Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 71, pp. 1–32. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu INTEGRABLE NONLINEAR PERTURBED HIERARCHIES OF NLS-MKDV EQUATION AND SOLITON SOLUTIONS QIULAN ZHAO, HONGBIAO CHENG, XINYUE LI, CHUANZHONG LI Abstract. We propose three spectral problems for NLS-mKdV equation by combining three integrable coupling ways. Then we obtain three nonlinear perturbation terms to derive three integrable nonlinear perturbed hierarchies of the NLS-mKdV equation. We proved the Lax integrability of the integrable nonlinear perturbed hierarchies. On the basis of a special orthogonal group, we prove the Liouville integrability of a third-order integrable nonlinear perturbed hierarchy of NLS-mKdV equation by deriving its bi-Hamiltonian structures. We build three Darboux matrices for constructing the Darboux transforma- tions of the first two equations. As applications of the Darboux transformation, we present explicit solutions of these equations, three-dimensional plots, and density profiles the evolution of solitary waves. 1. Introduction The nonlinear Schrödinger (NLS) equation has been derived in fields such as plasma physics, deep water waves, and nonlinear fiber optics; see [15, 20]. The modified Korteweg-de Vries (mKdV) equation appears in the description of van Alfvén waves in collisionless plasma, cosmic plasma, water waves, and so on; see [8, 25]. Both mKdV equation and the NLS equation are well-known for their physical and mathematical significance in nonlinear evolution models. The study of NLS-mKdV hierarchy is a significant topic in soliton theory. We consider the effect of perturbations so that their applicability can be extended to higher order nonlinearity or to larger amplitude waves. During the past few decades, there has been an increasing interest in the study of NLS-mKdV hierarchy, which was proposed as subalgebras of the loop algebra Ã1 in [6], qt = βrxx + 4βr(r2 − q2), rt = βqxx + 4βq(r2 − q2). (1.1) A system of integrable coupling system is a larger system include the original in- tegrable system as its sub-system; see [13]. A few ways to construct integrable coupling systems includes perturbations [11], creating new loop algebras [12], and enlarging spectral problem [22]. Integrable coupling makes integrable system more abundant and complex. Based on integrable coupling, the multi-component in- tegrable couplings of the NLS-mKdV hierarchy was proposed in [27]. The super 2020 Mathematics Subject Classification. 35Q51, 37K10. Key words and phrases. Integrable perturbed hierarchies; nonlinear perturbation terms; Darboux transformation; soliton solutions. ©2022. This work is licensed under a CC BY 4.0 license. Submitted April 29, 2022. Published October 13, 2022. 