A unified approach to the AKNS, DNLS, KP and mKP hierarchies in the anti-self-dual Yang-Mills reduction

A unified approac h to the AKNS, DNLS, KP and mKP hierarc hies in the an ti-self-dual Y ang-Mills reduction Shangsh uai Li 1 , 2 ∗ Ken-ic hi Maruno 3 † Da-jun Zhang 4 , 5 ‡ 1 Sc ho ol of Mathematics and Statistics, Ningb o Universit y , Ningb o 315211, China 2 Graduate School of Mathematics, Nagoy a Universit y , Nagoy a 464-8602, Japan 3 Departmen t of Applied Mathematics, F aculty of Science and Engineering, W aseda Universit y , T okyo 169-8555, Japan 4 Departmen t of Mathematics, Shanghai Universit y , Shanghai 200444, China 5 Newtouc h Center for Mathematics of Shanghai Universit y , Shanghai 200444, China Marc h 25, 2026 Abstract W e sho w a unified approac h to the Ablo witz-Kaup-Newell-Segur (AKNS) hierarc h y and the unreduced deriv ativ e nonlinear Schr¨ odinger (DNLS) hierarc hies (including the Kaup- New ell, Chen-Lee-Liu, Gerdjik o v-Iv ano v and a generalized DNLS), together with their m ulti- comp onen t extensions, in the framew ork of the anti-self-dual Y ang-Mills (ASD YM) reduc- tion. By restricting the gauge group to GL(2), the Kadomtsev-P etviashvili (KP) and mod- ified KP (mKP) hierarchies are form ulated in the ASDYM reduction via squared eigen- function symmetry constrain ts. In this case, the bilinearization of the generalized DNLS equations can also b e understoo d through this reduction. Finally , Gram-t ype exact solu- tions for the relev ant equations are presen ted in terms of quasi-determinan ts. Keyw ords: an ti-self-dual Y ang-Mills equation; reduction; gauge transformation; hierarc hy structure; exact solution 1 In tro duction The an ti-self-dual Y ang-Mills (ASDYM) equation is a fundamental mo del in quantum field theory and v arious branc hes of mathematical ph ysics [1]. Since the 1970s, the ASD YM equation has b een in tensively studied, revealing numerous prop erties, such as the connection to complex v ector bundles [2–4], the existence of instan ton solutions [5–8], the Painlev ´ e prop ert y [9, 10] and so on. As an integrable system, the ASD YM equation has b een in vestigated using a v ariet y of in tegrable metho ds, suc h as the in verse scattering transformation [11], B¨ ac klund transformation based on the Riemann-Hilbert problem [12], a direct method inspired b y the Sato theory [13], bilinear metho d [14–16], Darb oux transformation [17–20], direct linearization approach [21], bi-differen tial graded algebra approach [22] and so on. ∗ Email: lishangshuai@n bu.edu.cn † Email: kmaruno@waseda.jp ‡ Corresp onding author. Email: djzhang@staff.shu.edu.cn 1 The fact that the ASDYM equation can b e reduced to many low er-dimensional classical in- tegrable systems (see W ard’s conjecture in [23]) has motiv ated a trend in searching for suc h in tegrable reductions. F or example, the Korteweg-de V ries (KdV) equation and the non- linear Schr¨ odinger (NLS) equation can b e obtained from the SL(2) ASD YM equation [24]. The relationship b et w een the SL( N ) ASDYM equation and the Gel’fand-Dick ey hierarch y (whic h includes the KdV equation) has been established in [25]. F urthermore, reduction to the Kadom tsev-Petviash vili (KP) hierarc hy are p ossible b y allowing the gauge p oten tials to b e op erator-v alued [26]. Ablo witz et al. discussed the deriv ation of integrable hierarchies from the ASD YM hierarch y , pro viding examples such as the N -wa v e system, and the KdV, NLS, Dav ey- Stew artson (DS), and KP equations [27]. Ge et al. deriv ed reductions to the sine-Gordon and Liouville equations [28]. In addition to the KP and DS equations, the ASD YM equation admits reductions to other (2+1)-dimensional equations, including the mo dified KP (mKP) equation, a (2+1)-dimensional Gardner equation, a generalized DS equation [29], and the Calogero-Bogoy a vlenskii-Schiff (CBS) equation [30]. F urthermore, the W ard’s chiral mo del in (2+1) dimensions [31, 32] and a (2+1)-d relativistic-in v arian t field theory mo del [33] are obtainable from the J -matrix form ulation; a hierarc hy of (2+1)-d NLS equation and its relationship with the ASD YM equation hav e b een discussed in [34]. F or one-dimensional equations, the P ainlev ´ e equations arise through the reductions of con- formal symmetries [1, 35]. Besides, B¨ ac klund transformations of the ASD YM equation can be reduced to obtain some discrete integrable systems [36]. Moreov er, W ard’s conjecture also ex- tends to non-comm utative space [37]. Comprehensiv e reviews of the ASDYM reductions can b e found in [1, 38], along with their extensive references. A recent dev elopment is the formulation of the Cauc hy matrix schemes of the ASDYM equations [39–41], which facilitate connections b et w een the ASDYM equations and other low er- dimensional integrable systems. As a result, a reduction from the ASD YM matrix formulations to the F ok as-Lenells (FL) equation w as found in [42]. The FL equation is the first negative mem b er of the p oten tial Kaup-Newell (KN) hierarc hy [43–45], which is commonly kno wn as the (unreduced) deriv ativ e nonlinear Sc hr¨ odinger (DNLS) hierarc h y . The reduction from the ASD YM to the FL equation in [42] suggests the existence of further hidden structures that can be exploited for reductions. In this paper, w e aim to reveal these structures and establish the connection betw een the ASD YM hierarch y and other integrable hierarc hies, especially the Ablowitz-Kaup-New ell-Segur (AKNS) hierarch y , the unreduced DNLS hierarc hies (including the KN, Chen-Lee-Liu (CLL), Gerdjiko v-Iv ano v (GI) and a generalized DNLS), and the KP and mKP hierarc hies. W e will make use of the J -matrix form ulation and K -matrix form ulation of the ASD YM hierarch y (see equation (2.6) and (2.7)). Here we sp ecially men tion that by restricting the gauge group to GL(2), the KP hierarc h y and mKP hierarch y are resp ectiv ely defined b y the (1 , 1)-th entries of K and J , whic h will b e explained in Sec.4. It has long b een b eliev ed that the KP equations cannot b e derived from the ASDYM equations via reduction unless op erator-v alued gauge fields are p ermitted [30, 38]. In con trast, w e provide an op erator-free reduction to the KP and mKP hierarchies from the ASDYM equations. This paper is organized as follo ws: In Section 2, we briefly review the ASD YM hierarc hy and sev eral classical integrable hierarchies. The transformations b et ween these hierarchies, realized within the ASD YM framework, are established in Section 3. In Section 4, w e summarize the results for the GL(2) case, where we discuss the Lax pairs of the AKNS hierarch y , the reduction to the KP and mKP hierarchies, and the bilinear transformation of the generalized DNLS system. F or the completeness of this pap er, we pro vide the procedure of constructing exact 2 solutions in Section 5, and sho w ho w ASDYM solutions map to those of the reduced systems. The final part is devoted to the concluding remarks. 2 Preliminary: hierarc h y structures of some integrable systems In this section, w e will review the hierarch y structures of the ASD YM equations and sev eral represen tative integrable equations. 2.1 The ASD YM equations In this pap er, we mainly consider the ASDYM hierarch y of the following form. Let g b e the Lie algebra of a Lie group G . W e assume the following zero-curv ature system holds [46, 47]: [ ∂ n +1 − λ∂ n + A n +1 , ∂ m +1 − λ∂ m + A m +1 ] = 0 , (2.1) i.e., ∂ m A n +1 − ∂ n A m +1 = 0 , (2.2a) ∂ m +1 A n +1 − ∂ n +1 A m +1 − [ A n +1 , A m +1 ] = 0 , (2.2b) whic h is the compatible condition of the asso ciated linear system: ( ∂ n +1 − λ∂ n + A n +1 )Ψ = 0 , n ∈ Z , (2.3) or alternatively ( ∂ n − λ n ∂ 0 + n − 1 X i =0 λ i A n − i )Ψ = 0 , n ∈ Z + , (2.4a) ( ∂ n − λ n ∂ 0 − − 1 X i = n λ i A n − i )Ψ = 0 , n ∈ Z − . (2.4b) Here λ is the sp ectral parameter, Ψ = Ψ( λ ) is the eigenfunction and A n +1 ∈ g with n ∈ Z are gauge potentials. This system dep ends on a set of infinite indep enden t complex v ariables t := { t n } . W e denote the corresp onding partial deriv atives b y ∂ n := ∂ /∂ t n . The gauge potentials A n +1 can b e expressed in terms of either J -matrix or K -matrix via the definitions: A n +1 := − ( ∂ n +1 J ) J − 1 = ∂ n K. (2.5) Substituting this expression in to the zero-curv ature system (2.2) yields the follo wing t w o matrix form ulations of the ASDYM hierarc hy: • J -matrix formulation (the Y ang equation [48, 49]): ∂ m (( ∂ n +1 J ) J − 1 ) − ∂ n (( ∂ m +1 J ) J − 1 ) = 0 . (2.6) • K -matrix formulation (the Chalmers-Siegel equation [50, 51]): ∂ m +1 ∂ n K − ∂ n +1 ∂ m K − [ ∂ n K, ∂ m K ] = 0 . (2.7) Remark 1. The c omp atible expr ession (2.5) for the gauge p otentials { A j } is imp ortant in our r ese ar ch b e c ause b oth the J -matrix formulation (2.6) and the K -matrix formulation (2.7) ar e the c onse quenc es of (2.5) . In some r e ductions, we ne e d to start fr om the gauge p otential expr ession (2.5) r ather than the ab ove two e quations. We wil l se e the advantage in so doing. 