ G, [ G, ¯ A ]]= √ −∞ (cid:88) m,n = − (cid:34) (cid:16) u α m α n + 2 β m β n (cid:17) F m + n + + u α m β n F m + n − α m α n F m + n +1+ − uβ m β n F m + n − + uβ m β n F m + n − − (cid:35) ,
13! [ G, [ G, [ G, ¯ A ]]]= √ −∞ (cid:88) l,m,n = − (cid:34) − α l α m α n F l + m + n +2 + α l (cid:16) u α m α n + 2 β m β n (cid:17) F l + m + n +1 − uα l β m β n F l + m + n − uα l α m β n F m + n + l + + 2 uβ l β m β n F l + m + n − − β l α m α n F l + m + n +1 − + β l ( uα m α n + 4 β m β n ) F l + m + n − − uβ l β m β n F l + m + n − − (cid:35) ,
4! [ G, [ G, [ G, [ G, ¯ A ]]]]= 16 √ −∞ (cid:88) k,l,m,n = − (cid:34) − uα l α l α m β n F k + l + m + n +1 + 2 uα k β l β m β n F k + l + m + n +2 α k α l α m α n F k + l + m + n +2+ − { α k α l ( uα m α n + 4 β m β n ) + 4 β k β l α m α n } F k + l + m + n +1+ +2 { uα k α l β m β n + β k β l ( uα m α n + 4 β m β n ) } F k + l + m + n + − uβ k β l β m β n F k + l + m + n − − uβ k α l α m β n F k + l + m + n − + 4 uβ k β l β m β n F k + l + m + n − − (cid:35) , · · · and12! [ G, G x ] = −∞ (cid:88) m,n = − ( β m α n,x − α m β n,x ) F m + n + ,
13! [ G, [ G, G x ]] = 13 −∞ (cid:88) l,m,n = − ( β m α n,x − α m β n,x ) (cid:16) α l F l + m + n +1 + 2 β l F l + m + n − (cid:17) ,
14! [ G, [ G, [ G, G x ]]] = 16 −∞ (cid:88) k,l,m,n = − ( β m α n,x − α m β n,x ) (cid:16) β k β l F k + l + m + n + − α k α l F k + l + m + n +1+ (cid:17) , · · · (27)The order-by-order conditions of vanishing the coefficients of F n , F n − s lead to the expressionsfor the expansion coefficients of the gauge operator g as, O (cid:0) F (cid:1) : α − = 0 , O (cid:0) F − (cid:1) : α − = u x √ , O (cid:0) F − (cid:1) : α − = u xxx √ √ uu x , O (cid:0) F − (cid:1) : α − = u xxxxx √ √ u x u xx + 532 √ uu xxx + 1124 √ u u x , · · ·O (cid:0) F − − (cid:1) : β − = − u , O (cid:0) F − − (cid:1) : β − = − u xx − u , O (cid:0) F − − (cid:1) : β − = − u xxxx − uu xx −
332 ( u x ) − u , O (cid:0) F − − (cid:1) : β − = − u xxxxxx −
564 ( u xx ) − u x u xxx − uu xxxx − u ( u x ) − u u xx − u , · · · (28) hese immediately lead to the expression of the rotated spatial Lax component ˜ A as theexpansion coefficients of the in Eq. 25 are completely determined, γ = √ ,γ − = − u √ ,γ − = − u √ ,γ − = 116 √ (cid:0) ( u x ) − uu xx − u (cid:1) ,γ − = − uu xxxx √ − √ u ( u x ) − u u xx √ − √ u , · · · (29)in terms of the deformed solution u .Subsequently, the temporal Lax component ¯ B transforms to, ˜ B = g ¯ Bg − + g t g − = ¯ B + [ G, ¯ B ] + 12! [ G, [ G, ¯ B ]] + 13! [ G, [ G, [ G, ¯ B ]]] + · · · + G t + 12! [ G, G t ] + 13! [ G, [ G, G t ]] + · · · , (30)with a few of the lowest order commutators being, [ G, ¯ B ]= −∞ (cid:88) n = − (cid:104) − √ α n F n +2 − √ uα n F n +1 + f ( u ) √ α n F n − u x α n F n + − √ f ( u ) β n F n − − √ β n F n +1 − − √ uβ n F n − + √ f ( u ) β n F n − − (cid:105)
12! [ G, [ G, ¯ B ]]= −∞ (cid:88) m,n = − (cid:104) − u x α m α n F m + n +1 − f ( u ) √ α m (cid:18) α n + β n (cid:19) F m + n + 4 √ α m α n F m + n +2+ +4 √ uα m α n − β m β n ) F m + n +1+ − √ uβ m β n F m + n + + √ f ( u ) β m β n F m + n − − u x β m α n F m + n − − √ f ( u ) β m β n F m + n − − (cid:105) ,
3! [ G, [ G, [ G, ¯ B ]]]= −∞ (cid:88) l,m,n = − (cid:104) √ α l α m α n F l + m + n +3 + 2 √ α l ( uα m α n − β m β n ) F l + m + n +2 − √ uα l β m β n F l + m + n +1 + √ f ( u ) α l β m β n F l + m + n + 23 u x α l α m α n F l + m + n +1+ + 23 (cid:26) f ( u ) √ α l α m (2 α n − β n ) − u x β l β m α n (cid:27) F l + m + n + − √ f ( u ) β l β m β n F l + m + n − + 8 √ β l α m α n F l + m + n +2 − + 4 √ β l ( uα m α n − β m β n ) F l + m + n +1 − − √ uβ l β m β n F l + m + n − + 2 √ f ( u ) β l β m β n F l + m + n − − (cid:105) ,
14! [ G, [ G, [ G, [ G, ¯ B ]]]]= −∞ (cid:88) k,l,m,n = − (cid:104) u x α k α l α m α n F k + l + m + n +2 + 16 α k (cid:26) f ( u ) √ α l α m (2 α n − β n ) − u x β l β m α n (cid:27) × F k + l + m + n +1 − f ( u )3 √ α k β l β m β n F k + l + m + n − √ α k α l α m α n F k + l + m + n +3+ + √ { β k β l α m α n − α k α l ( uα m α n − β m β n ) } F k + l + m + n +2+ + 2 √ { uα k α l β m β n + β k β l ( uα m α n − β m β n ) } F k + l + m + n +1+ − (cid:26) f ( u ) √ α k α l β m β n + 4 √ uβ k β l β m β n (cid:27) F k + l + m + n + + √ f ( u )3 β k β l β m β n F k + l + m + n − + 13 u x β k α l α m α n F k + l + m + n +1 − + 13 β k (cid:26) f ( u ) √ α l α m (2 α n − β n ) − u x β l β m α n (cid:27) × F k + l + m + n − − √ f ( u ) β k β l β m β n F k + l + m + n − − (cid:105) , · · ·
12! [
G, G t ] = −∞ (cid:88) m,n = − ( β m α n,t − α m β n,t ) F m + n + ,
13! [ G, [ G, G t ]] = 13 −∞ (cid:88) l,m,n = − ( β m α n,t − α m β n,t ) (cid:16) α l F l + m + n +1 + 2 β l F l + m + n − (cid:17)
14! [ G, [ G, [ G, G t ]]] = 16 −∞ (cid:88) k,l,m,n = − ( β m α n,t − α m β n,t ) (cid:16) β k β l F k + l + m + n + − α k α l F k + l + m + n +1+ (cid:17) · · · (31)The rotated temporal Lax component will span both Kernal and Image of sl (2) to mandate ageneral form: B = −∞ (cid:88) n =0 (cid:2) a n F n − + b n F n + c n F n + (cid:3) , (32)wherein, a few of the nontrivial expansion coefficients are, a = 2 √ u,a − = − f ( u ) √ u xx √ √ u ,a − = − u √ f ( u ) + u xt √ u xxx √ √ uu xx + 32 √ u x ) + 136 √ u ,a − = u xxxt √ √ u t u x + uu xt ) + u xxxxxx √ √ u xx ) + 158 √ u x u xxx + 78 √ uu xxxx + 194 √ u ( u x ) + 338 √ u u xx + 136 √ u − f ( u )8 √ (cid:0) u xx + 3 u (cid:1) , · · · (33) b = 0 = b ,b − = − u t − u xxx − uu x ,b − = − u xxt − uu t f ( u )8 u x − u xxxxx − u x u xx − uu xxx − u u x , · · · (34) c = − √ ,c = − √ u,c − = 1 √ (cid:0) f ( u ) − u (cid:1) ,c − = f ( u )2 √ u − ( u x ) √ − uu xx √ − √ u , · · · (35)Preceding the gauge transformation, following Eq. 23, the curvature has the form: ¯ F tx = ¯ A t − ¯ B x + [ ¯ A, ¯ B ]= − (cid:20) u t + u xxx + 2 uu x − (cid:18) δH [ u ] δu + u xx (cid:19) x (cid:21) σ − − X σ ≡ −X F , (36)wherein the coefficient of σ − vanish following the deformed KdV equation. Then, following thegauge transformation yields the rotated curvature as: ¯ F tx → ˜ F tx = g ¯ F tx g − = ¯ F tx + [ G, ¯ F tx ] + 12! [ G, [ G, ¯ F tx ]] + 13! [ G, [ G, [ G, ¯ F tx ]]] + · · · , (37)wherein a few of the commutators have the forms: G, ¯ F tx ] = 2 X −∞ (cid:88) n = − α n F n + ,