1 2 Q. ZHAO, H. CHENG, X. LI, C. LI EJDE-2022/71 Hamiltonian structure of a super NLS-mKdV hierarchy is obtained by using su- per trace identity [26]. To further study NLS-mKdV hierarchy, researchers con- struct the completion of the NLS-mKdV integrable coupling systems and binary nonlinearization [18, 24]. There are many methods to obtain explicit solutions of integrable equations, for instance, Darboux transformation method [4, 14, 23], inverse scattering transformation [1, 2], Hirota method [7, 21], Bäcklund transfor- mation [16, 9], and so on. Darboux transformation is an efficient method for solving nonlinear partial differential equations. The Darboux transformation studies the explicit solution of an integrable system from a seed solution. We choose differ- ent seed solutions and analyze the relations among them. In 2019, the generalized super-NLS-mKdV equation was solved with Darboux transformation in [5]. Then they gave analytic solutions by using symbolic computations, and plot their graphs. It is important to derive a soliton hierarchy from its corresponding spectral problem. In 2009, Dong et al [3] studied the spectral problem Φx = UΦ, U = [ λ p+ q −p+ q −λ ] . (1.2) Starting from this equation, they constructed integrable couplings of NLS-mKdV hi- erarchy, and established their Hamiltonian structures and Super-Hamiltonian struc- tures. We enlarge the spectral problem (1.2) as follows Φx = UΦ, U = [ U U0 0 U ] =  λ p+ q λ r + s −p+ q −λ −r + s −λ 0 0 λ p+ q 0 0 −p+ q −λ  . (1.3) By using the perturbation technique, we would like to generalize the spectral prob- lem (1.3) Φx = U1Φ, U1 =  λ+ h p+ q λ r + s −p+ q −λ− h −r + s −λ 0 0 λ+ h p+ q 0 0 −p+ q −λ− h  , (1.4) where h = ε(qs− pr), ε is an arbitrary constant. This way of adding the nonlinear term h is a “completion process of integrable couplings” [17]. We construct a fourth- order spectral problem (1.4) by enlarging spectral problem and adding a perturbed term. In 2013, Ma [10] considered the spectral problem Φx = ÛΦ, Û =  0 q λ −q 0 −p −λ p 0  , (1.5) established a soliton hierarchy from zero curvature equation associated with so(3,R) and their Hamiltonian structures. Motivated by the spectral problem (1.5) , we construct the spectral problem Φx = U2Φ, U2 =  0 −p+ q λ+ h p− q 0 −p− q −λ− h p+ q 0  , (1.6) where h = ε(p2 + q2). We construct the third-order spectral problem (1.6) by enlarging the Lie algebra and adding a perturbed term. By enlarging (1.6), we EJDE-2022/71 INTEGRABLE NONLINEAR PERTURBED HIERARCHIES 3 obtain the spectral problem Φx = U3Φ with U3 =  0 −p+ q λ+ h 0 −r + s λ p− q 0 −p− q r − s 0 −r − s −λ− h p+ q 0 −λ r + s 0 0 0 0 0 −p+ q λ+ h 0 0 0 p− q 0 −p− q 0 0 0 −λ− h p+ q 0  , (1.7) where h = ε(qs+pr). In this article, starting from (1.4), (1.6) and (1.7) we propose three nonlinear integrable perturbed hierarchies of the NLS-mKdV equation. Our aim is to study explicit solutions of three nonlinear integrable perturbed hierarchies of NLS-mKdV equation by Darboux transformation method. Actually, it is difficult to select a Darboux matrix in integrable system and it is more difficult to obtain the explicit solution by Darboux transformation method in perturbed system than in unperturbed system. Moreover, three generalized forms of the NLS-mKdV equation are discussed together facilitates classification and comparisons. In addition, three- dimensional plots and density profiles of explicit solutions are visually presented to show their properties. This article is outlined as follows. In Section 