3 2.2 The AKNS equations The well-kno wn NLS equation is one of the most imp ortan t in tegrable equations, arising in a wide range of ph ysical con texts [52]: i u τ + u ζ ζ + 2 δ | u | 2 u = 0 , δ = ± 1 . (2.8) Here i 2 = − 1, | u | 2 = u ¯ u where bar represents complex conjugate, and the parameter δ distin- guishes b et w een the fo cusing ( δ = 1) and defo cusing ( δ = − 1) cases of the nonlinearity . No w, let us forget ab out the complex structure of (2.8) and consider the follo wing system (the NLS equation is reco v ered b y introducing ( ζ , τ ) := (i x, i t 2 ) as real co ordinates and requiring u := r = δ ¯ q ): r t 2 = r xx − 2 q r 2 , (2.9a) q t 2 = − q xx + 2 q 2 r , (2.9b) whic h is known as the unreduced NLS system, b elonging to the p ositiv e AKNS hierarch y (see [53] or Chapter 3.3 of [54]): r q ! t n = R n H 0 , R := ∂ x − 2 r ∂ − 1 x q − 2 r ∂ − 1 x r 2 q ∂ − 1 x q − ∂ x + 2 q ∂ − 1 x r ! , H 0 = r − q ! , (2.10) where ∂ − 1 x is the integration operator defined as ∂ − 1 x f = 1 2 ( R x −∞ − R ∞ x ) f ( x ′ )d x ′ , and n ∈ Z . The recursion op erator R can be also rewritten as R := σ 3 ∂ x + 2 − r q ! ∂ − 1 x  q r  , σ 3 := Diag(1 , − 1) . (2.11) This hierarch y can b e alternativ ely expressed in a recursiv e form r q ! t n +1 = R r q ! t n , r q ! t 0 = r − q ! , (2.12) whic h expands to the follo wing system: r t n +1 = r x,t n − 2 r ∂ − 1 x ( q r ) t n , r t 0 = r , (2.13a) q t n +1 = − q x,t n + 2 q ∂ − 1 x ( r q ) t n , q t 0 = − q . (2.13b) 2.3 The DNLS equations and gauge transformations The NLS-t ype equations with deriv ativ e nonlinearities are referred to as the deriv ativ e nonlinear Sc hr¨ odinger (DNLS) equations. One example of the DNLS equations, prop osed b y Gerdjik o v and Iv ano v (GI), is presented as [55]: i u τ + u ζ ζ + 2i δ u 2 ¯ u ζ + 2 | u | 4 u = 0 . (2.14) The unreduced GI system reads p t 2 = p xx − 2 p 2 q x − 2 p 3 q 2 , (2.15a) q t 2 = − q xx − 2 q 2 p x + 2 p 2 q 3 , (2.15b) 4 whic h yields the GI equation (2.14) by taking ( ζ , τ ) := (i x, i t 2 ) as real co ordinates and the conjugate reduction u := p = δ ¯ q . The abov e system is the first nonlinear p ositiv e mem ber of the GI hierarch y: p q ! t n +1 = R p q ! t n , p q ! t 0 = p − q ! , (2.16) where R is the recursion operator [56]: R := ∂ x − 2 p∂ − 1 x ( q x + 2 pq 2 ) − 2 p 2 + 2 p∂ − 1 x ( p x − 2 p 2 q ) − 2 q 2 + 2 q ∂ − 1 x ( q x + 2 pq 2 ) − ∂ x − 2 q ∂ − 1 x ( p x − 2 p 2 q ) ! . (2.17) The expansion of (2.16) yields p t n +1 = p x,t n − 2 p∂ − 1 x ( pq x ) t n − 2 p∂ − 1 x ( p 2 q 2 ) t n , p t 0 = p, (2.18a) q t n +1 = − q x,t n − 2 q ∂ − 1 x ( p x q ) t n + 2 q ∂ − 1 x ( p 2 q 2 ) t n , q t 0 = − q . (2.18b) Remark 2. We wil l show that the AKNS hier ar chy (2.13) and the GI hier ar chy (2.18) shar e the same q -function in our c onstruction. In fact, ther e is a R ic c ati-typ e Miur a tr ansformation b etwe en the two systems (cf. [57]): r = − p x + p 2 q . (2.19) The e quation (2.18b) is a dir e ct r esult of (2.13b) in light of (2.19) . Sinc e the t n +1 -derivative of r c an b e r epr esente d as r t n +1 = (2 pq − ∂ x ) p t n +1 + p 2 q t n +1 , (2.20) we ar e le d to (2.18a) by substituting (2.19) into (2.13a) and utilizing (2.18b) . This r elation also facilitate d the investigations on the FL e quation fr om the AKNS ne gative flow [58, 59], which was r e c ently r e alize d in the ASDYM r e duction [42]. The DNLS equations are of three t yp es, other than the DNLS equation of GI-t yp e (2.14), there are the Chen-Lee-Liu (CLL) DNLS equation [60]: i u τ + u ζ ζ − 2i δ | u | 2 u ζ = 0 , (2.21) and the Kaup-Newell (KN) DNLS equation [61]: i u τ + u ζ ζ − 2i δ ( | u | 2 u ) ζ = 0 . (2.22) The three models are all in tegrable and related via gauge transformations [62]. In fact, the metho d of gauge transformation can b e further applied to deriv e a generalized deriv ative non- linear Sc hr¨ odinger (GDNLS) equation. Through the transformation (here w e use u [GDNLS] and u [GI] to distinguish the v ariables in differen t DNLS equations): u [GDNLS] = u [GI] exp  i γ δ Z | u | 2 [GI] d ζ  , γ ∈ C , (2.23) one obtains the GDNLS equation (here we denote u = u [GDNLS] ) [63–65]: i u τ + u ζ ζ − 2i γ δ | u | 2 u ζ − 2i( γ − 1) δ u 2 ¯ u ζ + ( γ − 1)( γ − 2) | u | 4 u = 0 . (2.24) 5 It b ecomes the GI equation (2.14) when γ = 0, the CLL equation (2.21) when γ = 1 and the KN equation (2.22) when γ = 2. Nev ertheless, these three equations are t ypically in v estigated indep enden tly , as the gauge transformations linking them inv olv e complex integrations that b ecome difficult to compute in m ulti-soliton case. The hierarch y structure of (2.24) can b e inv estigated by applying the gauge transformation to (2.18). Inspired from (2.23), the gauge factor is introduced as s = exp( − ∂ − 1 x ( pq )) , (2.25) whic h satisfies the following dynamics s x s = − pq , (2.26a) s t n s = − ∂ − 1 x ( pq ) t n , (2.26b) s t n +1 s = − ∂ − 1 x ( p x q ) t n + ∂ − 1 x ( p 2 q 2 ) t n + pq t n . (2.26c) The new v ariables are introduced as ( u, v ) := ( s γ p, s − γ q ) , (2.27) whic h transforms the GI hierarc h y (2.18) to the GDNLS hierarc h y u t n +1 = u x,t n + γ u∂ − 1 x ( u x v ) t n + γ u x ∂ − 1 x ( uv ) t n + 2( γ − 1) u∂ − 1 x ( uv x ) t n − ( γ − 1)( γ − 2) u∂ − 1 x ( u 2 v 2 ) t n , (2.28a) v t n +1 = − v x,t n + γ v ∂ − 1 x ( uv x ) t n + γ v z ∂ − 1 x ( uv ) t n + 2( γ − 1) v ∂ − 1 x ( u x v ) t n + ( γ − 1)( γ − 2) v ∂ − 1 x ( u 2 v 2 ) t n , (2.28b) where u t 0 = u and v t 0 = − v . The GDNLS hierarc h y with recursive co ordinates can be generated b y the action of a recursion operator M on the preceding flow u v ! t n +1 = M u v ! t n , u v ! t 0 = u − v ! , (2.29) where the recursion operator M = {M ij } is given by M 11 = ∂ x + γ uv + ( γ − 2) u∂ − 1 x v x + γ u x ∂ − 1 x v − 2( γ − 1)( γ − 2) u∂ − 1 x uv 2 , (2.30a) M 12 = 2( γ − 1) u 2 − ( γ − 2) u∂ − 1 x u x + γ u x ∂ − 1 x u − 2( γ − 1)( γ − 2) u∂ − 1 x u 2 v , (2.30b) M 21 = 2( γ − 1) v 2 − ( γ − 2) v ∂ − 1 x v x + γ v x ∂ − 1 x v + 2( γ − 1)( γ − 2) v ∂ − 1 x uv 2 , (2.30c) M 22 = − ∂ x + γ uv + ( γ − 2) v ∂ − 1 x u x + γ v x ∂ − 1 x u + 2( γ − 1)( γ − 2) v ∂ − 1 x u 2 v . (2.30d) As a direct consequence of (2.28) and (2.29), the recursive formulations and recursion op er- ators for the CLL and KN hierarc hies are provided without proof. T aking γ = 1 in (2.28) and define ( ˜ p, ˜ q ) := ( u | γ =1 , v | γ =1 ) = ( sp, s − 1 q ) , (2.31) it gives rise to the CLL hierarc h y ˜ p t n +1 = ˜ p x,t n + ˜ p∂ − 1 x ( ˜ p x ˜ q ) t n + ˜ p x ∂ − 1 x ( ˜ p ˜ q ) t n , ˜ p t 0 = ˜ p, (2.32a) ˜ q t n +1 = − ˜ q x,t n + ˜ q ∂ − 1 x ( ˜ p ˜ q x ) t n + ˜ q x ∂ − 1 x ( ˜ p ˜ q ) t n , ˜ q t 0 = − ˜ q . (2.32b) 6 The recursion op erator for CLL hierarc h y is presen ted as e R := ∂ x + ˜ p ˜ q + ˜ p x ∂ − 1 x ˜ q − ˜ p∂ − 1 x ˜ q x ˜ p x ∂ − 1 x ˜ p + ˜ p∂ − 1 x ˜ p x ˜ q x ∂ − 1 x ˜ q + ˜ q ∂ − 1 x ˜ q x − ∂ x + ˜ p ˜ q + q x ∂ − 1 x ˜ p − ˜ q ∂ − 1 x ˜ p x ! . (2.33) T aking ( ˆ p, ˆ q ) := ( u | γ =2 , v | γ =2 ) = ( s 2 p, s − 2 q ) , (2.34) it yields the KN hierarch y ˆ p t n +1 = ˆ p x,t n + 2( ˆ p∂ − 1 x ( ˆ p ˆ q ) t n ) x , ˆ p t 0 = ˆ p, (2.35a) ˆ q t n +1 = − ˆ q x,t n + 2( ˆ q ∂ − 1 x ( ˆ p ˆ q ) t n ) x , ˆ q t 0 = − ˆ q , (2.35b) with the asso ciated recursion op erator b R := ∂ x + 2 ˆ p ˆ q + 2 ˆ p x ∂ − 1 x ˆ q 2 ˆ p 2 + 2 ˆ p x ∂ − 1 x ˆ p 2 ˆ q 2 + 2 ˆ q x ∂ − 1 x ˆ q − ∂ x + 2 ˆ p ˆ q + 2 ˆ q x ∂ − 1 x ˆ p ! . (2.36) The recursion op erator of KN hierarc h y can also b e written as b R := ∂ x + 2 ∂ x ( ˆ p∂ − 1 x ˆ q ) 2 ∂ x ( ˆ p∂ − 1 x ˆ p ) 2 ∂ x ( ˆ q ∂ − 1 x ˆ q ) − ∂ x + 2 ∂ x ( ˆ q ∂ − 1 x ˆ p ) ! , (2.37) or alternatively b R := σ 3 ∂ x + 2 ∂ x ˆ p ˆ q ! ∂ − 1 x  ˆ q ˆ p  , σ 3 := Diag(1 , − 1) . (2.38) 2.4 The KP hierarc h y and squared eigenfunction symmetry constrain t The in tegrability of the KP hierarch y can b e describ ed through utilizing the pseudo-differential op erator [66–68]. It is form ulated as the compatible condition of the linear system Lϕ = λϕ, L := ∂ x + u 2 ∂ − 1 x + u 3 ∂ − 2 x + · · · , (2.39a) ϕ t m = B m ϕ, B m := ( L m ) ≥ 0 , m ≥ 1 . (2.39b) The compatible condition of (2.39) indicates the Lax representation of