12! [ G, [ G, ¯ F tx ]] = X −∞ (cid:88) m,n = − (cid:0) α m α n F m + n +1 + 2 β m α n F m + n − (cid:1) ,
13! [ G, [ G, [ G, ¯ F tx ]]] = 23 X −∞ (cid:88) l,m,n = − (cid:16) β l β m α n F l + m + n + − α l α m α n F l + m + n +1+ (cid:17) ,
14! [ G, [ G, [ G, [ G, ¯ F tx ]]]]= X −∞ (cid:88) k,l,m,n = − (cid:104) α k β l β m α n F k + l + m + n +1 − α k α l α m α n F k + l + m + n +2 + 4 β k β l β m α n × F k + l + m + n − − β k α l α m α n F k + l + m + n +1 (cid:105)
15! [ G, [ G, [ G, [ G [ G, , ¯ F tx ]]]]]= X −∞ (cid:88) j,k,l,m,n = − (cid:104) α j α k α l α m α n F j + k + l + m + n +2+ − α j α k β l β m β n + β j β k α l α m α n ) × F j + k + l + m + n +1+ + 4 β j β k β l β m α n F j + k + l + m + n + (cid:105) , · · · (38)On the other hand, as ¯ F tx ∝ F , the general form of the rotated curvature will have the form, ˜ F tx = X (cid:88) n (cid:0) f n F n + f + n F n + + f − n F n − (cid:1) . (39)On comparing the two expressions of the rotated curvature, one obtains a few of its nontrivialexpansion coefficients in terms of the system variables as, f = − ,f − = 0 = f − ,f − = ( u x ) ,f − = u xxx
64 + 332 uu x , · · · (40) + − = 0 ,f + − = u x √ ,f + − = u xxx √ √ uu x ,f + − = u xxxxx √ √ u x u xx + 516 √ uu xxx + 1516 √ u u x , · · · (41) f −− = 0 ,f −− = − uu x √ ,f −− = − uu xxx √ − u x u xx √ − u u xx √ ,f −− = − uu xxxxx √ − u x u xxxx √ − u xx u xxx √ − √ uu x u xx − √ u x ) − √ u u x − √ u u xxx , · · · (42)The rotated curvature, on the other hand, can directly be obtained from the correspondingLax pair (cid:16) ˜ A, ˜ B (cid:17) as, ˜ F tx = ˜ A t − ˜ B x + [ ˜ A, ˜ B ] ≡ (cid:88) n (cid:104) ( γ n,t − c n,x ) F n + − (cid:32) b n,x − γ n (cid:88) m a m F m +1 (cid:33) F n − (cid:32) a n,x + 2 γ n (cid:88) m b m F m − (cid:33) F n − (cid:105) . (43)It can be seen that the projection of the rotated curvature onto the Image sub-sector definedby the generators F n + , to which ˜ A was rotated exclusively, is linear in the coefficient. This willbe crucial for defining charges for this system. In order to demonstrate the deviation from integrability, based on the QI deformation, it ispertinent to evaluate quantities which would have represent conservation or have themselvesbe conserved for the undeformed system. The deliberate gauge transformation in Eq. 24rotates one (spatial) Lax component to an Abelian sub-algebra spanned by { F n + } which is aparticular Drinfeld-Sokolov reduction. This essentially isolates the corresponding contributionto the curvature F tx from any nonlinearity arising from the general non-commutation of theLax components. For an undeformed system, owing to its integrability manifesting as thevanishing of the curvature at each spectral order, the contribution to the curvature from the iearized sub-algebra yields a very simple ‘continuity’ relation: γ n,t − c n,x = 0 following Eq. 41.This enables one to construct conserved charges: Q n = (cid:90) x γ n → dQ n dt = (cid:90) x ( γ n,t − c n,x ) ≡ , (44)subjected to feasible boundary behavior of the coefficients c n as they exclusively depend on thesolution u which is local for all the purposes of interest. As the curvature does not vanish forthe deformed system, on comparing its two expressions in Eq.s 39 and 43, these charges turnout to be non-conserved in general: dQ n dt = (cid:90) x ( γ n,t − c n,x ) = (cid:90) x X f + n ≡ Γ n , (45)exclusively because the anomaly X is non-zero. However, a subset of them can still be conserveddepending on particular values of the expansion coefficients f + n . For example, trivially, thecharge Q − is identically conserved following Eq.s 41 . In general, for a given deformed solution u , the coefficients γ n can be well-localized for a particular set { n } resulting in a constant valuefor the corresponding Q n s. Finally the anomaly X itself can have certain overall symmetrywhich will yield a vanishing derivative for Q n s for a particular subset { f + n } , a topic that willbe illustrated upon in the next section. All these possibilities may render the system quasi-integrable having a subset of conserved charges, since being functions of a localized deformedsolution u of the system they are expected to vanish asymptotically.As discussed in Ref. [20], though for a particular kind of quasi-deformations, the anomalouscharges regain conservation for the deformed solution u being well-localized; either solitonicor even multi-solitonic but well-separated. However, such solutions were subjected to a fullnumerical treatment which is beyond the scope of the present work. Instead we focus on variousparticular forms of the deformed Hamiltonian H [ u ] leading to classes of possible quasi-KdVsystems, which are explicated in the next section. In order to explore the details for quasi-integrability of the KdV system, we now utilize the Z symmetry of sl (2) loop algebra, which has strong similarity with that corresponding tothe quasi-NLS system [16] and agrees with the particular quasi-KdV systems in Re. [20]. Ithas been found that the anomaly function and the relevant expansion coefficients must possesdefinite space-time parities for quasi-conservation of the charges. Although the exact reasonbehind this is not known clearly, it might closely be related to the Abelianization approachitself. The Z transformation is a combination of the order 2 automorphism of sl (2) loopalgebra: This particular conservation is essentially a statement of the deformed KdV equation being satisfied, as one cancheck by substituting for γ − and c − . This further serves as a testament to the locality of u , which was assumed forits vanishing on the spatial boundary, as Q − needs to