2, we obtain a fourth-order in- tegrable perturbed hierarchy of NLS-mKdV equation by zero curvature equation, and explicit solutions by using Darboux transformation method. Also we present plots of these explicit solutions. In Section 3, we prove the Liouville integrability of a third-order integrable perturbed hierarchy of NLS-mKdV equation. We study explicit solutions of the first two nontrivial equations by using Darboux transfor- mation, and plot their graphs. In Section 4, we obtain a sixth-order integrable perturbed hierarchy of NLS-mKdV equation on the basis of a sixth-order spectral problem. Also we obtain explicit solutions by using Darboux transformations, and present three-dimensional plots, and density profiles of explicit solutions. In Section 5, we give some conclusions. 2. Fourth-order nonlinear integrable perturbed hierarchy of NLS-mKdV equation and their explicit solutions 2.1. Fourth-order nonlinear integrable perturbed hierarchy of NLS-mKdV equation. To obtain a fourth-order nonlinear integrable perturbed hierarchy of NLS-mKdV equation, we consider the stationary zero curvature equation [19] as- sociated with spectral problem (1.4), V1,x = [U1, V1] = U1V1 − V1U1, (2.1) with Φt = V1Φ, V1 =  a b+ c d f + g b− c −a f − g −d 0 0 a b+ c 0 0 b− c −a  . (2.2) 4 Q. ZHAO, H. CHENG, X. LI, C. LI EJDE-2022/71 Obviously, the above equation becomes ax = 2pb− 2qc, bx = 2λc− 2pa+ 2hc, cx = 2λb− 2qa+ 2hb, dx = 2pf − 2qg + 2rb− 2sc, fx = 2λg + 2λc+ 2hg − 2pd− 2ra, gx = 2λf + 2λb+ 2hf − 2qd− 2sa. (2.3) By assuming the Laurent series expansions V1 = ∞∑ i=0  ai bi + ci di fi + gi bi − ci −ai fi − gi −di 0 0 ai bi + ci 0 0 bi − ci −ai λ−i. (2.4) Substituting (2.4) into (2.3), and comparing the powers of coefficients of λ, we arrive at am+1,x = 2pbm+1 − 2qcm+1, bm+1 = 1 2 cm,x + qam − hbm, cm+1 = 1 2 bm,x + pam − hcm, dm+1,x = 2pfm+1 − 2qgm+1 + 2rbm+1 − 2scm+1, fm+1 = 1 2 gm,x − 1 2 cm,x + (s− p)am + hcm + qdm − hfm, gm+1 = 1 2 fm,x − 1 2 bm,x + (r − q)am + hbm + pdm − hgm, (2.5) and a0x = 0, b0 = 0, c0 = 0, d0x = 0, f0 = 0, g0 = 0. We take the initial values a0 = α and d0 = β, where α, β are arbitrary constants. The values of first few terms are calculated as follows a1 = 0, b1 = αq, c1 = αp, d1 = 0, f1 = α(s− q) + βq, g1 = α(r − p) + βp, a2 = 1 2 α(p2 − q2), b2 = 1 2 αpx − αhq, c2 = 1 2 αqx − αhp, d2 = αpr − αqs+ α(q2 − p2) + 1 2 β(p2 − q2), f2 = 1 2 αrx − αpx + 1 2 βpx + 2αhq − αhs− βhq, g2 = 1 2 αsx − αqx + 1 2 βqx + 2αhp− αhr − βhp, b3 = 1 4 αqxx− αhpx + αh2q + 1 2 αq(p2 − q2 − 1 2 αphx), c3 = 1 4 αpxx− αhqx + αh2p+ 1 2 αp(p2 − q2 − 1 2 αqhx), a3 = 1 2 α(pqx − qpx) + αh(q2 − p2), EJDE-2022/71 INTEGRABLE NONLINEAR PERTURBED HIERARCHIES 5 f3 = 1 4 αsxx− 3 4 αqxx+ 1 4 βqxx+ 3 2 αhxp+ 3αhpx − 1 2 αhxr − 1 2 αhrx − 1 2 βhxp − βhpx − 1 2 αhrx − 3αh2q + αh2s+ βh2q + αpqr − αq2s+ 3 2 αq(q2 − p2) + 1 2 βq(p2 − q2) + 1 2 αs(p2 − q2), g3 = 1 4 αrxx− 3 4 αpxx+ 1 4 βpxx+ 3 2 αhxq + 3αhqx − 1 2 αhxs− 1 2 αhsx − 1 2 βhxq − βhqx − 1 2 αhsx − 3αh2p+ αh2r + βh2p+ αp2r − αpqr + 3 2 αp(q2 − p2) + 1 2 βp(p2 − q2) + 1 2 αr(p2 − q2), d3 = 1 2 α(psx − spx) + 3 2 α(qpx − qpx) + 1 2 β(pqx − qpx) + 1 2 α(rqx − qrx) + 