KP hierarch y ∂ t m L = [ B m , L ] , (2.40a) ∂ t m B n − ∂ t n B m + [ B n , B m ] = 0 , (2.40b) whic h gives rise to the KP hierarch y (here we denote ˆ u = u 2 ) ˆ u t 1 = K 1 = ˆ u x , (2.41a) ˆ u t 2 = K 2 = ˆ u y , (2.41b) ˆ u t 3 = K 3 = 1 4 ˆ u xxx + 3 ˆ u ˆ u x + 3 4 ∂ − 1 x ˆ u y y , (2.41c) · · · · · · The linear system (2.39) and its adjoint form are closely related to the AKNS hierarc h y (2.10) under the squared eigenfunction symmetry constraint. W e explain it b y in tro ducing the inner pro duct ⟨ f ( x ) , g ( x ) ⟩ := Z ∞ −∞ f ( x ) g ( x )d x, (2.42) 7 where f ( x ) and g ( x ) tend to zero fast enough when | x | → ∞ . It follo ws that the adjoint op erator of O , denoted as O ∗ , is defined b y ⟨ f ( x ) , O g ( x ) ⟩ = ⟨ O ∗ f ( x ) , g ( x ) ⟩ . (2.43) Th us, apart from (2.39), one can alternativ ely consider its adjoint system L ∗ ϕ ∗ = λϕ ∗ , ϕ ∗ t m = − B ∗ m ϕ ∗ , (2.44) where we use ϕ ∗ to denote the eigenfunction of (2.44). It can b e pro ved that ( ϕϕ ∗ ) x is a symmetry of the KP hierarc hy , conv entionally called the squared eigenfunction symmetry [69, 70]. Since ˆ u x is also a symmetry of the KP hierarch y , one ma y consider a symmetry constraint ˆ u x + ( ϕϕ ∗ ) x = 0, which indicates that ˆ u = − ϕϕ ∗ = − r q, ( r , q ) := ( ϕ, ϕ ∗ ) . (2.45) It turns out that under the abov e constrain t, the pseudo-differential operator L can b e repre- sen ted as L = ∂ x − r ∂ − 1 x q . (2.46) This leads the linear system ϕ t n = B n ϕ and ϕ ∗ t n = − B ∗ n ϕ ∗ to a constrained form r t n = B n r =  ( ∂ x − r ∂ − 1 x q ) n  ≥ 0 r , (2.47a) q t n = − B ∗ n q = −  ( − ∂ x + q ∂ − 1 x r ) n  ≥ 0 q , (2.47b) whic h w as pro v ed to be the p ositiv e AKNS hierarch y (2.10) with n ≥ 1 [71–74]. Based on the ab o v e connection, once we hav e a solution pair ( r , q ) of the AKNS hierarch y with n ≥ 1, then ˆ u = − q r pro vides a solution for the KP hierarch y (2.41). Remark 3. Base d on the Miur a tr ansformation (2.19) and the gauge tr ansformation (2.25) , the KP solution c an b e pr ovide d by (se e also in [74, 75]) ˆ u = ˜ p x ˜ q = p x q − p 2 q 2 = − q r , (2.48) wher e ( p, q ) satisfy the GI hier ar chy (2.18) and ( ˜ p, ˜ q ) satisfy the CLL hier ar chy (2.32) . 2.5 The mKP hierarc h y and squared eigenfunction symmetry constrain t Lik ewise, there is a similar result for the mKP hierarch y . The mKP hierarc hy is go verned b y the linear system L ψ = λψ , L := ∂ x + v 0 + v 1 ∂ − 1 x + · · · , (2.49a) ψ t m = B m ψ , B m := ( L m ) ≥ 1 , m ≥ 1 , (2.49b) where the compatibility gives rise to the mKP hierarch y (here we denote ˆ v = v 0 ) ˆ v t 1 = K 1 = ˆ v x , (2.50a) ˆ v t 2 = K 2 = ˆ v y , (2.50b) ˆ v t 3 = K 3 = 1 4 ˆ v xxx − 3 2 ˆ v 2 ˆ v x + 3 2 ˆ v x ∂ − 1 x ˆ v y + 3 4 ∂ − 1 x ˆ v y y , (2.50c) · · · · · · 8 Equation (2.50c) is kno wn as the mKP equation. The adjoin t system of (2.49) is giv en b y L ∗ ψ ∗ = λψ ∗ , ψ ∗ t m = −B ∗ m ψ ∗ , (2.51) where ψ ∗ is the asso ciated eigenfunction. By imp osing the symmetry constraint [74] ˆ v = ψ ψ ∗ = ˜ p ˜ q , ( ˜ p, ˜ q ) := ( ψ , ψ ∗ ) , (2.52) the op erator L can b e represented as L = ∂ x + ˜ p∂ − 1 x ˜ q ∂ x , (2.53) and meanwhile, the coupled system ψ t n = B n ψ and ψ ∗ t n = −B ∗ n ψ ∗ is restricted to [76] ˜ p t n = B n ˜ p =  ( ∂ x + ˜ p∂ − 1 x ˜ q ∂ x ) n  ≥ 1 ˜ p, (2.54a) ˜ q t n = −B ∗ n ˜ q = −  ( − ∂ x + ∂ x ˜ q ∂ − 1 x ˜ p ) n  ≥ 1 ˜ q . (2.54b) This gives rise to the CLL hierarc h y (for n ≥ 1) ˜ p ˜ q ! t n = e R n e H 0 , e H 0 = ˜ p − ˜ q ! , (2.55) where the recursion op erator e R is giv en b y (2.33). This fact indicates that once we hav e a solution pair ( ˜ p, ˜ q ) of the CLL hierarch y with n ≥ 1, then ˆ v = ˜ p ˜ q yields a solution for the mKP hierarc hy (2.50). 3 The an ti-self-dual Y ang-Mills reduction In the previous section, w e briefly review ed the hierarc h y structures for the ASD YM equation along with several classical integrable systems. Though these hierarc hies ha ve man y closed con- nections, they are often treated separately . F or example, the Riccati-type Miura transformation (2.19) reveals the connection b et w een the AKNS hierarc hy and the GI hierarch y , but it is diffi- cult to determine p from giv en ( r , q ) b y using (2.19). On the other hand, once w e ha ve a solution of the GI hierarc hy , the associated solutions for the GDNLS hierarch y can b e calculated in prin- ciple by using the gauge transformation (2.23). Ho w ever, the gauge factor in (2.25) inv olv es an in tegration, whic h mak es the calculation complicated in m ulti-soliton case. In this section, w e demonstrate how the AKNS and DNLS hierarchies emerge from the gauge p oten tial expression (2.5) and the K -matrix formulation (2.7). As a result, these complex transformations find a p erfect realization in the same framew ork of the ASD YM reduction. 3.1 Dimensional reduction condition The ASD YM equations are equipp ed with a group structure, which is usually characterized b y G = GL( M , C ), where M ≥ 2. Here w e assume M = m 1 + m 2 and partition the J -matrix and K -matrix as 2 × 2 blo c k matrices: J = ( J 11 ) m 1 × m 1 ( J 12 ) m 1 × m 2 ( J 21 ) m 2 × m 1 ( J 22 ) m 2 × m 2 ! , K = ( K 11 ) m 1 × m 1 ( K 12 ) m 1 × m 2 ( K 21 ) m 2 × m 1 ( K 22 ) m 2 × m 2 ! . (3.1) 9 In this sense, the expansion (2.5) indicates ∂ n +1 J = − ( ∂ n K ) J , whic h yields ∂ n +1 J 11 = − ( ∂ n K 11 ) J 11 − ( ∂ n K 12 ) J 21 , (3.2a) ∂ n +1 J 12 = − ( ∂ n K 11 ) J 12 − ( ∂ n K 12 ) J 22 , (3.2b) ∂ n +1 J 21 = − ( ∂ n K 21 ) J 11 − ( ∂ n K 22 ) J 21 , (3.2c) ∂ n +1 J 22 = − ( ∂ n K 21 ) J 12 − ( ∂ n K 22 ) J 22 . (3.2d) The dimensional reduction is imp osed by assuming the follo wing constraint on K -matrix: ∂ 0 K = [ K, E ] , (3.3) where E := 1 m 1 + m 2 m 2 ( I m 1 × m 1 ) 0 0 − m 1 ( I m 2 × m 2 ) ! . (3.4) Explicitly , this implies the expansion of ∂ 0 K and ∂ 1 J = − ( ∂ 0 K ) J gives rise to ∂ 0 K = 0 − K 12 K 21 0 ! , (3.5) and ∂ 1 J = K 12 J 21 K 12 J 22 − K 21 J 11 − K 21 J 12 ! . (3.6) Under the reduction (3.3), the K -matrix formulation (2.7) with m = 0 simplifies to ∂ 1 ∂ n K − [ ∂ n +1 K, E ] − [ ∂ n K, [ K , E ]] = 0 . (3.7) Expressing this equation in terms of each entry of K , we obtain the following system ∂ 1 K 11 = K 12 K 21 , ∂ 1 K 22 = − K 21 K 12 , (3.8a) ∂ 1 ∂ n K 12 = − ∂ n +1 K 12 − ( ∂ n K 11 ) K 12 + K 12 ( ∂ n K 22 ) , (3.8b) ∂ 1 ∂ n K 21 = ∂ n +1 K 21 + ( ∂ n K 22 ) K 21 − K 21 ( ∂ n K 11 ) . (3.8c) F urthermore, by assuming J 11 to b e in vertible, the t 1 -deriv ativ e of J 21 J − 1 11 indicates ∂ 1 ( J 21 J − 1 11 ) = − K 21 − ( J 21 J − 1 11 ) K 12 ( J 21 J − 1 11 ) , (3.9) where relations (3.6) are used in the calculation. The t wo relations, i.e. (3.7) and (3.9), will rev eal the construction of the matrix AKNS and GI hierarchies in the GL( M , C ) ASDYM reduction framework. Let us proceed. 3.2 Reduction to the matrix AKNS and GI hierarc hies Theorem 1. F or the GL( M , C ) ASD YM hier ar chy (2.7) that satisfies the c onstr aint (3.3) , set x := t 1 as the sp acial variable and define ( R, Q ) := ( K 21 , − K 12 ) . (3.10) Then, ( R, Q ) satisfy the r e cursive r epr esentation of the matrix AKNS hier ar chy R t n +1 = R x,t n − R∂ − 1 x ( QR ) t n − ∂ − 1 x ( RQ ) t n R, R t 0 = R, (3.11a) Q t n +1 = − Q x,t n + Q∂ − 1 x ( RQ ) t n + ∂ − 1 x ( QR ) t n Q, Q t 0 = − Q, (3.11b) wher e R and Q T ar e m 1 × m 2 matrix-value d functions. 