be a constant. Additionally it may be related to the conservedenergy of the system [20] as observed earlier in Ref.s [13, 14, 16]. ( F n ) = F n , Σ( F n − ) = − F n − and Σ( F n + ) = F n + , (46)and parity: P : (˜ x, ˜ t ) → ( − ˜ x, − ˜ t ); with ˜ x = x − x and ˜ t = t − t (47)about a particular point ( x , t ) in space-time, which can very well be chosen to be the origin.These transformations mutually commute, as they work in two different spaces ( i. e. , groupand coordinate subspaces). Thus, Ω (cid:0) ¯ A (cid:1) = − ¯ A, Ω = Σ P , (48)for u being parity-even, which makes sense as we are interested in localized (solitonic) struc-tures . The KdV equation is parity-invariant to begin with, and its quasi-modification (Eq. 17)is also the same, subjected to the explicit deformation(s) to be introduced in the next section .More intuitively, as the quasi-deformed systems are known to support single-soliton structuressimilar to those of the undeformed systems [13, 14, 16, 20] and since the well-known bright anddark KdV solitons are parity-even, it is sensible to expect that u could be such [20].The use of Z transformation felicitate the asymptotic vanishing of the integral of X f + n sin a general way, thereby ensuring conservation of the corresponding charges Q n s [16]. Forthis purpose the generator F n + serves as the semi-simple element that splits the sl (2) loopalgebra into two sub-sectors. We have already achieved it in subsection 2.2 through the gaugetransformation, a general version of which can be expressed as, g = exp −∞ (cid:88) n = − G n , (49)where G n is any linear combination of the generators F n , F n − that rotates to the sub-sectordefined by F n + . Then the gauge-rotated spatial connection, takes the form, ¯ A → ˜ A = g ¯ A g − + g x g − , (50)which continues to be a eigenstate of Ω . To see this, considering the BCH expansions as in Eq.26, we can identify contributions to ˜ A = (cid:80) n ˜ A n for different n (powers of spectral parameter λ ) as, One can very well identify ( x , t ) as the centre of such a solitonic structure. Practically it amounts to having X x odd in derivatives, which it is. For the given Lax pair in Eq. 21. A = ¯ A , ˜ A − = (cid:2) G − , ¯ A (cid:3) + ¯ A − + G − x , ˜ A − = (cid:2) G − , ¯ A (cid:3) + (cid:2) G − , ¯ A − (cid:3) + 12! (cid:2) G − , (cid:2) G − , ¯ A (cid:3)(cid:3) + G − x + 12! (cid:2) G − , G − x (cid:3) , ˜ A − = (cid:2) G − , ¯ A (cid:3) + (cid:2) G − , ¯ A − (cid:3) + 12! (cid:16) (cid:2) G − , (cid:2) G − , ¯ A (cid:3)(cid:3) + (cid:2) G − , (cid:2) G − , ¯ A (cid:3)(cid:3) (cid:17) + 12! (cid:2) G − , (cid:2) G − , ¯ A − (cid:3)(cid:3) + 13! (cid:2) G − , (cid:2) G − , (cid:2) G − , ¯ A (cid:3)(cid:3)(cid:3) + G − x + 12! (cid:16) (cid:2) G − , G − x (cid:3) + (cid:2) G − , G − x (cid:3) (cid:17) + 13! (cid:2) G − , (cid:2) G − , G − x (cid:3)(cid:3) , ... (51) where ¯ A = ¯ A + ¯ A − , ¯ A = √ F , ¯ A − = − u √ (cid:0) F − − F − − (cid:1) . Wherein terms with numerical prefixes are separated according to their individual grades (pow-ers of λ ). We can immediately conclude that Ω (cid:16) ˜ A (cid:17) = − ˜ A . From the second equation, Ω (cid:16) ˜ A − (cid:17) ≡ − (cid:2) Ω (cid:0) G − (cid:1) , ¯ A (cid:3) − ¯ A − + Ω (cid:0) G − x (cid:1) , (52)which when added back to the second equation yields, (1 + Ω) (cid:16) ˜ A − (cid:17) = (cid:2) (1 − Ω) (cid:0) G − (cid:1) , ¯ A − ∂ x (cid:3) . (53)In the above, the LHS is exclusively in the sl (2) sub-sector defined by F n + whereas the RHSis excluded of it. Thus both the sides identically vanishes, with the RHS non-trivially leadingto Ω (cid:0) G − (cid:1) = G − . One can similarly proceed from the third of Eq.s 51 onward to obtain Ω ( G n ) = G n ∀ n ∈ Z − , eventually leading to Ω ( g ) = g . This finally implies Ω (cid:16) ˜ A (cid:17) = − ˜ A from Eq. 50.Therefore the Z automorphism is preserved for the spatial connection under the Abelianisinggauge transformation. One can check this explicitly from the expansion coefficients obtained forthe particular case of ˜ A in the last section, which is manifested through their parity propertiesas, P ( α n ) = − α n and P ( β n ) = β n . (54)This eventually implies definite parity properties of the coefficients f + n of the rotated curvaturein the linearized sub-sector defined by F n + s. To see this we utilize the Killing form of the sl (2) loop algebra [16]: K ( F n F m ) = 2 δ m + n, , K (cid:0) F n ± F m ± (cid:1) = 2 δ m + n +1 , , K (cid:0) F n F m ± (cid:1) = 0 = K (cid:0) F n + F m − (cid:1) ;wherein K ( • ) = − i π (cid:73) dλλ T r ( • ) . (55) hen from Eq.s 36 and 39 one can express the expansion coefficients of ˜ F tx in the linearizedsub-sector as, f + n ≡ K (cid:0) − gF gF − n − (cid:1) ∀ n ∈ Z + . (56)Since the Killing form is invariant under Ω , P (cid:0) f + n (cid:1) ≡ Ω (cid:0) f + n (cid:1) = 12 K (cid:0) − Ω( g )Ω (cid:0) F (cid:1) Ω (cid:0) g − (cid:1) Ω (cid:0) F − n − (cid:1)(cid:1) ≡ − f + n . (57)One can explicitly check this to be true from the particular expressions obtained for f + n s in theprevious section. Therefore, from Eq. 45, Q n ( t = ˜ t ) − Q n ( t = − ˜ t ) = (cid:90) ˜ t − ˜ t (cid:90) ˜ x − ˜ x X f + n ≡ , (58)for the anomaly function X being parity-even. In the above, ± (˜ x, ˜ t ) refers to spatiotemporalinfinity where all the charges are supposed to vanish for sensible quasi-integrability, therebyensuring the same for the deformation under consideration. This general parity properties ofthe system agrees in detail with those obtained in Ref. [20] for the RLW and mRLW systems.Therein analytical derivation and numerical evolution of one-, two- and three-soliton solutionsof these quasi-KdV systems display the parity-evenness distinctly. Apart from when theywere interacting the corresponding anomaly and expansion coefficients display the exact parityproperties obtained here in order to conserve the charges. Following the close relation