3αh(p2 − q2) + βh(q2 − p2) + 2h2, . . . Now, taking V1 = m∑ i=0  ai bi + ci di fi + gi bi − ci −ai fi − gi −di 0 0 ai bi + ci 0 0 bi − ci −ai λm−i+  δm 0 0 0 0 −δm 0 0 0 0 δm 0 0 0 0 −δm  . Then the corresponding zero curvature equation U1,t − V1,x + [U1, V1] = 0 (2.6) gives pt = 2bm+1 + 2qδm, qt = 2cm+1 + 2pδm, rt = 2bm+1 + 2fm+1 + 2sδm, st = 2cm+1 + 2gm+1 + 2rδm, ht = δmx. (2.7) From this equation, we can obtain δmx = ht = ε(qts+ qst − ptr − prt) = ε[(2cm+1 + 2pδm)s+ q(2cm+1 + 2gm+1 + 2rδm) − (2bm+1 + 2qδm)r − p(2bm+1 + 2fm+1 + 2sδm)] = ε(2cm+1s+ 2gm+1q − 2bm+1r − 2fm+1p+ 2cm+1q − 2bm+1p) = −ε(am+1,x + dm+1,x). Thus we introduce δm = −ε(am+1 + dm+1), (2.8) 6 Q. ZHAO, H. CHENG, X. LI, C. LI EJDE-2022/71 and then we have generated a fourth-order nonlinear integrable perturbed hierarchy of NLS-mKdV equation pt = 2bm+1 − 2εq(am+1 + em+1), qt = 2cm+1 − 2εp(am+1 + em+1), rt = 2bm+1 + 2fm+1 − 2εs(am+1 + em+1), st = 2cm+1 + 2gm+1 − 2εr(am+1 + em+1). (2.9) When m = 1, by setting ε = 1, (2.9) becomes pt = αpx − (2α− 2β)q(p2 − q2), qt = αqx − (2α− 2β)p(p2 − q2), rt = αrx − αpx + βpx + (2α− 2β)(p2s− qqr), st = αsx − αqx + βqx + (2α− 2β)(pqs− q2r). (2.10) When m = 2, (2.9) becomes pt = 1 2 αqxx − 2αhpx + 2αh2q + αh2q + αq(p2 − q2)− αphx − 4q(α(qpx − pqx) + 1 2 α(psx − spx) + 1 2 β(pqx − qpx) + 1 2 α(rqx − qrx) + 2αh(p2 − q2 + βh(q2 − p2) + 2h2)), qt = 1 2 αpxx − 2αhqx + 2αh2p+ αh2p+ αp(p2 − q2)− αqhx − 4p(α(qpx − pqx) + 1 2 α(psx − spx) + 1 2 β(pqx − qpx) + 1 2 α(rqx − qrx) + 2αh(p2 − q2 + βh(q2 − p2) + 2h2)), rt = 1 2 αsxx − 2αqxx + 1 2 βqxx + 2αphx + 4αhpx − αrhx − αhr − βphx − 2βhpx − αhrx − 4αh2q + 2αh2s+ 2βh2q + 2αpqr − 2αq2s+ 2αq(p2 − q2) + βq(p2 − q2) + αs(p2 − q2) − 4s(α(qpx − pqx) + 1 2 α(psx − spx) + 1 2 β(pqx − qpx) + 1 2 α(rqx − qrx) + 2αh(p2 − q2 + βh(q2 − p2) + 2h2)), st = 1 2 αrxx − 2αpxx + 1 2 βpxx + 2αqhx + 4αhqx − αshx − αhs − qhx − 2βhqx − αhsx − 4αh2p+ 2αh2r + 2βh2p+ 2αp2r − 2αpqs + 2αp(p2 − q2) + βp(p2 − q2) + αr(p2 − q2)− 4r(α(qpx − pqx) + 1 2 α(psx − spx) + 1 2 β(pqx − qpx) + frac12α(rqx − qrx) + 2αh(p2 − q2 + βh(q2 − p2) + 2h2)), (2.11) by setting ε = 1. Therefore, we obtain (2.9) which is a Lax integrable. In the next two subsections, we try to obtain explicit solutions of (2.10) and (2.11) by the Darboux transformation method. EJDE-2022/71 INTEGRABLE NONLINEAR PERTURBED HIERARCHIES 7 2.2. N-fold Darboux transformation of (2.10) and its explicit solutions. We will construct the Darboux transformation of (2.10). Firstly, we introduce the gauge transformation Φ̃ = T1Φ. (2.12) The original Lax pair under Darboux transformation are transformed into the new Lax pair Φ̃x = Ũ1(p̃, q̃, r̃, s̃, λ)φ̃, Φ̃t = Ṽ1(p̃, q̃, r̃, s̃, λ)φ̃, φ̃ =  φ̃1 φ̃2 φ̃3 φ̃4  , (2.13) where Ũ1 = (T1,x + T1U1)T−11 , Ṽ1 = (T1,t + T1V1)T−11 . (2.14) For this, we consider the Darboux matrix T1 =  A11 A12 A13 A14 A21 A22 A23 A24 0 0 A11 A12 0 0 A21 A22  , (2.15) where A11 = λN + N−1∑ i=0 A (i) 11λ i, A12 = N−1∑ i=0 A (i) 12λ i, A13 = N−1∑ i=0 A (i) 13λ i, A14 = N−1∑ i=0 A (i) 14λ i, A21 = N−1∑ i=0 A (i) 21λ i, A22 = λN + N−1∑ i=0 A (i) 22λ i, A23 = N−1∑ i=0 A (i) 23λ i, A24 = N−1∑ i=0 A (i) 24λ i . We define Φ =  ϕ1 ψ1 ϕ2 