10 Pr o of. This theorem is a direct result of (3.8). In fact, (3.8a) indicates K 11 = ∂ − 1 x ( K 12 K 21 ) and K 22 = − ∂ − 1 x ( K 21 K 12 ). Inserting them into (3.8b) and (3.8c) and expressing the results in terms of ( R , Q ) using (3.10), we get (3.11). The system (3.11) appears to be a non-comm utative v ersion of (2.13), and can b e viewed as the m ulti-comp onen t AKNS hierarch y as w ell. F or example, taking n = 1 in (3.11) yields a matrix form of (2.9) R t 2 = R xx − 2 RQR, (3.12a) Q t 2 = − Q xx + 2 QRQ. (3.12b) Defining ( ζ , τ ) := (i x, i t 2 ) and imposing U := R = δ Q † , where ‘ † ’ denotes the complex conjugate transp ose, it giv es rise to the matrix NLS equation i U τ + U ζ ζ + 2 δ U U † U = 0 , δ = ± 1 . (3.13) In particular, when U is a t w o-dimensional v ector, it b ecomes the Manak o v system [77]; when U is a 2 × 2 symmetric matrix, it becomes an integrable model of the spin-1 Gross-Pitaevskii equation that describ es spinor Bose-Einstein condensates [78–80]. T o construct the matrix GI hierarch y , by defining P := J 21 J − 1 11 (3.14) and using (3.10), from the relation (3.9) we hav e P x = − R + P QP , (3.15) whic h is the matrix version of the Miura transformation (2.19) and connects the matrix AKNS and GI hierarchies. Then w e present the following theorem. Theorem 2. F or the GL( M , C ) ASDYM hier ar chy, we c onsider the c omp atible gauge p otential expr ession (2.5) and the K -matrix formulation (2.7) that satisfy the c onstr aint (3.3) wher e M = m 1 + m 2 . Setting x := t 1 as the sp acial variable and defining ( P , Q ) := ( J 21 J − 1 11 , − K 12 ) , (3.16) we have the r e cursive r epr esentation of the matrix GI hier ar chy P t n +1 = P x,t n − ∂ − 1 x ( P Q x + P QP Q ) t n P − P ∂ − 1 x ( Q x P + QP QP ) t n , P t 0 = P , (3.17a) Q t n +1 = − Q x,t n − ∂ − 1 x ( QP x − QP QP ) t n Q − Q∂ − 1 x ( P x Q − P QP Q ) t n , Q t 0 = − Q, (3.17b) wher e P and Q T ar e m 1 × m 2 matrix-value d functions. Pr o of. W e make use of the Miura transformation (3.15) which we rewrite as R = − P x + P QP. (3.18) Inserting it into (3.11b) yields (3.17a) directly . T o get (3.17b), we calculate the t n +1 -deriv ativ e of P b y using (3.2), which gives rise to P t n +1 = − R t n − P ∂ − 1 x ( QR ) t n − ∂ − 1 x ( RQ ) t n P − P Q t n P . Then we substitute (3.18) in to the ab o ve relation, and we are led to P t n +1 = P x,t n − P ( QP ) t n − ( P Q ) t n P − P ∂ − 1 x ( − QP x + QP QP ) t n − ∂ − 1 x ( − P x Q + P QP Q ) t n P = P x,t n − ∂ − 1 x ( P Q x + P QP Q ) t n P − P ∂ − 1 x ( Q x P + QP QP ) t n , whic h is exactly (3.17b). W e complete the proof. 11 3.3 Reduction to the v ector DNLS hierarc hies 3.3.1 The scalar case T o inv estigate the multi-component extensions of other DNLS hierarc hies, we first realize the gauge transformation of the scalar GI hierarch y in the ASD YM framew ork, where m 1 = m 2 = 1. By Theorem 2, the v ariables of the scalar GI hierarc h y are defined as ( p, q ) := ( J 21 J − 1 11 , − K 12 ) . (3.19) Noticing that from (3.6) we hav e ∂ x J 11 = K 12 J 21 , since p and q are scalars, we hav e q p = − K 12 J 21 J − 1 11 = − ( ∂ x J 11 ) J − 1 11 = − (ln J 11 ) x . (3.20) This indicates that the gauge factor (2.25) is determined by s = exp  − ∂ − 1 x ( q p )  = exp( ∂ − 1 x (ln J 11 ) x ) = J 11 , (3.21) whic h agrees with (2.26a) and (3.20). No w that p, q and the gauge factor s are formulated in terms of the ASD YM v ariables J and K , so are the DNLS v ariables defined in (2.27), (2.31) and (2.34). In the following w e summarize the realizations of the DNLS hierarchies in the ASDYM framew ork in T able 1. T able 1: Realizations of the DNLS hierarchies in the ASDYM framew ork The DNLS hierarchies Realizations in the ASDYM framew ork The GI hierarch y (2.18) ( p, q ) = ( J 21 J − 1 11 , − K 12 ) The CLL hierarch y (2.32) ( ˜ p, ˜ q ) = ( J 21 , − J − 1 11 K 12 ) The KN hierarch y (2.35) ( ˆ p, ˆ q ) = ( J 21 J 11 , − J − 2 11 K 12 ) The GDNLS hierarch y (2.28) ( u, v ) = ( J 21 J γ − 1 11 , − J γ 11 K 12 ) The gauge factor (2.25) s = J 11 3.3.2 The v ector case The ab o v e discussion in the scalar case can b e extended to the v ector case, if w e still assume J 11 to be a scalar function while P and Q defined in (3.16) are a column v ector and a row v ector resp ectiv ely . This setting corresponds to the partition M = m 1 + m 2 where m 1 = 1 , m 2 = m . W e now construct the v ector GDNLS hierarch y . Theorem 3. F or the GL( M , C ) ASDYM hier ar chy, we c onsider the c omp atible gauge p otential expr ession (2.5) and the K -matrix formulation (2.7) that satisfy the c onstr aint (3.3) wher e m 1 = 1 , m 2 = m . Set x := t 1 as the sp acial variable and define ( U, V ) := ( J 21 J γ − 1 11 , − J γ 11 K 12 ) . (3.22) 12 Note that in this c ase J 11 is a sc alar while U and V T ar e m -th or der c olumn ve ctors. Then we have the r e cursive r epr esentation of the ve ctor GDNLS hier ar chy U t n +1 = U x,t n + γ ∂ − 1 x ( U x V ) t n U + γ U x ∂ − 1 x ( V U ) t n + ( γ − 1) U ∂ − 1 x ( V x U ) t n + ( γ − 1) ∂ − 1 x ( U V x ) t n U + ( γ − 1) ∂ − 1 x ( U V U V ) t n U − ( γ − 1) 2 U ∂ − 1 x ( V U V U ) t n , U t 0 = U, (3.23a) V t n +1 = − V x,t n + γ V ∂ − 1 x ( U V x ) t n + γ ∂ − 1 x ( V U ) t n V x + ( γ − 1) ∂ − 1 x ( V U x ) t n V + ( γ − 1) V ∂ − 1 x ( U x V ) t n − ( γ − 1) V ∂ − 1 x ( U V U V ) t n + ( γ − 1) 2 ∂ − 1 x ( V U V U ) t n V , V t 0 = − V . (3.23b) Pr o of. The pro of is provided in Appendix A. The system (3.23) app ears to b e the vectorial generalization of (2.28). By defining ( ξ , τ ) := (i x, i t 2 ) as real coordinates and imposing V = δ U † , it reduces to the v ector GDNLS equation i U τ + U ζ ζ − 2i γ δ U ζ U † U − 2i( γ − 1) δ U U † ζ U + ( γ − 1)( γ − 2) U ( U † U ) 2 = 0 . (3.24) Assuming U = ( u 1 , u 2 , . . . , u m ) T , one rewrites (3.24) into its explicit form ( j = 1 , . . . , m ) i u j,τ + i u j,ζ ζ − 2i γ δ u j,ζ m X k =1 | u k | 2 − 2i( γ − 1) δ u j m X k =1 ¯ u k,ζ u k + ( γ − 1)( γ − 2) u j m X k =1 | u k | 4 = 0 . (3.25) Similar to the scalar GDNLS equation, the vector GDNLS equation (3.24) b ecomes the vector GI DNLS equation when γ = 0 (see equation (12) in [81]) i U τ + U ζ ζ − 2i δ U U † ζ U + 2( U † U ) 2 U = 0 , (3.26) and the vector CLL DNLS equation when γ = 1 (see equation (10) in [81]) i U τ + U ζ ζ − 2i δ U ζ U † U = 0 , (3.27) as well as the v ector KN DNLS equation when γ = 2 (see equation (2) in [81]) i U τ + U ζ ζ − 4i δ U ζ U † U − 2i δ U U † ζ U = 0 . (3.28) F or more details ab out the multi-component DNLS equations, one can also refer to [82–85] and the references therein. 4 Additional commen ts on GL(2) gauge group When the gauge group is restricted to b e G = GL(2), the matrix AKNS and v ector DNLS hierarc hies b ecome scalar systems in the ASDYM reduction framew ork. In this case, more fruitful results would emerge. 4.1 Lax represen tation of the AKNS system The first result is the realization of the Lax represen tation of the AKNS system in the ASDYM framew ork. This provides an alternative wa y to obtain the AKNS hierarc h y from the ASDYM 13 reduction. T o ac hieve that, in addition to the dimensional reduction constrain t (3.3), w e assume ∂ 0 Ψ = σ 3 Ψ. Then the ASDYM linear system (2.4) with positive n is written as Ψ x =  λσ 3 − [ K, 1 2 σ 3 ]  Ψ , Ψ t n = λ n σ 3 − n − 1 X i =0 λ i K t n − i − 1 ! Ψ . (4.1) According to (3.8), the K -matrix can b e expressed with resp ect to (3.10) in scalar case, i.e. ( r , q ) := ( K 21 , − K 12 ) , (4.2) whic h reads K = − ∂ − 1 x ( q r ) − q r ∂ − 1 x ( q r ) ! . (4.3) Substituting (4.3) into (4.1) gives rise to the Lax pair for the AKNS hierarch y (2.13), in which the x -deriv ativ e of Ψ is Ψ x = λ − q − r − λ ! Ψ , (4.4) whic h is exactly the AKNS sp ectral problem. F or the time part, when n = 2, the t 2 -deriv ativ e of Ψ expands to Ψ t 2 =  λ 2 σ 3 − λ [ K, 1 2 σ 3 ] − K x  Ψ = λ 2 + q r − λq + q x − λr − r x − λ 2 − q r ! Ψ . (4.5) The compatible condition of (4.4) and (4.5) yields the unreduced NLS system (2.9). When taking n = 3, we ha ve Ψ t 3 =  λ 3 σ 3 − λ 2 [ K, 1 2 σ 3 ] − λK x − K t 2  Ψ , = λ 3 + λq r − q x r + q r x − λ 2 q + λq x − q xx + 2 q 2 r − λ 2 r − λr x − r xx + 2 q r 2 − λ 3 − λq r − q r x + q x r ! Ψ , (4.6) whic h along with (4.4) giv es rise to the coupled system for mKdV equation r t 3 = r xxx − 6 q rr x , (4.7a) q t 3 = q xxx − 6 q rq x . (4.7b) 4.2 Reduction to the KP and mKP hierarc hies In Sec.2.4 and 2.5 w e hav e review ed the KP and mKP hierarc hies along with the asso ciated squared eigenfunction symmetry constrain ts. Since b oth the AKNS and CLL hierarchies can b e formulated in the ASD YM reduction framew ork, the KP and mKP hierarc hies can be also obtained in the same framework: • The KP hierarc h y from the AKNS hierarc h y: ˆ u = − q r = K 12 K 21 = K 11 ,x . • The mKP hierarc h y from the CLL hierarc h y: ˆ v = ˜ q ˜ p = q p = − K 12 J 21 J − 1 11 = − (ln J 11 ) x . Let us summarize the ab o ve constructions in the follo wing theorem. 14 Theorem 4. F or the GL( 2 , C ) ASD YM hier ar chy, we c onsider the J -matrix formulation (2.6) and the K -matrix formulation (2.7) that satisfy the c onstr aint (3.3) . Set ( x, y ) := ( t 1 , t 2 ) as the sp e cial variables. Then, u := K 11 ,x satisfies the KP hier ar chy (2.41) , and v = − (ln J 11 ) x satisfies the mKP hier ar chy (2.50) . W e ma y address more words to explain ho w the KP hierarch y is constructed from the AKNS hierarc hy . In fact, in light of the symmetry construct (2.45), the positive AKNS hierarch y (2.10) can recov er ϕ t n = B n ϕ, ϕ ∗ t n = − B ∗ n ϕ ∗ , ( n = 1 , 2 , · · · ) . The KP hierarc hy is defined from the compatibilit y of { ϕ t n = B n ϕ } for differen t n . Lik ewise, the mKP hierarch y can b e obtained from the CLL hierarc h y and the constraint (2.52). It is believed that the KP equation has not y et b een derived from the ASD YM equations b y reduction in the known literature unless the existence of op erator-v alued gauge fields is allo wed [30, 38]. In this section, we hav e pro vided op erator-free reductions for b oth the KP and the mKP hierarchies. In the form ulation K 11 and J 11 are used, which b elong to the entries of the K -matrix and J -matrix with the constrain t (3.3). 