betweenthe parity property and quasi-integrability of various systems [13, 14, 16, 15, 17] includingKdV-like systems [20] the preceding treatment strengthens the validity of our general quasi-deformation approach to the KdV-system.For the quasi-NLS system [16] the anomaly needed to be parity-odd. For the quasi-KdV it iscrucial to obtain parity-even X instead, which essentially means the Hamiltonian H [ u ] needsto be modified judiciously. The undeformed Hamiltonian in Eq. 13 is parity-even. Thus, fromthe expression of the anomaly (Eq. 16), we need a parity-even extension to H [ u ] to obtain aparity-even X . For example, with a deformation of the form, H [ u ] → H [ u ] = (cid:90) ∞−∞ (cid:20) u x − u + (cid:15)F ( u ) (cid:21) , (cid:15) ∈ N , (59)where F ( u ) = uu xx we obtain X = (cid:15)u xx which is parity-even as required. In particular, thiswill ensure conservation of Q − along with the trivially conserved Q − locally and asymptoticconservation of all Q n s in general given u is parity-even. To the latter end, the correspondingdeformed KdV equation looks like, u t + 6 uu x + (1 − (cid:15) ) u xxx = 0 , (60)which essentially is a scaling of the undeformed system and thereby is integrable. Though thisparticular one is a somewhat trivial deformation, it is to be noted that the proposed scheme or quasi-deformation can lead to integrable deformations as a subclass. Moreover, some quasi-deformed models may asymptotically go to a scaled version of the undeformed model insteadof the exact one . This equation supports single-soliton solutions of the form, u = c (cid:20)(cid:114) c − (cid:15) ) ( x − ct − x + ct ) (cid:21) , c > , (61)moving with speed c . This is expected as the choice for F ( u ) is nothing but a total derivativeaway from the first term in H [ u ] . Non-trivially and more importantly, however, this providesan opportunity to construct a hierarchy of higher-order/degree extensions of KdV, with differentchoices of F ( u ) . For demonstration, we consider the following two: F ( u ) = 32 m ( m − u m and F ( u ) = 34 uu (2 n ) , with m ∈ Z · · · ; n = 1 , , · · · , (62)where m is ordinary power and n is the order of space derivatives, leading to the higher-derivative equations, u t + 6 uu x + u xxx = (cid:15)u m − u x and u t + 6 uu x + u xxx = (cid:15)u (2 n +1) , (63)respectively. In particular, for n = 1 we obtain the system in Eq. 60. It may be worthwhile tostudy such extensions (deformations) of the KdV system which could admit more complicatedsolitonic structures than that in Eq. 61. It should be pointed out that not all of them could bequasi-integrable; beyond the obvious parity-count it should be crucial that such system posseslocalized solutions with proper asymptotic properties as explained above.One prospect is to deform H [ u ] in such a way X forms a total derivative when multipliedwith f n + s. This will automatically ensure quasi-conservation for u being localized; but this maynot be possible for multiple orders n . Such a system may not support localized solutions at all.More importantly the deformation part may not be linearly isolated in the Hamiltonian, likethe ‘potential deformations’ in Ref.s [13, 14, 16]. Then to identify X a order-by-order approachneeds to be adopted which, however, does not mean that (cid:15) needs to be small [16, 24, 17]. In thenext section we consider this approach utilizing the weak-coupling mapping to NLS system. It is not always possible to obtain solutions for arbitrary quasi-deformed systems like those inEq.s 63. However, since the QI parameter (cid:15) is independent, an order-by-order expansion canperturbatively lead to the quasi-deformed solution as seen for other systems [13, 14, 16, 15, 17].The correspondence between the KdV and the NLS system at the solution level in a weak-coupling limit [8] further supports this expectation. This correspondence materializes throughthe following parameterization of the NLS amplitude: Or even to a non-holonomic version as observed for the NLS system [24]. = ε (cid:16) ϕe iθ + ¯ ϕe − iθ (cid:17) + ε k (cid:16) ϕ e i θ + ¯ ϕ e − i θ (cid:17) − ε k | ϕ | ; where , (64) θ = k x + ω t, < ε (cid:28) , ω = k (cid:54) = 0 . On substituting the above mapping in the KdV equation 9, equating terms with phase e ± iθ at O (cid:0) ε (cid:1) , one arrives at the NLS equation, ϕ T + i k ϕ XX + i k | ϕ | ϕ = 0 , (65)and its complex conjugate with respect to the new coordinates, T = ε t and X = ε (cid:0) x + 3 k t (cid:1) . (66)In Eq. 65, the ‘time’-derivative term comes from that of the KdV, the second derivative termcomes from the third derivative term of the same and the non-linear term comes from its coun-terpart in KdV. Such direct correspondence, though approximate, strengthens the perturbativeapproach to obtain QI KdV system. One should keep in mind this analogical approach intro-duces another expansion parameter ε . This shows how the effect of quasi-deformation in onesector effects the other. The single bright soliton solution of a quasi-NLS system is of the form[13], φ d = (cid:104) (2 + (cid:15) ) / ρ sech (cid:110) (1 + (cid:15) ) ρ (cid:16) ˜ X − V ˜ T (cid:17)(cid:111)(cid:105) (cid:15) exp (cid:20)(cid:18) ρ − V (cid:19) ˜ T + V X (cid:21) , ˜ X = X − X , ˜ T = T − T and ρ, V, X , T ∈ R + ⊗ R ⊗ R ⊗ R . (67)For the QID parameter (cid:15) → one regains the undeformed soliton. Both these solutions areplotted in Fig.s 1a and 1c showing that the localized nature prevails over QID. The weak-coupling map of Eq. 64 yields a soliton train-like solution for the KdV system (Fig. 1b) thatgets distorted over QID (Fig. 1d) in the NLS sector. Tough it is not a priory guarantied thatthe weak-coupling map will persist over quasi-deformation it should be noted that the mapitself is an approximation. Never the less, since the mapped KdV soliton train only displaysminor local distortions over QID in the NLS sector, it can strongly be expected that quasi-KdVsystem can be obtained that supports localized solutions having a few conserved charges.It should be more assuring to show that a proposed quasi-deformation of the KdV equationmaps to a known quasi-NLS solution. For this purpose, we consider the case of Eq.s 63 with m = 3 which essentially amounts the scaling of the nonlinear term as, u t + u xxx + (6 − (cid:15) ) uu x = 0 , (68)that maintains integrability as a special case of quasi-modification. This deformed KdV systeminvariably maps to a quasi-NLS system. From Eq.s 64 the modified NLS equation is obtainedas , In the map of Eq. 64 [8], the non-linear terms of KdV and NLS systems map exclusively to each-other. Therefore,any scaling of the one in the KdV equation, like that in Eq. 68, corresponds to the same scaling of the similar termin the NLS equation. a) Undeformed single NLS soliton. (b) Undeformed mapped KdV soliton train.