ψ2 ϕ3 ψ3 ϕ4 ψ4  . (2.16) From (2.15) and (2.16), we have Φ̃ = T1Φ =  A11 A12 A13 A14 A21 A22 A23 A24 0 0 A11 A12 0 0 A21 A22   ϕ1 ψ1 ϕ2 ψ2 ϕ3 ψ3 ϕ4 ψ4  =  A11ϕ1 +A12ϕ2 +A13ϕ3 +A14ϕ4 A11ψ1 +A12ψ2 +A13ψ3 +A14ψ4 A21ϕ1 +A22ϕ2 +A23ϕ3 +A24ϕ4 A21ψ1 +A22ψ2 +A23ψ3 +A24ψ4 A11ϕ3 +A12ϕ4 A11ψ3 +A12ψ4 A21ϕ3 +A22ϕ4 A21ψ3 +A22ψ4  . (2.17) So there exists γj (j = 1, 2) satisfying γ (1) j ϕ̃+ γ (2) j ψ̃ = 0. (2.18) 8 Q. ZHAO, H. CHENG, X. LI, C. LI EJDE-2022/71 Substituting (2.18) into (2.17), we obtain γ (1) j (A11ϕ1 +A12ϕ2 +A13ϕ3 +A14ϕ4) + γ (2) j (A11ψ1 +A12ψ2 +A13ψ3 +A14ψ4) = 0, γ (1) j (A21ϕ1 +A22ϕ2 +A23ϕ3 +A24ϕ4) + γ (2) j (A21ψ1 +A22ψ2 +A23ψ3 +A24ψ4) = 0, γ (1) j (A11ϕ3 +A12ϕ4) + γ (2) j (A11ψ3 +A12ψ4) = 0, γ (1) j (A21ϕ3 +A22ϕ4) + γ (2) j (A23ψ3 +A24ψ4) = 0. From the above equalities, we obtain A11 +A12ω (1) j +A13ω (2) j +A14ω (3) j = 0, A21 +A22ω (1) j +A23ω (2) j +A24ω (3) j = 0, A11ω (2) j +A12ω (3) j = 0, A21ω (2) j +A22ω (3) j = 0, (2.19) where ω (1) j = γ (1) j ϕ2 + γ (2) j ψ2 γ (1) j ϕ1 + γ (2) j ψ1 , ω (2) j = γ (1) j ϕ3 + γ (2) j ψ3 γ (1) j ϕ1 + γ (2) j ψ1 , ω (3) j = γ (1) j ϕ4 + γ (2) j ψ4 γ (1) j ϕ1 + γ (2) j ψ1 . Furthermore, the following equations are obtained by (2.19) N−1∑ i=0 A (i) 11λ i j + ω (1) j N−1∑ i=0 A (i) 12λ i j + ω (2) j N−1∑ i=0 A (i) 13λ i j + ω (3) j N−1∑ i=0 A (i) 14λ i j = −λNj , N−1∑ i=0 A (i) 21λ i j + ω (1) j N−1∑ i=0 A (i) 22λ i j + ω (2) j N−1∑ i=0 A (i) 23λ i j + ω (3) j N−1∑ i=0 A (i) 24λ i j = −ω(1) j λNj , ω (2) j N−1∑ i=0 A (i) 11λ i j + ω (3) j N−1∑ i=0 A (i) 12λ i j = −ω(2) j λNj , ω (2) j N−1∑ i=0 A (i) 21λ i j + ω (3) j N−1∑ i=0 A (i) 22λ i j = −ω(3) j λNj . Proposition 2.1. Matrix Ũ1 is of the same type as U1 defined by (1.4), i.e, Ũ1 can be written as Ũ1 =  λ+ q̃s̃− p̃r̃ p̃+ q̃ λ r̃ + s̃ −p̃+ q̃ −λ− q̃s̃− p̃r̃ −r̃ + s̃ −λ 0 0 λ+ q̃s̃+ p̃r̃ p̃+ q̃ 0 0 p̃+ q̃ −λ− q̃s̃+ p̃r̃  , (2.20) EJDE-2022/71 INTEGRABLE NONLINEAR PERTURBED HIERARCHIES 9 in which the transformation formulae between old and new potentials are defined by p̃ = p−A(N−1) 12 −A(N−1) 21 , q̃ = q −A(N−1) 21 +A (N−1) 21 , r̃ = r −A(N−1) 12 −A(N−1) 14 −A(N−1) 21 −A(N−1) 23 , s̃ = s−A(N−1) 12 −A(N−1) 14 +A (N−1) 21 +A (N−1) 23 . (2.21) Proof. Setting (T1,x + T1U1)T ∗1 =  Γ11 Γ12 Γ13 Γ14 Γ21 Γ22 Γ23 Γ24 0 0 Γ11 Γ12 0 0 Γ21 Γ22  , where T−11 = T ∗1 det(T1) , (2.22) we deduce (T1,x + T1U1)T ∗1 = (detT1)O(λ), (2.23) where O(λ) =  O (1) 11 λ+O (0) 11 O (0) 12 O (1) 13 λ O (0) 14 O (0) 21 O (1) 22 λ+O (0) 22 O (0) 23 O (1) 24 λ 0 0 O (1) 11 λ+O (0) 11 O (0) 12 0 0 O (0) 21 O (1) 22 λ+O (0) 22  . (2.24) Substituting (2.23) into (2.22), we obtain T1,x + T1U1 = O(λ)T1. (2.25) From this equation, by equating the coefficients of λN and λN+1, we find O (1) 11 = 1, O (0) 12 = p+ q − 2A (N−1) 12 = p̃+ q̃, O (0) 14 = r + s− 2A (N−1) 12 − 2A (N−1) 14 = r̃ + s̃, O (0) 21 = −p+ q + 2A (N−1) 21 = −p̃+ q̃, O (0) 23 = −r + s+ 2A (N−1) 21 + 2A (N−1) 23 = −r̃ + s̃, O (1) 24 = −1, O (1) 13 = 1, O (1) 22 = −1, O (0) 11 = qs− pr = q̃s̃− p̃r̃, O (0) 22 = −qs+ pr = −q̃s̃+ p̃r̃. We see that Ũ1 = O(λ). The proof is complete. � Proposition 2.2. The matrix Ṽ (1) 1 is of the same type as V (1) 1 defined by (2.2); therefore, Ṽ (1) 1 can be rewritten as Ṽ (1) 1 = αλ+ δ1 αp̃+ αq̃ βλ α(s̃− r̃ + p̃+ q̃) + β(p̃+ q̃) αq̃ − αp̃ −αλ− δ1 α(s̃− q̃ − r̃ + p̃) + β(q̃ − p̃) −βλ 0 0 αλ+ δ1 αp̃+ αq̃ 0 0 αq̃ − αp̃ −αλ− δ1  . 