4.3 Bilinear transformation for the GDNLS equation The bilinear transformation for ASDYM equations has b een given by [15, 16]: J := G f , J − 1 := S f , K := H f , (4.8) whic h transforms (2.5) into the following bilinear equations D n +1 G · S = − 2 D n H · ( f I ) , (4.9) where GS = f 2 I and I is the identit y matrix. Here D n is the Hirota’s bilinear deriv ativ e [86] D n f ( t n ) · g ( t n ) = ( ∂ n − ∂ n ′ ) f ( t n ) g ( t n ′ ) | n ′ = n . (4.10) F or the GL(2) ASDYM equations, where G and H are 2 × 2 matrices, we partition them as G = { g ij } and H = { h ij } (1 ≤ i, j ≤ 2). Thus the en tries in J and K can b e rewritten as J ij = g ij f , K ij = h ij f . (4.11) According to the bilinear transformation (4.8) and the ASDYM reduction framew ork, w e pro- p ose the bilinear transformation for GDNLS system as ( u, v ) = ( J 21 J γ − 1 11 , − J − γ 11 K 12 ) = g γ − 1 11 g 21 f γ , − f γ − 1 h 12 g γ 11 ! , (4.12) whic h perfectly agrees with the construction in [87] by Kak ei, Sasa and Satsuma. This fact pro vides an independent supp ort for the bilinear transformations in the ASD YM reductions. The GDNLS hierarch y con tains all three t yp es of the DNLS hierarchies, their bilinear trans- formations are summarized as follows: • The GI hierarc h y (2.18): ( p, q ) = ( J 21 J − 1 11 , − K 12 ) = ( g 21 /g 11 , − h 12 /f ). • The CLL hierarc h y (2.32): ( ˜ p, ˜ q ) = ( J 21 , − J − 1 11 K 12 ) = ( g 21 /f , − h 12 /g 11 ). 15 • The KN hierarc h y (2.35): ( ˆ p, ˆ q ) = ( J 21 J 11 , − J − 2 11 K 12 ) = ( g 11 g 21 /f 2 , − f h 12 /g 2 11 ). It is in teresting to note that the FL equation has the same formu lation as the GI system in this unified reduction framework (see [88], Eq.(2.18)), although the FL system is derived as a p oten tial form in the negativ e order KN hierarc hy . Our construction also explains wh y the bilinear transformations of the DNLS equations hav e different tau functions [89, 90]. 5 Sp ecial soliton solutions In Section 3, we hav e realized the AKNS hierarc hy and the DNLS hierarc hies in the ASD YM reduction framework. In this part, we will show the ASD YM solutions consequen tly give rise to the NLS and DNLS solutions. 5.1 Solutions of the ASD YM While explicit solutions to the GL( M , C ) ASDYM equations hav e b een constructed in sev- eral w orks [16, 20, 40], we here presen t one kind of Gram-t yp e solutions expressed by quasi- determinan ts. Briefly speaking, the concept of quasi-determinan t is a non-comm utativ e gener- alization of the ratio b et w een a matrix and its submatrix. Consider a partitioned blo c k matrix H of the following form: H = A B C d ! , (5.1) where A ∈ C N × N is a square inv ertible matrix, d ∈ C m 1 × m 2 , with B and C b eing matrices of compatible dimensions. The quasi-determinant of H with resp ect to the blo c k d (which is framed) is given by the formula | H | [ d ] :=      A B C d      = d − C A − 1 B . (5.2) F or details ab out prop erties of quasi-determinan ts, one can refer to [91, 92]. T o present ASDYM solutions w e first recall a result on solutions to the Sylvester equation. Lemma 1. [93] F or given squar e matric es A, B , C of the same or der, the Sylvester e quation AX − X B = C (5.3) has a unique matrix solution X if A and B do not shar e eigenvalues. Then we hav e the following. Theorem 5. L et G = GL ( M , C ) , we assume J [0] and K [0] to b e se e d solutions that satisfy A [0] n +1 := − ( ∂ n +1 J [0] )( J [0] ) − 1 = ∂ n K [0] . (5.4) Supp ose c onstant matric es Ξ , Λ ∈ C N × N do not shar e eigenvalues, η and θ satisfy the fol lowing line ar differ ential system: ∂ n +1 η − Ξ( ∂ n η ) = η A [0] n +1 , (5.5a) ∂ n +1 θ − ( ∂ n θ )Λ = − A [0] n +1 θ . (5.5b) 16 Intr o duc e a squar e matrix Ω as a solution of the Sylvester e quation ΞΩ − ΩΛ = η θ, (5.6) fr om which Ω is uniquely define d b e c ause Ξ and Λ do not shar e eigenvalues. Then, the fol lowing c onstructions J =      Ω Ξ − 1 η θ I      J [0] , K = −      Ω η θ 0      + K [0] (5.7) satisfy the c omp atible expr ession (2.5) for the gauge p otentials. Pr o of. W e leav e the pro of with details in Appendix B. A similar one is av ailable in Section 3 of [41]. W e are going to present explicit forms for the elements in volv ed in (5.7). By setting J [0] = I and K [0] = 0, i.e., A [0] n +1 = 0, the ASDYM solution in v acuum state is go verned by the following Cauc hy matrix structure (see also in Section 2.3 of [40]): ΞΩ − ΩΛ = η θ, ∂ n η = Ξ n ( ∂ 0 η ) , ∂ n θ = ( ∂ 0 θ )Λ n . (5.8) F or conv enience, we c ho ose Ξ and Λ to b e diagonal matrices, and th us the system (5.8) is satisfied by the following construction: Ξ := Diag( ξ 1 , · · · , ξ N ) , η := ( ϕ is ( L ( ξ i , t ))) N × M , (5.9a) Λ := Diag( λ 1 , · · · , λ N ) , θ := ( ψ sj ( L ( λ j , t ))) M × N , (5.9b) Ω = (Ω ij ) N × N , Ω ij = P M s =1 ϕ is ψ sj ξ i − λ j , (5.9c) where ϕ is and ψ sj are arbitrary functions of the follo wing linear phase factors respectively: L ( ξ i , t ) = X n ∈ Z ξ n i t n , L ( λ j , t ) = X n ∈ Z λ n j t n . (5.10) In this case, the explicit form ulae of matrix formulations are given by J =      Ω Ξ − 1 η θ I      = I − θ Ω − 1 Ξ − 1 η , K = −      Ω η θ 0      = θ Ω − 1 η . (5.11) 5.2 Realization of dimensional reduction T o meet with the constraint (3.3), w e imp ose the initial condition on the t 0 -flo ws of η and θ : ∂ 0 η = η E , ∂ 0 θ = − E θ , (5.12) where E is the blo c k matrix defined in (3.4). In fact, taking deriv ative ∂ 0 on the Sylv ester equation in (5.8) yields Ξ( ∂ 0 Ω) − ( ∂ 0 Ω)Λ = η E θ − η E θ = 0 , (5.13) whic h leads to ∂ 0 Ω = 0. Th us, from (5.11) w e hav e ∂ 0 K = ( ∂ 0 θ )Ω − 1 η + θ Ω − 1 ( ∂ 0 θ ) = − E K + K E = [ K , E ] , (5.14) 17 whic h recov ers the constrain t (3.3). T o get explicit expressions for J ij and K ij ( i, j = 1 , 2), by reform ulating η and θ in to η = ( η 1 , η 2 ) , η 1 = ( ϕ is ) N × m 1 , η 2 = ( ϕ is ) N × m 2 , (5.15a) θ = θ 1 θ 2 ! , θ 1 = ( ψ sj ) m 1 × N , θ 2 = ( ψ sj ) m 2 × N , (5.15b) the elements in the block matrices J and K in (3.1) are respectively given b y J 11 = I − θ 1 Ω − 1 Ξ − 1 η 1 , J 12 = − θ 1 Ω − 1 Ξ − 1 η 2 , (5.16a) J 21 = − θ 2 Ω − 1 Ξ − 1 η 1 , J 22 = I − θ 2 Ω − 1 Ξ − 1 η 2 , (5.16b) as well as K 11 = θ 1 Ω − 1 η 1 , K 12 = θ 1 Ω − 1 η 2 , (5.17a) K 21 = θ 2 Ω − 1 η 1 , K 22 = θ 2 Ω − 1 η 2 . (5.17b) In addition, the initial condition (5.12) indicates that ϕ is and ψ sj are exponential functions of the following forms: ϕ is = A is exp 1 m 1 + m 2 X n ∈ Z ξ n i t n ! , 1 ≤ s ≤ m 1 , (5.18a) ϕ is = A is exp − 1 m 1 + m 2 X n ∈ Z ξ n i t n ! , m 1 + 1 ≤ s ≤ m 1 + m 2 , (5.18b) ψ sj = B sj exp − 1 m 1 + m 2 X n ∈ Z λ n j t n ! , 1 ≤ s ≤ m 1 , (5.18c) ψ sj = B sj exp 1 m 1 + m 2 X n ∈ Z λ n j t n ! , m 1 + 1 ≤ s ≤ m 1 + m 2 , (5.18d) where A is , B sj ∈ C are initial phase factors. Inserting them in to (5.15) and (5.9) one can get explicit forms of η j , θ j and Ω, and then get J ij and K ij from (5.16) and (5.17), resp ectiv ely . 5.3 Solutions of the fo cusing NLS eq uation and GI DNLS equation As examples, w e sho w in the following that ho w solutions of the focusing NLS equation (2.8) | δ =1 and the fo cusing GI DNLS equation (2.14) | δ =1 can b e obtained from the abov e solutions pre- sen ted in Sec.5.2. T o get solutions for these tw o equations, we set G = GL(2) and define ( ζ , τ ) := (i t 1 , i t 2 ) to b e real coordinates, which indicates m 1 = m 2 = 1 and (5.18) becomes ϕ i 1 = A i 1 exp  − i ξ i 2 ζ − i ξ 2 i 2 τ  , ϕ i 2 = A i 2 exp  i ξ i 2 ζ + i ξ 2 i 2 τ  , (5.19a) ψ 1 j = B 1 j exp i λ j 2 ζ + i λ 2 j 2 τ ! , ψ 2 j = B 2 j exp − i λ j 2 ζ − i λ 2 j 2 τ ! . (5.19b) η j , θ j and Ω are correspondingly defined from (5.15) and (5.9c). 