(c) Quasi-deformed NLS single soliton. (d) Quasi-deformed mapped KdV soliton train. Figure 1: The effect of quasi-deformation on the weak-coupling map from NLS to KdV solutions.The NLS single soliton (a) maps to a KdV soliton train (b), a property retained over the quasi-deformation though the effect of quasi-deformation is clear (c and d). Herein, (cid:15) = 1 . , V = 1 = ρ , X = 0 = T ε = 0 . and ω = 1 = k . 21 T + i k ϕ XX + i (cid:16) − (cid:15) (cid:17) k | ϕ | ϕ = 0 . (69)It is easy to see that, (cid:16) − (cid:15) (cid:17) | ϕ | ≈ δδ | ϕ | V ( | ϕ | ) with V ( | ϕ | ) ≡ | ϕ | − ˜ (cid:15) ) , ˜ (cid:15) = (cid:15) , (70)following the physical fact that the density | ϕ | is sufficiently small in the weak-coupling limit.The above identification in terms of the NLS potential V ( ϕ ) qualifies the obtained NLS systemas a quasi-deformed one [16]. As long as the mapping prevails, the deformation of Eq. 68should represent a quasi-KdV system, which it trivially is, whose solutions can be obtainedfrom those of the quasi-NLS ones as done above. More concretely this mapping between the twodeformed systems justifies the Hamiltonian deformation approach to the quasi-KdV system.The appearance of essentially the same sl (2) algebra for the quasi-NLS system [16], though thelinearizing sub-sector being different, further strengthens this line of argument.Further confirmation of this assertion is obtained at the solution level. The single-solitonsolution for the scaled KdV equation in Eq. 68 has the form, u = c (cid:20) (cid:112) cβ ( x − x − βc ( t − t )) (cid:21) , where β = 1 − (cid:15) (cid:15), c > , (71)which essentially amounts to a variable scaling of the undeformed system. The correspondingone-soliton solution for the NLS system of Eq. 69 is, ϕ ( X, T ) = K sech (cid:104) Λ K (cid:16) Λ ˜ X − V ˜ T (cid:17)(cid:105) exp (cid:20) i V ˜ X + i (cid:0) Λ K − V (cid:1) ˜ T (cid:21) ; (72) where Λ = i (cid:114) βk , Λ = − i √ k , K ∈ R + , with similar variable scaling. These exact solutions could be obtained since both the deformedequations correspond to modification of the self-coupling strength maintaining integrabilitywhich may not be the case in general. However it strongly suggests that a quasi-KdV sys-tem can be obtained in the usual way that is known to work for other systems. Indeed thenecessary algebraic framework obtained in the previous sections set up the conditions for aquasi-integrable KdV system. The anomaly X and deformed solution u being parity-evenshould be enough for an actual quasi-KdV system. Choosing different forms of F ( u ) to obtain a parity-even X ensures asymptotic (or quasi)integrability, which also includes some integrable systems. For example a choice of F ( u ) = κ u + κ u results in a parity-even X . However, it leads to Gardner or mKdV equationsdepending on the values of κ , which are completely integrable and further supports kink-typesolutions. As a non-trivial example, instead of an extension to the Hamiltonian like F ( u ) , we onsider a power-modification of the nonlinear term therein by the QID parameter (cid:15) in thesame spirit of the quasi-NLS system [16] as, H Def1 [ u ] ≡ (cid:90) ∞−∞ dx (cid:18) u x − u (cid:15) (cid:19) . (73)It amounts to deforming the KdV nonlinearity through power-scaling of the amplitude u → u (cid:15) . As the nonlinearity has directly been effected the corresponding equation, u t + u xxx + 6 uu x = 4 uu x − (cid:15) )(2 + 3 (cid:15) ) u (cid:15) u x , (74)becomes relatively difficult to solve. We have numerically obtained a few solutions using Math-ematica 8 for different values of (cid:15) in Fig.s 2 which represent deviations from the undeformedstructure. As expected for finite (cid:15) the solutions do not posses definite parity which even-tually distorts the parity of the anomaly function X . This results in non-conserved chargesat any finite time. However, these deformed solutions are still localized, which could meanstrong interactions or radiation-effected solitonic structures [20], strongly suggesting asymp-totic conservation of the same charges. To have a better idea about this system we undertakean order-by-order expansion of this system in terms of (cid:15) . Though this parameter need notbe small always, such an order-expansion is valid for the dependence of the solution on theparameter (cid:15) being analytic [16]. For the particular case, such an expansion of the anomalytakes the form, X = 2 u − (cid:15) ) u (cid:15) ≈ − (cid:15)u (1 + 3 ln( u )) + O ( (cid:15) ) , (75)which mirrors the parity of the solution. It provides a logarithmic nonlinearity at first order in (cid:15) making the evaluation of quasi-corrections at that order quite difficult, especially when u (cid:28) .As an approximation, for finite u and (cid:15) (cid:28) , a localized solution has the form u app = c sech φ where φ satisfies the following approximate expression: x − c t ≈ − √ c (1 − (cid:15) ) φ + 2 √ c (cid:15) (cid:16) ln c φ (cid:17) coth φ. (76)Clearly, it goes to the usual KdV bright soliton for (cid:15) = 0 . A plot for the deformed solution u app in Fig. 3 depicts a soliton train-like structure. Such structures strongly suggest asymptoticintegrability of the system if not actual integrability.In order to evaluate the charges and analyze their (quasi-)conservation we consider an order-by-order expansion in (cid:15) of all the quantities of the deformed system. We assume quantities likethe solution u , anomaly X , charge Q n etc to be fairly well-behaved functions of (cid:15) and thus canbe expanded as power-series in the same. The deformed solution can thus be expanded as, u = u + (cid:15)u + (cid:15) u + · · · , (77)with u satisfying the undeformed KdV equation. Similarly the anomaly and rate of change ofcharges can be expanded as, X = (cid:15) X + (cid:15) X + · · · and Γ n = (cid:15) Γ n + (cid:15) Γ n + · · · (78) x u d (a) (cid:15) =0 x (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:180) (cid:180) u d (b) (cid:15) = 0 . . x (cid:45) (cid:180) (cid:45) (cid:180) (cid:180) (cid:180) u d (c) (cid:15) = 0 . . x (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:180) (cid:180) u d (d) (cid:15) = 0 . . Figure 2: Numerical solutions u d corresponding to the deformation u → u (cid:15) in the Hamilto-nian. The very localized and parity-even single-soliton structure for (cid:15) = 0 (2a) gets significantlydistorted (2b, 2c, 2d) even for small values (cid:15) . They still remain fairly localized suggesting asymptoticconservation of the corresponding charges. These plots are evaluated at time t = 1 .24igure 3: Approximate soliton train-like solution for power-scaling u → u (cid:15) of the nonlinear termin the Hamiltonian with (cid:15) = 0 . and c = 1 . The zero-order contribution to the anomaly vanishes as it corresponds to the undeformed systemand so does that to Γ n . For the anomaly in Eq. 75 the O ( (cid:15) ) contribution to the deformedequation has the form: u ,t + u ,xxx + 6 ( u u ) x = − (10 + 12 ln u ) u u ,x , (79)The solution u of this equation, with u being the KdV 1-soliton solution, is depicted in Fig.4. It does not have definite parity unlike the parity-even undeformed solution in Fig. 2a, thusyielding a first-order deformed solution u d = u + (cid:15)u without definite parity. Consequently,the anomaly for this particular case can be expanded as, X = − (cid:15) { u ) } u + 3 (cid:15) { u u + (2 u + u ) u ln( u ) } + O (cid:0) (cid:15) (cid:1) , (80)which is no longer parity-even at O (cid:0) (cid:15) (cid:1) . As for the rate of change of the charges Γ n the O ( (cid:15) ) depends only on u after expanding the coefficients f + n s in (cid:15) and thus vanishes. One cantrivially check this for the KdV single soliton u = ( c / (cid:2) √ c ( x − c t ) / (cid:3) as f + n containonly odd derivatives of u (Eq.s 41). As a demonstration, the next order contribution for n = 2 is, Γ = 12 √ (cid:90) x [ X u ,x + X u ,x ] since f + − = 12 √ u + (cid:15)u ) x , (81)where X and X can be read off of Eq. 80. Clearly, Γ (cid:54) = 0 since it contains u and thus Q − will not be conserved eventually.In principle these deviations from integrability can be calculated exactly for all n . The O ( (cid:15) ) contributions to X and the integrand in Γ n are entirely constituted of the undeformedsolution u whereas the O (cid:0) (cid:15) (cid:1) contributions contain only u in addition. The corrections to .75 9.80 9.85 9.90 9.95 10.00 x (cid:180) (cid:180) (cid:180) u Figure 4: O ( (cid:15) ) correction to the deformed KdV solution. It clearly deviates from even-paritystructure eventually leading to non-conserved charges. the undeformed solution u n s can successively be calculated from the O ( (cid:15) n ) contribution tothe parent equation 74 after evaluating u n − previously and thus, eventually, all the deformedcharges and their rates can be obtained up to all orders in principle. A good convergence of thenet sum of these contributions should ensure quasi-conservation but it requires a good deal ofnumeric simulation which is beyond this work. However, such confirmation has already beenobtained for particular quasi-KdV systems [20]. It is fruitful to compare the implications of quasi-deformation obtained thus far with thosefrom nonholonomic deformation of the KdV and as well as that of the NLS systems. Thenonholonomic deformation is practically obtained through extending the temporal Lax com-ponent with local functions of various grade by infusing particular powers of the spectralparameter λ into them, which does not effect the time-evolution of the system. This incor-porates an inhomogeneous extension to the original differential equation, with higher orderdifferential constraints imposed on the deformation functions, obtained through retaining thezero-curvature condition. Thus the deformed system still stays integrable . Nonholonomic de-formation had been well-analyzed for KdV and coupled complex KdV systems [18], from bothloop-algebraic and AKNS approaches, and in case of NLS systems it has recently been shownthat the nonholonomic deformation is locally different from quasi deformation as the latterleads to non-integrable, but asymptotically they may converge [24].The quasi-deformation is usually applied at the level of functions of the independent variable[13, 14, 16, 15] or that of a functional as in the present case for KdV, that deforms the Laxcomponent itself without effecting at the other spectral sectors. This generally yields a non-zerocurvature. Therefore, both the deformations are fundamentally different. However, since thequasi-conserved charges asymptotically are conserved a quasi-deformed system may convergeto a nonholonomic one asymptotically. In the present case the deformed solutions maintain ocalization. Further, in the present case, the logarithmic contribution ( e. g. in Eq. 75)becomes subdominant at large distances for a localized deformed solution u leaving a pureKdV-type system with scaled constants which should be integrable. A similar property wasobserved for the NLS system in Ref. [24]. Considering the weak coupling map between thesetwo systems and their quasi-deformations (Sec. 4) one could expect the quasi-KdV system toconverge to a nonholonomic variant of the KdV system.In that regime it may be possible to interpret the present power-series expansion in (cid:15) aslocal constraints characterizing a nonholonomic system, that are identified with constraints.The present quasi-KdV system further supports single- and multi-soliton-type solutions whichcan very well converge to their ideal counterparts asymptotically. Thus it will be interestingto identify such systems with order-by-order relations (‘constraints’) by evaluating asymptoticform of the exact solution for quasi-KdV and other systems. It is seen that a comprehensive quasi-integrable deformation of the KdV system is indeedpossible, provided the loop-algebraic generalization [18] has been considered. As the KdVequation is not dynamical neither in the sense of Galilean (like NLS) nor Lorentz (like SG)systems, the deformation has to be performed at an off-shell level ( i. e. , without using theEOM). The available Hamiltonian formulation of KdV system comes to rescue in this ab-initio treatment, wherein the Lax construction in the SU (2) representation has been utilized toobtain the standard Abelianization that utilized the inherent sl (2) loop algebra. In the sub-space where the Abelianization manifests, anomalous conservation laws were obtained. Theanomaly function and the coefficients of the rotated Lax components are shown to have definiteparity properties given the deformed solution being a parity eigenstate, subsequently ensuringasymptotic conservation of the anomalous charges implying quasi-integrability.As particular cases, both local extensions as well as power-deformation in the Hamiltoniandensity are considered. The prior allows for constructing a scaled KdV at the simplest level,with single-soliton profile, as well as families of higher-derivative extensions to the same withsome of them possibly being quasi-integrable, with at least one conserved charge. This is in-tuitively allowed, following the weak-coupling correspondence between KdV and NLS systems.The compatibility of the present deformation with that of QI NLS system has been obtained,followed by corresponding one-soliton solutions with variable scaling. The deformed KdV so-lution corresponding to the quasi-NLS soliton appeared to have similar properties to the KdVsoliton train that corresponds to the undeformed NLS soliton. In case of power deformationof the type u → u (cid:15) in the Hamiltonian the situation becomes more complicated with possi-ble singularities. Single soliton-like localized structures are still supported by these deformedsystems. Further, an order-by-order expansion in the deformation parameter led to localizedsolutions. Although they may not posses definite parity, asymptotically they are expected toyield conservation of the charges. It will be worthwhile to numerically analyze particular stablesolutions to this deformed KdV and higher derivative systems, and to study their behavior withthose from QI NLS system when the weak correspondence is valid. We aspire to imply thesame for complex coupled KdV formalism in the future. he present approach of quasi-deformation can be extended to further related systems, in-cluding KdV-type hierarchies, mKdV and their non-local counterparts. An obvious generaliza-tion would be that of the coupled complex KdV system of Eq.s 7 could be more challengingowing to the requirement of a constant semi-simple sl (2) element in the spatial Lax component.A suitable form of deformation of this component could be, A → (cid:16) ¯ qλ + 1 (cid:17) σ + − ( u − λ ) σ − ≡ ¯ q − q √ F − + ¯ q + q √ F − − + √ F , (82)under the sl (2) representation of Eq.s 20. However, this includes a scaling of the conjugatefield ¯ q by the spectral parameter which violates its relative grading with respect to q in A toyield a KdV system, the latter being apparent from the following equivalent Lax pair: A → ¯ qλ σ + − λqσ − and B → (¯ qq x − q ¯ q x ) σ − λ (cid:0) ¯ q xx + 2 q ¯ q (cid:1) σ + + λ (cid:0) u xx + 2¯ qq (cid:1) σ − , (83)with explicit spectral dependence, yielding the same undeformed KdV system. This hampersthe Abelianization scheme, more so as the Lax pair of Eq.s 5 (or Eq. 83) is a direct consequenceof the AKS hierarchy [18]. However the quasi-deformation of this complex coupled KdV systemmay be possible through some non-trivial Lax representation; which may directly be obtainedthrough brute-force numerical calculations.The aim of the present work was to obtain a first-principle quasi-deformation formalism ofthe KdV system which corresponds to deformation of the corresponding Hamiltonian. Thishas led to various possible deformed structures which may be quasi-integrable. The particularcase of RLW and mRLW systems are known to be so [20] and multi-solitonic structures werenumerically obtained in conformity. However, pure analytic determination ( e. g. by Hirotamethod) of definite-parity localized solutions of these systems are yet to be achieved. Theperturbative approach of the present work supply some insight into the possible solutions andtheir localization is found to be very much possible. A full-on numerical simulation of suchsolutions is beyond the scope of this work. We expect to take up this task in the near future.We further aspire to analyze similar systems like mKdV and other hierarchies of the KdVsystem for possible quasi-deformation and to extend this study to non-local systems. Acknowledgement:
PG is grateful to Jun Nian and Vasily Pestun for interesting discussions.The authors are also grateful to Professors Luiz. A. Ferreira, Wojtek J. Zakrzewski and BettiHartmann for their encouragement, various useful discussions and critical reading of the draftduring the initial phases of this work.