18 5.3.1 Solutions of the fo cusing NLS equation In light of Theorem 1 and (5.17), we ha ve ( r , q ) = ( θ 2 Ω − 1 η 1 , − θ 1 Ω − 1 η 2 ) , (5.20) whic h provide solutions to the follo wing unreduced NLS system: i r τ + r ζ ζ + 2 q r 2 = 0 , i q τ − q ζ ζ − 2 q 2 r = 0 . (5.21) T o achiev e solutions for the focusing NLS equation, we in tro duce the conjugate constraints as Λ = Ξ † , A i 1 = ¯ B 1 i , A i 2 = ¯ B 2 i , ( i = 1 , 2 , · · · N ) . (5.22) It follows that θ 1 = η † 1 , θ 2 = η † 2 , (5.23) where η j = ( ϕ 1 j , ϕ 2 j , · · · , ϕ N j ) T , j = 1 , 2 , (5.24) with { ϕ kj } given in (5.19a). Substituting them in to the Sylvester equation (5.6) yields ΞΩ − ΩΞ † = η 1 η † 1 + η 2 η † 2 . (5.25) Comparing it with its complex conjugate transpose and making use of Lemma 1, w e hav e Ω = − Ω † . Finally w e reach to q = − η † 1 Ω − 1 η 2 = ( η † 2 Ω − 1 η 1 ) † = ¯ r . (5.26) Th us the abov e q solves the focusing NLS equation i q τ + q ζ ζ + 2 | q | 2 q = 0 . (5.27) 5.3.2 Solutions of the fo cusing GI DNLS equation Lik ewise, by Theorem 2 and (5.17), we ha ve ( p, q ) =  J 21 J 11 , − K 12  =  − θ 2 Ω − 1 Ξ − 1 η 1 1 − θ 1 Ω − 1 Ξ − 1 η 1 , − θ 1 Ω − 1 η 2  (5.28) that satisfy the unreduced GI system i p τ + p ζ ζ + 2i p 2 q ζ + 2 p 3 q 2 = 0 , i q τ − q ζ ζ + 2i q 2 p ζ − 2 p 2 q 3 = 0 . (5.29) Then we imp ose the conjugate constraints Λ = Ξ † , A i 1 = ¯ B 1 i , A i 2 = − ¯ ξ i ¯ B 2 i , ( i = 1 , 2 , · · · , N ) . (5.30) First, it allows us to hav e θ 1 = η † 1 , θ 2 = − η † 2 Ξ † , (5.31) 19 where η j is giv en in (5.24). Second, b y substituting (5.30) and (5.31) into the Sylv ester equation (5.6), we are led to ΞΩ − ΩΞ † = η 1 η † 1 − η 2 η † 2 Ξ † . (5.32) Then, subtracting (5.32) † from (5.32) and making use of Lemma 1 sho w that Ω = − Ω † + η 2 η † 2 , (5.33) whic h provides the relation b et ween Ω and Ω † for the GI DNLS equation. Note that after substituting (5.33) in to the term ΩΞ † in (5.6), we arrive at ΞΩ + Ω † Ξ † = η 1 η † 1 , (5.34) whic h indicates a conjugate constrain t b et w een Ω and Ξ. Th us we can rewrite the J 11 , J 21 and K 12 in terms of only η j , Ω and Ξ: J 11 = 1 − η † 1 Ω − 1 Ξ − 1 η 1 , J 21 = η † 2 Ξ † Ω − 1 Ξ − 1 η 1 , K 12 = η † 1 Ω − 1 η 2 . (5.35) Then, through a direct calculation, w e ha ve K † 12 J 11 = η † 2 Ω −† η 1 − η † 2 Ω −† η 1 η † 1 Ω − 1 Ξ − 1 η 1 = η † 2 Ω −† η 1 − η † 2 (Ω −† Ξ + Ξ † Ω − 1 )Ξ − 1 η 1 = − J 21 , (5.36) where A −† denotes the inv erse of A † , and s = J 11 yields J 11 J † 11 = (1 − η † 1 Ω − 1 Ξ − 1 η 1 )(1 − η † 1 Ξ −† Ω −† η 1 ) = 1 − η † 1 (Ω − 1 Ξ − 1 + Ξ −† Ω −† ) η 1 + η † 1 Ω − 1 Ξ − 1 η 1 η † 1 Ξ −† Ω −† η = 1 − η † 1 (Ω − 1 Ξ − 1 + Ξ −† Ω −† − Ω − 1 Ξ − 1 − Ξ −† Ω −† ) η 1 = 1 . (5.37) Th us we arriv e at p = J 21 J − 1 11 = − K † 12 = ¯ q , whic h giv es rise to the GI equation i p τ + p ζ ζ + 2i p 2 ¯ p ζ + 2 | p | 4 p = 0 . (5.38) And u = s γ p = ( ¯ s ) − γ ¯ q solv es the GDNLS equation (2.24) with δ = 1, where s = J 11 is giv en in (5.35). 6 Concluding remarks In this pap er, we ha ve deriv ed the matrix AKNS hierarch y , the matrix GI hierarc hy , the vector GDNLS hierarc h y (whic h includes the GI, KN, and CLL hierarchies as special cases), and the scalar KP and mKP hierarchies as reductions of the ASD YM equations. These reductions were ac hieved by utilizing the compatible expression (2.5) for the gauge p oten tials { A j } , along with the resulting equations (2.6) and (2.7)—sp ecifically , the J -matrix and K -matrix formulations of the ASD YM equations. All the dimensional reductions are achiev ed under the same constraint (3.3) imposed on K , and the ab o ve mentioned low er dimensional hierarc hies are formulated using the en tries of K and J . This approach pro vides a unified framework for understanding these integrable hierarchies, their Miura-type links, and the bilinearization of DNLS equations. The v ariable form ulations in GL(2) gauge group are the following: • The AKNS hierarc h y (2.13): ( r, q ) = ( K 21 , − K 12 ). 20 • The GI hierarc h y (2.18): ( p, q ) = ( J 21 J − 1 11 , − K 12 ). • The gauge factor b et ween DNLS hierarc hies (2.25): s = J 11 . • The GDNLS hierarc h y (2.28): ( u, v ) = ( J 21 J γ − 1 11 , − J − γ 11 K 12 ). • The KP hierarc h y (2.41): u = K 11 ,x . • The mKP hierarc h y (2.50): v = − (ln J 11 ) x . Among them, the AKNS and GI hierarchies admit matrix v ersion in GL( m 1 + m 2 ) gauge group, while the GDNLS hierarc hy admits a v ector v ersion in GL(1 + m ) gauge group. Note that the v ector GDNLS hierarc hy reduces to the GI, CLL and KN when γ = 0 , 1 , 2, respectively . F urthermore, it is noteworth y that we hav e deriv ed the KP and mKP hierarc hies via op erator- free ASDYM reductions, in con trast to previous approac hes [30, 38]. Within this unified framework, not only equations, but also soliton solutions of the ASDYM equations can b e systematically mapped to those of the reduced hierarc hies. The fact about solutions has b een demonstrated in Section 5. W e presented Gram-t yp e solutions in terms of quasi-determinan ts in Theorem 5 and its pro of is provided in Appendix B. Soliton solutions as sp ecial case of Theorem 5 w ere presented in Sec.5.1 as w ell. Explicit form ulae of K and J that meet the constraint (3.3) hav e b een given in Sec.5.2, which provide soliton solutions for all the lo wer dimensional hierarchies obtained through the reductions. In addition, we also provided t wo examples to sho w further complex reductions of solutions for the fo cusing NLS and GI DNLS equations. W e end up this pap er with mentioning some p oten tial topics inspired from our current w ork. Firstly , in con trast to traditional ASDYM reduction tec hniques [1, 38], our metho d allo ws a direct corresp ondence b et ween the v ariables of the ASDYM equations and those of the resulting in tegrable hierarc hies in reductions. This means many kno wn solutions of the ASD YM equations might b e used to generated solutions for the reduced integrable hierarchies obtained in this pap er, and vice v ersa. Secondly , it is w ell known that the ASD YM equation has a special t yp e solutions, namely , instan tons. The solutions we ha ve presen ted in Theorem 5 do not include suc h type of solutions. Whether the reduction links are helpful in constructing instan tons will b e considered in the future. Thirdly , the reductions to the DNLS hierarchies and soliton solution formulations presented in Sec.5.2 implies the Cauch y matrix structures of the DNLS t yp e equations, which hav e not yet b een revealed in the literature. W e will in v estigate these structures elsewhere. In addition, the reductions also imply possibility to get Bogo y avlenskii’s t yp e breaking soliton solutions. This will be considered separately . Finally , while satisfactory discretizations for the ASDYM equation are currently lacking, but most of the ab o ve listed lo wer dimensional equations hav e been discretized. The results of this pap er should b e helpful in finding an integrable discrete analogue of the ASDYM equation. Considering that the ASD YM equation has v ery rich structures in differential geometry , a successful discretization for the ASD YM equation will bring more insight for differential geometry from a discrete persp ectiv e. Ac kno wledgments The work of SSL is supp orted by the JSPS Ov erseas Research F ellowships (P25326). The w ork of KIM is supp orted by the JSPS grant (JP22K03441, 23K22407) and the JSPS Bilateral Program (JPJSBP120247414). The work of DJZ is supported by the NSF C gran t (12271334, 1241540016). 21 A Pro of of Theorem 3 In the vector case, the gauge factor s is defined as (cf.(2.25)) s = exp  − ∂ − 1 x ( QP )  , (A.1) where P and Q T are m -th order column v ector, defined in (3.16). Similar to the scalar case presen ted in in Sec.3.3.1, in the v ector case, we also ha ve QP = − (ln J 11 ) x and s = J 11 . Thus w e can compute the dynamics of s with respect to P and Q : s x s = − QP , s t n s = − ∂ − 1 x ( QP ) t n , (A.2a) s t n +1 s = − ∂ − 1 x ( QP ) t n +1 = − ∂ − 1 x ( QP x ) t n + ∂ − 1 x ( QP QP ) t n + Q t n P , (A.2b) where to get the last equation we ha ve made use of (3.17). With these in hand, w e then calculate the deriv ativ es of U and V defined in (3.22). Since J 11 is a scalar, we ha ve U = J 21 J γ − 1 11 = s γ P . Th us we find U x = s γ P x + γ s γ − 1 s x P = s γ ( P x − γ P QP ) , where we hav e used (A.2a). Similarly , we hav e U x = s γ ( P x − γ P QP ) , U x,t n = s γ ( P x,t n − γ ( P QP ) t n − γ ( P x − γ P QP ) ∂ − 1 x ( QR ) t n ) , U t n +1 = s γ ( P t n +1 − γ P ∂ − 1 x ( QP x ) t n + γ P ∂ − 1 x ( QP QP ) t n + γ P Q t n P ) , as well as the deriv atives of V = − J − γ 11 K 12 = s − γ Q : V x,t n = s − γ ( Q x,t n + γ ( QP Q ) t n + γ ( Q x + γ QP Q ) ∂ − 1 x ( QR ) t n ) , V t n +1 = s − γ ( Q t n +1 + γ ∂ − 1 x ( QP x ) t n Q − ∂ − 1 x ( QP QP ) t n Q − Q t n P Q ) . As for the nonlinear terms in (3.23a), they are expressed in terms of P and Q as ∂ − 1 x ( U x V ) t n U = s γ ∂ − 1 x ( P x Q − γ P QP Q ) t n P , U x ∂ − 1 x ( V U ) t n = s γ ( P x − γ P QP ) ∂ − 1 x ( QP ) t n , U ∂ − 1 x ( V x U ) t n = s γ P ∂ − 1 x ( Q x P + γ QP QP ) t n , ∂ − 1 x ( U V U V ) t n U = s γ ∂ − 1 x ( P QP Q ) t n P , U ∂ − 1 x ( V U V U ) t n = s γ P ∂ − 1 x ( QP QP ) t n . Then substituting the ab o v e exp ressions in to the b oth sides of (3.23a) and making use of (3.17a), w e find (3.23a) holds. The pro of for (3.23b) can b e done along the same line. W e skip presen ting details. B Pro of of Theorem 5 The pro of consists of the follo wing lemmas. Lemma 2. [41, 94] Consider the quasi-determinant (5.2) and supp ose the derivative of A has the de c omp osition ∂ n A = n X l =1 E l F l , (B.1) 22 wher e ∂ n is the differ ential op er ator with r esp e ct to t n , E l and F l ar e matric es of c omp atible dimensions. Then, the derivative of the quasi-determinant is given by ∂ n      A B C d      = ∂ n d +      A B ∂ n C 0      +      A ∂ n B C 0      + n X l =1      A E l C 0           A B F l 0      . (B.2) Lemma 3. The matrix Ω define d in (5.6) satisfies the fol lowing differ ential r e curr enc e r elations: ∂ n +1 Ω = ( ∂ n η ) θ + ( ∂ n Ω)Λ = Ξ( ∂ n Ω) − η ( ∂ n θ ) , n ∈ Z . (B.3) Pr o of. W e b egin by taking t n +1 -deriv ativ e of the Sylv ester equation (5.6), which giv es rise to Ξ( ∂ n +1 Ω) − ( ∂ n +1 Ω)Λ = ( ∂ n +1 η ) θ + η ( ∂ n +1 θ ) . Substituting the disp ersion relations ∂ n +1 η = Ξ( ∂ n η ) and ∂ n +1 θ = ( ∂ n θ )Λ into the right-hand side yields Ξ( ∂ n +1 Ω) − ( ∂ n +1 Ω)Λ =Ξ( ∂ n η ) θ + η ( ∂ n θ )Λ + Ξ( ∂ n Ω)Λ − Ξ( ∂ n Ω)Λ =Ξ(( ∂ n η ) θ + ( ∂ n Ω)Λ) − (Ξ( ∂ n Ω) − η ( ∂ n θ ))Λ . Noticing that ( ∂ n η ) θ + ( ∂ n Ω)Λ = Ξ( ∂ n Ω) − η ( ∂ n θ ) is nothing but the result of the t n -deriv ativ e of (5.6), we are led to the recursiv e relation (B.3). Lemma 4. L et (5.7) b e written as J = W J [0] , K = Z + K [0] , (B.4) wher e W and Z ar e define d by W =      Ω Ξ − 1 η θ I      , Z = −      Ω η θ 0      . (B.5) Then, these matric es satisfy the fol lowing r elation: ∂ n +1 W + ( ∂ n Z ) W = [ W , A [0] n +1 ] . (B.6) Pr o of. By using the deriv ative form ula (B.2) of quasi-determinant and the recursiv e relation (B.3), we hav e ∂ n +1 W =      Ω Ξ − 1 η ∂ n +1 θ 0      +      Ω Ξ − 1 ∂ n +1 η θ 0      +      Ω ∂ n η θ 0           Ω Ξ − 1 η θ 0      +      Ω ∂ n Ω θ 0           Ω Ξ − 1 η Λ 0      . F or the first t w o terms on the right-hand side, substituting (5.5) in to them yields      Ω Ξ − 1 η ∂ n +1 θ 0      =      Ω Ξ − 1 η ( ∂ n θ )Λ 0      − A [0] n +1      Ω Ξ − 1 η θ 0      , and      Ω Ξ − 1 ∂ n +1 η θ 0      =      Ω ∂ n η θ 0      + A [0] n +1      Ω Ξ − 1 η θ 0      . 23 Th us, ∂ n +1 W can be rewritten as ∂ n +1 W =      Ω Ξ − 1 η ( ∂ n θ )Λ 0      +      Ω ∂ n Ω θ 0           Ω Ξ − 1 η Λ 0      +      Ω ∂ n η θ 0      W + [ W, A [0] n +1 ] . (B.7) On the other hand, the direct expansion of − ( ∂ n U ) V indicates − ( ∂ n Z ) W =      Ω ∂ n η θ 0      W +      Ω η ∂ n θ 0           Ω Ξ − 1 η θ I      +      Ω ∂ n Ω θ 0           Ω η I 0           Ω Ξ − 1 η θ I      . Then, noting that the following relation holds for an arbitrary matrix C with a suitable size,      Ω η C 0           Ω Ξ − 1 η θ I      = − C Ω − 1 η + C Ω − 1 η θ Ω − 1 Ξ − 1 η = − C Ω − 1 η + C (Ω − 1 Ξ − ΛΩ − 1 )Ξ − 1 η =      Ω Ξ − 1 η C Λ 0      , th us we ha ve − ( ∂ n Z ) W =      Ω ∂ n η θ 0      W +      Ω Ξ − 1 η ( ∂ n θ )Λ 0      +      Ω ∂ n Ω θ 0           Ω Ξ − 1 η Λ 0      . (B.8) Com bining (B.7) and (B.8), we are led to (B.5). Finally , based on (B.4) and (5.4), through a direct calculation w e hav e ∂ n +1 J + ( ∂ n K ) J = ( ∂ n +1 W ) J [0] − W A [0] n +1 J [0] + ( ∂ n Z ) W J [0] + A [0] n +1 W J [0] = ( ∂ n +1 W + ( ∂ n Z ) W − [ W, A [0] n +1 ]) J [0] , whic h v anishes due to (B.6). Thu s, J and K satisfy (2.5) and we complete the pro of. References [1] L.J. Mason, N.M.J. W o o dhouse, Integrabilit y , Self-Duality , and Twistor Theory , Oxford Universit y Press, Oxford, New Y ork, (1996). [2] S.K. Donaldson, Anti-self-dual Y ang-Mills connections o v er complex algebraic surfaces and stable v ector bundles, Pro c. Lond. Math. So c., 50 (1985) 1-26. [3] R.S. W ard, On self-dual gauge fields, Phys. Lett. A, 61 (1977) 81-82. [4] C.H. Gu, On classical Y ang-Mills fields, Phys. Rep., 80 (1981) 257-331. [5] M.F. A tiyah, R.S. W ard, Instan tons and algebraic geometry , Commun. Math. Phys., 55 (1977) 117-124. [6] F. Wilczek, Geometry and interactions of instantons. In D.R. Stump and D.H. W eingarten, editors, Quark Confinement and Field Theory , John Wiley and Sons, New Y ork, (1977) 211-219. [7] M.F. Atiy ah, N.J. Hitchin, V.G. Drinfield, Y.I. Manin, Construction of instantons, Ph ys. Lett. A, 65 (1978) 185-187. [8] E.F. Corrigan, D.B. F airlie, R.G. Y ates, P . Go ddard, The construction of self-dual solutions to SU(2) gauge theory , Commun. Math. Phy s., 58 (1978) 223-240. 24 [9] M. Jimbo, M.D. Krusk al, T. Miwa, Painlev ´ e test for the self-dual Y ang-Mills equation, Phys. Lett. A, 92 (1982) 59-60. [10] R.S. W ard, The P ainlev´ e prop ert y for the self-dual gauge-field equations, Phys. Lett. A, 102 (1984) 279-282. [11] A.A. Belavin, V.E. Zakharov, Y ang-Mills equations as inv erse scattering problem, Phys. Lett. B, 73 (1978) 53-57. [12] K. Ueno, Y. Nak amura, T ransformation theory for an ti-self-dual equations and the Riemann-Hilbert problem, Phys. Lett. B, 109 (1982) 273-278. [13] K. T ak asaki, A new approach to the self-dual Y ang-Mills equations, Comm un. Math. Phys., 94 (1984) 35-59. [14] N. Sasa, Y. Ohta, J. Matsukidaira, Bilinear form approach to the self-dual Y ang-Mills equations and integrable systems in (2+1)-dimension, J. Ph ys. So c. Japan, 67 (1998) 83-86. [15] Y. Ohta, J.J.C. Nimmo, C.R. Gilson, A bilinear approac h to a Pfaffian self-dual Y ang-Mills equation, Glasgo w Math. J., 43A (2001) 99-108. [16] Y. Ohta, A recen t topic on rogue wa v e, RIMS Kˆ okyˆ uroku Bessatsu, B96 (2024) 055-063. [17] J.J.C. Nimmo, C.R. Gilson, Y. Ohta, Applications of Darb oux transformations to the self-dual Y ang-Mills equations, Theor. Math. Ph ys., 122 (2000) 239-246. [18] M. Hamanak a, S.C. Huang, New soliton solutions of an ti-self-dual Y ang-Mills equations, J. High. Energ. Phys., 2020 (2020) 101 (18pp). [19] M. Hamanak a, S.C. Huang, Multi-soliton dynamics of anti-self-dual gauge fields, J. High. Energ. Ph ys., 2022 (2022) 039 (19pp). [20] S.C. Huang, On Soliton Solutions of the An ti-Self-Dual Y ang-Mills Equations from the Perspective of Integrable Systems (PhD thesis), Nagoy a Universit y , (2021). [21] S.S. Li, D.J. Zhang, Direct linearization of the SU(2) anti-self-dual Y ang-Mills equation in v arious spaces, J. Geom. Phys., 207 (2025) 105351 (19pp). [22] F. M ¨ uller-Hoissen, F r¨ olic her-Nijenh uis geometry and integrable matrix PDE systems, Nonlinearity , 38 (2025) 095005 (21pp). [23] R.S. W ard, In tegrable and solv able systems, and relations among them, Philos. T rans. R. Soc. Lond. A, 315 (1985) 451-457. [24] L.J. Mason, G.A.J. Sparling, Nonlinear Schr¨ odinger and Kortew eg-de V ries are reductions of self- dual Y ang-Mills, Phys. Lett. B, 137 (1989) 29-33. [25] I. B ak as, D.A. Depireux, The origin of gauge symmetries in in tegrable systems of the KdV type, In t. J. Mo d. Phys. A, 7 (1992) 1767-1792. [26] J. Schiff, Self-dual Y ang-Mills and the Hamiltonian structures of in tegrable systems, arXiv:hep.th/ 9211070, (1992). [27] M.J. Ablowitz, S. Chakrav art y , L.A. T akhta jan, A self-dual Y ang-Mills hierarch y and its reductions to integrable systems in 1+1 and 2+1 dimensions, Commun. Math. Phys., 158 (1993) 289-314. [28] M.L. Ge, L. W ang, Y.S. W u, Canonical reduction of self-dual Y ang-Mills theory to sine-Gordon and Liouville theories, Phys. Lett. B, 335 (1994) 136-142. [29] S. Chakra v arty , S.L. Kent, E.T. Newman, Some reductions of the self-dual Y ang-Mills equations to in tegrable systems in 2+1 dimensions, J. Math. Phys., 36 (1995) 763-772. [30] J. Schiff, In tegrability of Chern-Simons-Higgs v ortex equations and a reduction of the self-dual Y ang-Mills equations to three dimensions, NA TO ASI Ser. B, 278 (1992) 393-405. 