References [1] D. J. Korteweg and G. de Vries,
On the change of form of long waves advancing in arectangular canal and on a new type of long stationary waves , Philos. Mag. (1895)422–443.
2] A. Das,
Integrable models , World Scientific, Singapore (1989).[3] N. J. Zabusky and M. D. Kruskal,
Interaction of ‘Solitons’ in a collisionless plasma andthe recurrence of initial states , Phys. Rev. Lett. (1965) 240.[4] B. Malomed, Nonlinear Schrödinger Equations , in Scott, Alwyn (ed.), Encyclopedia ofNonlinear Science, New York: Routledge (2005) pp. 639–643 and references therein.[5] V. I. Arnold,
Mathematical Methods of Classical Mechanics, second edition, Graduate Textsin Mathematics 60 , Springer, New York, (1989).[6] L. L. Chau and W. Nahm,
Differential Geometric Methods in Theoretical Physics: Physicsand Geometry: Nato Science Series B , Springer Science and Business Media, Philadelphia(2013).[7] P. Lax,
Integrals of nonlinear equations of evolution and solitary waves , Comm. PureApplied Math. (1968) 467-490.[8] G. Schneider, Approximation of the Korteweg–de Vries equation by the NonlinearSchrödinger equation , J. Diff. Eqn. (1998) 333.[9] J. Nian,
Note on Nonlinear Schrödinger Equation, KdV Equation and 2D TopologicalYang-Mills-Higgs Theory , Int. J. Mod. Phys. A (2019) 1950074.[10] P. G. Drazin, Solions , London Mathematical Society Lecture Note Series Cambridge:Cambridge University Press (1983) and references therein.[11] T. Tao,
Why are solitons stable? , Bull. Am. Math. Soc. (2009) 1–33.[12] S. Ferreira, L. Girardello and S. Sciuto, An infinite set of conservation laws of the super-symmetric sine-gordon theory , Phys. Lett. B, (1978) 303.[13] L. A. Ferreira and W. J. Zakrzewski, The concept of quasi-integrability: a concrete example ,JHEP, (2011) 130.[14] L. A. Ferreira, G. Luchini and W. J. Zakrzewski, The Concept of Quasi-Integrability ,Nonlinear and Modern Mathematical Physics AIP Conf. Proc., (2013) 43.[15] K. Abhinav and P. Guha,
Quasi-Integrability in Supersymmetric Sine-Gordon Models , EPL (2016) 10004.[16] L.A. Ferreira, G. Luchini and W. J. Zakrzewski,
The concept of quasi-integrability formodified non-linear Schrödinger models , JHEP (2012) 103.[17] K. Abhinav, I. Mukherjee and P. Guha, Non-holonomic and Quasi-integrable deformationsof the AB Equation , arXiv:2008.05775 [math-ph] (To be published).[18] P. Guha,
Nonholonomic deformation of generalized KdV-type equations , J. Phys. A: Math.Theor. (2009) 345201.[19] M. Dunajski, Integrable Systems
ISlecture _ notes _ . pdf [20] F. ter Braak, L. A. Ferreira and W.J. Zakrzewski, Quasi-integrability of deformations ofthe KdV equation , Nuclear Physics B (2019) 49–94.[21] D.H. Peregrine,
Calculations of the development of an undular bore , J. Fluid Mech. (1966) 321–330.
22] T. B. Benjamin, J. L. Bona and J. J. Mahoney,
Model equations for long waves in nonlineardispersive systems , Philos. Trans. R. Soc. A (1972) 47–78.[23] J. D. Gibbon, J. C. Eilbeck and R. K. Dodd,
A modified regularized long-wave equationwith an exact two-soliton solution , J. Phys. A (1976) L127–L130.[24] P. Guha and I. Mukherjee, Analysis and comparative study of non-holonomic and quasi-integrable deformations of the nonlinear Schrödinger equation , Nonlinear Dyn. (2020)1179.[25] M. Adler and P. van Moerbeke, Completely Integrable Systems, Euclidean Lie Algebras,and Curves , Adv. Math. (1980) 267.[26] W. W. Symes, Systems of Toda type, inverse spectral problems, and representation theory ,Invent. Math. (1980) 13.[27] A. Fordy and P. P. Kulish, Nonlinear Schrödinger equations and simple Lie algebras
Com-mun. Math. Phys. (1983) 427.[28] P. Guha, Adler–Kostant–Symes construction, bi-Hamiltonian manifolds, and KdV equa-tions, Jour. Math. Phys. (1997) 5167.[29] P.Guha and I. Mukherjee, Hierarchies and Hamiltonian Structures of the NonlinearSchrödinger Family Using Geometric and Spectral Techniques , Disc. Cont. Dyn. Sys. B (2019) 1677.(2019) 1677.