25 [31] R.S. W ard, In tegrability of c hiral equations with torsion term, Nonlinearity , 1 (1988) 671-679. [32] R.S. W ard, Soliton solutions in an integrable c hiral model in 2+1 dimensions, J. Math. Ph ys., 29 (1988) 386-389. [33] S.V. Manako v, V.E. Zakharov, Three-dimensional mo del of relativistic-in v arian t field theory , in te- grable by the inv erse scattering transform, Lett. Math. Ph ys., 5 (1981) 247-253. [34] S. Kak ei, T. Ikeda, K. T ak asaki, Hierarch y of (2+1)-dimensional nonlinear Sc hr¨ odinger equation, self-dual Y ang-Mills equation, and toroidal Lie algebras, Ann. Henri Poincar ´ e, 3 (2002) 817-845. [35] L.J. Mason, N.M.J. W o o dhouse, Self-dualit y and the Painlev ´ e transcendents. Nonlinearit y , 6 (1993) 569-581. [36] G.B. Benincasa, The Anti-Self-Dual Y ang-Mills Equations and Discrete Integrable Systems (PhD thesis), Universit y College London, (2018). [37] M. Hamanak a. Non-commutativ e W ard’s conjecture and integrable systems, Nucl. Ph ys. B, 741 (2006) 368-389. [38] M.J. Ablo witz, S. Chakra v art y , R.G. Halburd, In tegrable systems and reductions of the self-dual Y ang-Mills equations, J. Math. Ph ys., 44 (2003) 3147-3173. [39] S.S. Li, C.Z. Qu, X.X. Yi, D.J. Zhang, Cauch y matrix approach to the SU(2) self-dual Y ang-Mills equation, Stud. Appl. Math., 148 (2022) 1703-1721. [40] S.S. Li, C.Z. Qu, D.J. Zhang, Solutions to the SU( N ) self-dual Y ang-Mills equation, Physica D, 453 (2023) 133828 (17pp). [41] S.S. Li, M. Hamanak a, S.C. Huang, D.J. Zhang, Soliton resonances in four dimensional W ess- Zumino-Witten mo del, Phys. Rev. D, 111 (2025) 086023 (21pp). [42] S.S. Li, S.Z. Liu, D.J. Zhang, F rom the self-dual Y ang-Mills equation to the F ok as-Lenells equation, Stud. Appl. Math., 155 (2025) e70126 (21pp). [43] A.S. F ok as, On a class of physically imp ortan t integrable equations, Physica D, 87 (1995) 145-150. [44] J. Lenells, Exactly solv able mo del for nonlinear pulse propagation in optical fib ers, Stud. Appl. Math., 123 (2009) 215-232. [45] J. Lenells, A.S. F ok as. On a no vel integrable generalization of the nonlinear Sc h¨ odinger equation, Nonlinearit y , 22 (2009) 11-27. [46] Y. Nak am ura, T ransformation groups acting on a self-dual Y ang-Mills hierarch y , J. Math. Phys., 29 (1988) 244-248. [47] K. T ak asaki, Hierarc hy structure in integrable systems of gauge fields and underlying Lie algebras, Comm un. Math. Ph ys., 127 (1990) 225-238. [48] C.N. Y ang, Condition of self-duality for SU(2) gauge fields on Euclidean four-dimensional space, Ph ys. Rev. Lett., 38 (1977) 1377-1379. [49] K. P ohlmeyer, On the Lagrangian theory of an ti-self-dual fields in four-dimensional Euclidean space, Comm un. Math. Ph ys., 72 (1980) 37-47. [50] A. Park es, A cubic action for self-dual Y ang-Mills, Phys. Lett. B, 286 (1992) 265-270. [51] G. Chalmers, W. Siegel, Self-dual sector of QCD amplitudes, Phy s. Rev. D, 54 (1996) 7628-7633. [52] V.E. Zakharov, A.B. Shabat, Exact theory of t wo-dimensional self-fo cusing and one-dimensional self-mo dulation of wa ve in nonlinear media, Sov. Phys. JETP , 34 (1972) 62-69. [53] M.J. Ablo witz, D.J. Kaup, A.C. Newell, H. Segur, The inv erse scattering transform-F ourier analysis for nonlinear problems, Stud. Appl. Math., 53 (1974) 249-315. [54] D.Y. Chen, Introduction to Soliton Theory (in Chinese), Science Press, Beijing, (2006). 26 [55] V.S. Gerdjiko v, I. Iv anov, A quadratic p encil of general type and nonlinear evolution equations. I I. Hierarc hies of Hamiltonian structures, Bulg. J. Phys., 10 (1983) 130-143. [56] E.G. F an, Integrable evolution systems based on Gerdjiko v-Iv ano v equations, bi-Hamiltonian struc- ture, finite-dimensional integrable systems and N -fold Darb oux transformation, J. Math. Phys., 41 (2000) 7769-7782. [57] S. Kakei, T. Kikuchi, Affine lie group approach to a deriv ative nonlinear Schrodinger equation and its similarity reduction, Int. Math. Res. Not., 2004 (2004) 4181-4209. [58] R.S. Y e, Y. Zhang, A v ectorial Darboux transformation for the F ok as-Lenells system, Chaos Solitons F ractals, 169 (2023) 113233 (7pp). [59] F. M ¨ uller-Hoissen, R.S. Y e, Miura transformation in bidifferential calculus and a vectorial Darb oux transformation for the F ok as-Lenells equation, J. Phys. A: Math. Theor., 58 (2025) 295201 (27pp). [60] H.H. Chen, Y.C. Lee, C.S. Liu, Integrabilit y of nonlinear Hamiltonian systems by in verse scattering metho d, Phys. Scr., 20 (1979) 490. [61] D.J. Kaup, A.C. New ell, An exact solution for a deriv ativ e nonlinear Sc hr¨ odinger equation, J. Math. Ph ys., 19 (1978) 798-801. [62] M. W adati, K. Sogo, Gauge transformations in soliton theory , J. Ph ys. So c. Japan, 52 (1983) 394- 398. [63] A. Kundu, Landau-Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Sc hr¨ odinger-type equations, J. Math. Phys., 25 (1984) 3433-3438. [64] A. Kundu, Exact solutions to higher-order nonlinear equations through gauge transformation, Ph ys- ica D, 25 (1987) 399-406. [65] P .A. Clarkson, C.M. Cosgrov e, Painlev ´ e analysis of the non-linear Sc hr¨ odinger family of equations, J. Phys. A: Math. Gen., 20 (1987) 2003-2024. [66] M. Sato, Soliton equations as dynamical systems on infinite dimensional Grassmann manifold, RIMS Kˆ oky ˆ uroku, 439 (1981) 30-46. [67] E. Date, M. Jimbo, M. Kashiwara, T. Miw a, T ransformation groups for soliton equations – Euclidean Lie algebras and reduction of the KP hierarch y , Publ. Res. Inst. Math. Sci., 18 (1982) 1077-1110. [68] Y. Ohta, J. Satsuma, D. T ak ahashi, T. T okihiro, An elementary in tro duction to Sato theory , Prog. Theor. Phys. Suppl., 94 (1988) 210-241. [69] J. Matsukidaira, J. Satsuma, W. Strampp, Conserved quan tities and symmetries of KP hierarch y , J. Math. Phys., 31 (1990) 1426-1434. [70] W. Oevel, Darboux theorems and W ronskian formulas for integrable systems: I. Constrained KP flo ws, Physica A, 195 (1993) 533-576. [71] B.A. Konopelchenk o, J. Sidorenk o, W. Strampp, (1+1)-dimensional in tegrable systems as symmetry constrain ts of (2+1)-dimensional systems, Phys. Lett. A, 157 (1991) 17-21. [72] Y. Cheng, Y.S. Li, The constraint of the Kadomtsev-P etviash vili equation and its sp ecial solutions, Ph ys. Lett. A, 157 (1991) 22-26. [73] B. Xu, A unified approach to recursion op erators if the reduced (1+1)-dimensional Hamiltonian systems, Inv erse Probl., 8 (1992) L13-L20. [74] Y. Cheng, Y.S. Li, Constrain ts of the 2+1 dimensional in tegrable soliton systems, J. Phys. A: Math. Gen., 25 (1992) 419-431. [75] A. Dimakis, F. M ¨ uller-Hoissen, F rom AKNS to deriv ativ e NLS hierarchies via deformations of asso ciativ e pro ducts, J. Phys. A: Math. Gen., 39 (2006) 14015-14033. 27 [76] D.Y. Chen, k -constraint for the mo dified Kadomtsev-P etviash vili system, J. Math. Ph ys., 43 (2002) 1956-1965. [77] S.V. Manako v, On the theory of tw o-dimensional stationary self-fo cusing of electromagnetic wa v es, So v. Phys. JETP , 38 (1974) 248-253. [78] J. Ieda, T. Miyak aw a, M. W adati, Exact analysis of soliton dynamics in spinor Bose-Einstein con- densates, Phys. Rev. Lett., 93 (2004) 194102 (4pp). [79] Y. Kaw aguchi, M. Ueda, Spinor Bose-Einstein condensates, Phys. Rep., 520 (2012) 253-381. [80] S.S. Li, D.J. Zhang, Cauc hy matrix structure and solutions of the spin-1 Gross-Pitaevskii equations, Comm un. Nonlinear Sci. Numer. Simulat., 129 (2024) 107705 (14pp). [81] P .J. Olv er, V.V. Sokolo v, Non-abelian integrable systems of the deriv ative nonlinear Sc hr¨ odinger t yp e, In verse Probl., 14 (1998) L5-L8. [82] A.P . F ordy , Deriv ativ e nonlinear Schr¨ odinger equations and Hermitian symmetric spaces, J. Phys. A: Math. Gen., 17 (1984) 1235-1245. [83] M. Hisak ado, M. W adati, Gauge transformation among generalised nonlinear Schr¨ odinger equations, J. Phys. So c. Japan, 63 (1994) 3962-3966. [84] T. Tsuc hida, M. W adati, New integrable systems of deriv ativ e nonlinear Schr¨ odinger equations with m ultiple comp onen ts, Phys. Lett. A, 257 (1999) 53-64. [85] T. Tsuc hida, M. W adati, Complete in tegrability of deriv ativ e nonlinear Schr¨ odinger-type equations, In verse Probl., 15 (1999) 1363-1373. [86] R. Hirota, The Direct Metho d in Soliton Theory , Cambridge Universit y Press, (2004). [87] S. Kak ei, N. Sasa, J. Satsuma, Bilinearization of a generalized deriv ativ e nonlinear Schr¨ odinger equation, J. Phys. So c. Japan, 64 (1995) 1519-1523. [88] S.Z. Liu, J. W ang, D.J. Zhang, The F ok as-Lenells equations: Bilinear approach, Stud. Appl. Math., 148 (2022) 651-688. [89] A. Nak amura, H.H. Chen, Multi-soliton solutions of a deriv ative nonlinear Schr¨ odinger equation, J. Ph ys. So c. Japan. 49 (1980) 813-816. [90] J. W ang, H. W u, Rational solutions with zero background and algebraic solitons of three deriv ative nonlinear Schr¨ odinger equations: bilinear approac h, Nonlinear Dyn., 109 (2022) 3101-3111. [91] I. Gel’fand, V. Retakh, Determinan ts of matrices o v er noncommutativ e rings, F unct. Anal. Appl., 25 (1991) 91-102. [92] I. Gel’fand, S. Gel’fand, V. Retakh, R. Wilson, Quasideterminants, Adv. Math., 193 (2005) 56-141. [93] J. Sylvester, Sur l’equation en matrices px = xq , C. R. Acad. Sci. P aris, 99 (1884) 67-76, 115, 116. [94] C.R. Gilson, J.J.C. Nimmo, On a direct approach to quasideterminant solutions of a noncommuta- tiv e KP equation, J. Phys. A: Math. Theor., 40 (2007) 3839-3850. 28