10 Q. ZHAO, H. CHENG, X. LI, C. LI EJDE-2022/71 Proof. Setting (T1,t + T1V (1) 1 )T ∗1 =  Ξ11 Ξ12 Ξ13 Ξ14 Ξ21 Ξ22 Ξ23 Ξ24 0 0 Ξ11 Ξ12 0 0 Ξ21 Ξ22  , where T−11 = T ∗1 det(T1) . (2.26) Now we define P =  P (1) 11 λ+ P (0) 11 P (0) 12 P (1) 13 λ P (0) 14 P (0) 21 P (1) 22 λ+ P (0) 22 P (0) 23 P (1) 24 λ 0 0 P (1) 11 λ+ P11(0) P (0) 12 0 0 P (0) 21 P (1) 22 λ+ P (0) 22  . (2.27) Noting that (T1,t + T1V1)T ∗1 = (detT1)P (λ), we have T1,t + T1V (1) 1 = P (λ)T1. (2.28) By comparing the coefficients of λ in (2.28), we obtain P (1) 11 = α, P (1) 13 = β, P (1) 22 = −α, P (1) 24 = −β, P (0) 11 = δ = δ̃, P (0) 22 = −δ = −δ̃, P (0) 14 = α(s+ r − p− q) + β(p+ q)− 2βA (N−1) 12 − 2αA (N−1) 14 = α(s̃+ r̃ − p̃− q̃) + β(p̃+ q̃), P (0) 23 = α(s− q − r + p) + β(q − p) + 2αA (N−1) 23 + 2βA (N−1) 21 = α(s̃− q̃ − r̃ + p̃) + β(q̃ + q̃), P (0) 21 = α(q − p) + 2αA (N−1) 21 = α(q̃ − p̃), P (0) 12 = α(q + p) + 2αA (N−1) 12 = α(q̃ + p̃). The proof is complete. � To obtain the explicit solutions of (2.10), we choose the seed solution p = q = 0, r = s = 1. The spectral problems become Φx = U1Φ, U1 =  λ 0 λ 2 0 −λ 0 −λ 0 0 λ 0 0 0 0 −λ  , (2.29) and Φt = V (1) 1 Φ, V (1) 1 =  αλ 0 βλ 2α 0 −αλ 0 −βλ 0 0 αλ 0 0 0 0 −αλ  . (2.30) EJDE-2022/71 INTEGRABLE NONLINEAR PERTURBED HIERARCHIES 11 Solving these spectral problems, we obtain ϕ =  ϕ1 ϕ2 ϕ3 ϕ4  =  − 1 λe −λx−λαt −((βt+ x)λ− 1)e−αλt−λx 0 e−λx−αλt  , ψ =  ψ1 ψ2 ψ3 ψ4  =  ((βt+ x)λ+ 1)eλx+αλt e−λx−λαt eαλt+λx 0  . (2.31) In particular, when N = 1, T1 =  λ+A (0) 11 A (0) 12 A (0) 13 A (0) 14 A (0) 21 λ+A (0) 22 A (0) 23 A (0) 24 0 0 λ+A (0) 11 A (0) 12 0 0 A (0) 21 λ+A (0) 22  . (2.32) If we set ω (1) j = ϕ2 + γjψ2 ϕ1 + γjψ1 , ω (2) j = ϕ3 + γjψ3 ϕ1 + γjψ1 , ω (3) j = ϕ4 + γjψ4 ϕ1 + γjψ1 , (2.33) from Φ̃ = T1Φ, we obtain A (0) 11 +A (0) 12 ω (1) j +A (0) 13 ω (2) j +A (0) 14 ω (3) j = −λj , A (0) 21 +A (0) 22 ω (1) j +A (0) 23 ω (2) j +A (0) 24 ω (3) j = −λjω(1) j , A (0) 11 ω (2) j +A (0) 12 ω (3) j = −λjω(2) j , A (0) 21 ω (2) j +A (0) 22 ω (3) j = −λjω(3) j . (2.34) The solution to this equations is A (0) 12 = ∆A (0) 12 ∆′ , A (0) 14 = ∆A (0) 14 ∆ , A (0) 21 = ∆A (0) 21 ∆ , A (0) 23 = ∆A (0) 23 ∆ , (2.35) here, ∆ =  1 ω (1) 1 ω (2) 1 ω (3) 1 1 ω (1) 2 ω (2) 2 ω (3) 2 1 ω (1) 3 ω (2) 3 ω (3) 3 1 ω (1) 4 ω (2) 4 ω (3) 4  , ∆A (0) 21 =  −λ1ω(1) 1 ω (1) 1 ω (2) 1 ω (3) 1 −λ2ω(1) 2 ω (1) 2 ω (2) 2 ω (3) 2 −λ3ω(1) 3 ω (1) 3 ω (2) 3 ω (3) 3 −λ4ω(1) 4 ω (1) 4 ω (2) 4 ω (3) 4  , ∆A (0) 14 =  1 ω (1) 1 ω (2) 1 −λ1 1 ω (1) 2 ω (2) 2 −λ2 1 ω (1) 3 ω (2) 3 −λ3 1 ω (1) 4 ω (2) 4 −λ4  , ∆A (0) 23 =  1 ω (1) 1 −λ1ω(1) 1 ω (3) 1 1 ω (1) 2 −λ2ω(1) 2 ω (3) 2 1 ω (1) 3 −λ3ω(1) 3 ω (3) 3 1 ω (1) 4 −λ4ω(1) 4 ω (3) 4  , ∆′ = [ ω (2) 1 ω (3) 1 ω (2) 2 ω (3) 2 ] , ∆A (0) 12 = [ ω (2) 1 −λ1ω(2) 1 ω (2) 2 −λ2ω(2) 2 ] . (2.36) 12 Q. ZHAO, H. CHENG, X. LI, C. LI EJDE-2022/71 Accordingly we gain the explicit solutions of (2.10) as p̃ = −A(0) 12 −A (0) 21 , q̃ = −A(0) 21 +A (0) 21 , r̃ = 1−A(0) 12 −A (0) 14 −A (0) 21 −A (0) 23 , s̃ = 1−A(0) 12 −A (0) 14 +A (0) 21 +A (0) 23 . (2.37) Remark 2.3. By choosing suitable parameters, the left column displays the space- time distributions and the right column displays the density profiles different time for components p, q, r, s. Figure 1 shows that the bell soliton (a, b) and kink soliton (c, d) formed, respectively, by the one-soliton solutions and propagate along the negative direction of x-axis in the process of evolution. (a) p = q (b) p = q (c) r = s (d) r = s Figure 1. Soliton solutions of (2.10) and density plots with α = 1, β = 1, λ1 = −0.3, λ2 = 0.3, λ3 = −0.01, λ4 = 0.01, γ1 = −0.3, γ2 = 0.3, γ3 = −0.1, γ4 = 0.1. 2.3. Darboux transformation of (2.11) and its explicit solutions. We use the same Darboux matrix (2.15) to solve the second nonlinear equation (2.11). Proposition 2.4. The matrix Ṽ (2) 1 has the same form as V1 defined by (2.2) Ṽ (2) 1 =  V11 V12 V13 V14 V21 V22 V23 V24 0 0 V11 V12 0 0 V21 V22  , (2.38) where V11 = αλ2 + δ1λ+ 1 2 α(p̃2 − q̃2), V21 = αq̃λ− αp̃λ+ 1 2 αp̃x − αhq̃ − 1 2 αp̃x + αhp̃, V12 = αq̃λ+ αp̃λ+ 1 2 αp̃x − αhq̃ + 1 2 αp̃x − αhp̃, V22 = −αλ2 − δ1λ− 1 2 α(p̃2 − q̃2), V13 = βλ2 + αp̃r̃ − αq̃s̃+ α(q̃2 − p̃2) + 1 2 β(q̃2 − p̃2), V24 = −βλ2 − αp̃r̃ + αq̃s̃− α(q̃2 − p̃2)− 1 2 β(q̃2 − p̃2), EJDE-2022/71 INTEGRABLE NONLINEAR PERTURBED HIERARCHIES 13 V14 = α(s̃+ r̃ − q̃ − p̃)λ+ β(p̃+ q̃)λ+ 1 2 αr̃x − αp̃x + 1 2 βp̃x + 2αhq̃ − αhs̃− βhq̃ + 1 2 αs̃x − αq̃x + 1 2 βq̃x + 2αhp̃− αhr̃ − βhp̃, V23 = α(s̃− q̃ − r̃ + p̃)λ+ β(q̃ − p̃)λ+ 1 2 αr̃x − αp̃x + 1 2 βp̃px + 2αhq̃ − αhs̃− βhq̃ − 1 2 αs̃x + αq̃x − 1 2 βq̃x − 2αhp̃+ αhr̃ + βhp̃. Proof. Setting (T1,t + T1V (2) 1 )T ∗1 =  Θ11 Θ12 Θ13 Θ14 Θ21 Θ22 Θ23 Θ24 0 0 Θ11 Θ12 0 0 Θ21 Θ22  , (2.39) we have T−11 = T ∗1 / det(T1). Therefore, T1,t + T1V (2) 1 = Q(λ)T1, (2.40) where Q(λ) has columns 1 and 2: Q (2) 11 λ 2 +Q (1) 11 λ+Q (0) 11 Q (1) 12 λ+Q (0) 12 Q (1) 21 λ+Q (0) 21 Q (2) 22 λ 2 +Q (1) 22 λ+Q (0) 22 0 0 0 0  and columns 3 and 4: Q (2) 13 λ 2 +Q (1) 13 λ+Q (0) 13 Q (1) 14 λ+Q (0) 14 Q (1) 23 λ+Q (0) 23 Q (2) 24 λ 2 +Q (1) 24 λ+Q (0) 24 Q (2) 11 λ 2 +Q (1) 11 λ+Q (0) 11 Q (1) 12 λ+Q (0) 12 Q (1) 21 λ+Q (0) 21 Q (2) 22 λ 2 +Q (1) 22 λ+Q (0) 22  . Comparing the coefficients of λ in (2.40), we have Q (2) 11 = α, Q (2) 13 = β, Q (2) 22 = −α, Q (2) 24 = −β, Q (1) 11 = δ = δ̃, Q (1) 22 = −δ = −δ̃, Q (1) 13 = 0, Q (1) 14 = α(s+ r − p− q) + β(p+ q)− 2βA (N−1) 12 − 2αA (N−1) 14 = α(s̃+ r̃ − p̃− q̃) + β(p̃+ q̃), Q (1) 23 = α(s− q − r + p) + β(q − p) + 2αA (N−1) 23 + 2βA (N−1) 21 = α(s̃− q̃ − r̃ + p̃) + β(q̃ + q̃), Q (1) 21 = α(q − p) + 2αA (N−1) 21 = α(q̃ − p̃), Q (1) 12 = α(q + p) + 2αA (N−1) 12 = α(q̃ + p̃), Q (1) 24 = 0, Q (0) 11 = 1 2 α(p2 − q2) + α(q − p)A(N−1) 12 − α(p+ q)A (N−1) 21 + 2αA (N−1) 12 A (N−1) 21 = 1 2 α(p̃2 − q̃2), 14 Q. ZHAO, H. CHENG, X. LI, C. LI EJDE-2022/71 Q (0) 22 = −1 2 α(p2 − q2)− α(q − p)A(N−1) 12 + α(p+ q)A (N−1) 21 − 2αA (N−1) 12 A (N−1) 21 = 1 2 α(q̃2 − p̃2), Q (0) 12 = 1 2 α(px −A(N−1) 21,x −A(N−1) 21,x )− αh(q −A(N−1) 12 −A(N−1) 21 ) + 1 2 α(qx −A(N−1) 21,x +A (N−1) 21,x )− αh(p−A(N−1) 12 +A (N−1) 21 ) = 1 2 αp̃x − αhq̃ + 1 2 αq̃x − αhp̃, Q (0) 21 = 1 2 α(px −A(N−1) 21,x −A(N−1) 21,x )− αh(q −A(N−1) 12 −A(N−1) 21 ) − 1 2 α(qx −A(N−1) 21,x +A (N−1) 21,x ) + αh(p−A(N−1) 12 +A (N−1) 21 ) = 1 2 αp̃x − αhq̃ − 1 2 αq̃x + αhp̃. It is easy to see that Ṽ (2) 1 = Q(λ). The proof is complete. � To obtain the explicit solutions of (2.11), we choose the seed solution p = q = 0, r = s = 1. Then, the spectral problems become Φx = U1Φ, U1 =  λ 0 λ 2 0 −λ 0 −λ 0 0 λ 0 0 0 0 −λ  , (2.41) and Φt = V (2) 1 Φ, V (2) 1 =  αλ2 0 βλ2 2αλ 0 −αλ2 0 −βλ2 0 0 αλ2 0 0 0 0 −αλ2  . (2.42) Solving these spectral problems, we obtain ϕ =  ϕ1 ϕ2 ϕ3 ϕ4  =  e−λx−λ 2αt(tλ+ x)λ e−αλ 2t−λx 0 e−λx−αλ 2t  , ψ =  ψ1 ψ2 ψ3 ψ4  =  ((βtλ2 + x)λ)eλx+αλt e−λx−λ 2αt eαλ 2t+λx 0  . (2.43) When N = 1, the explicit solutions of (2.11) is p̃ = −A(0) 12 −A (0) 21 , q̃ = −A(0) 21 +A (0) 21 , r̃ = −A(0) 12 −A (0) 14 −A (0) 21 −A (0) 23 , s̃ = −A(0) 12 −A (0) 14 +A (0) 21 +A (0) 23 . (2.44) EJDE-2022/71 INTEGRABLE NONLINEAR PERTURBED HIERARCHIES 15 Remark 2.5. By choosing suitable parameters in solution (2.11), we provide four figures to analyze the transmission form of soliton solutions. In Figure 2(a, b), we can find that three sets of the parallel solitons, but they have different heights. Three single solitons keep at a certain interval which stably propagate and have no interaction. In Figure 2(c, d), solitons do not stably propagate. That is to say, the difference between Figure 2(a, b) and Figure 2(c, d) under the influence of the same parameter. (a) p = q (b) p = q (c) r = s (d) r = s Figure 2. Soliton solutions of (2.11) and density plots with α = 1, β = 1, λ1 = 0.05, λ2 = −0.05, λ3 = 0.1, λ4 = −0.1, γ1 = 0.2, γ2 = −0.2, γ3 = 2, γ4 = −2. 3. Third-order nonlinear integrable perturbed hierarchy of NLS-mKdV equation and their explicit solutions 3.1. Third-order nonlinear integrable perturbed hierarchy of NLS-mKdV equation. To obtain a third-order nonlinear integrable perturbed hierarchy of NLS-mKdV equation, we consider the auxiliary problem of the spectral problem (1.6) as follows Φt = V2Φ, V2 =  0 b− c a −b+ c 0 −b− c −a b+ c 0  . (3.1) Solving the stationary zero curvature equation (2.1), we obtain ax = 2pb− 2qc, bx = λc+ hc− pa, cx = −λb− hb+ qa. (3.2) By substituting expansions V2 = ∞∑ i=0  0 bi − ci ai −bi + ci 0 −bi − ci −ai bi + ci 0 λ−i (3.3) into (3.2), we obtain am+1,x = 2pbm+1 − 2qcm+1, bm+1 = −cm,x + qam − hbm, cm+1 = bm,x + pam − hcm, (3.4) 16 Q. ZHAO, H. CHENG, X. LI, C. LI EJDE-2022/71 and the initial conditions a0x = 2pb0− 2qc0, b0 = 0, c0 = 0. Now we choose a0 = α an arbitrary constant. Starting from the above initial values, we obtain a1 = 0, b1 = αq, c1 = αp, a2 = α(−p2 − q2), b2 = −px − αhq, c2 = αqx − αhp, . . . . Setting V2,m = m∑ i=0  0 bi − ci ai −bi + ci 0 −bi − ci −ai bi + ci 0 λm−i. We have V2,mx − [U, V2,m] =  0 bm+1 + cm+1 0 −bm+1 − cm+1 0 bm+1 − cm+1 0 −bm+1 + cm+1 0  . By considering Φt = V2Φ with V2 = V2,m + ∆2,m. We take the correction term ∆2,m =  0 0 δm 0 0 0 −δm 0 0  , where δm = −2ε(am+1 + dm+1). Then the corresponding zero curvature equation (2.6) give rise to the following a third-order nonlinear integrable perturbed hierarchy of NLS-mKdV equation pt = −bm+1 + 1 2 εqam+1, qt = cm+1 − 1 2 εpam+1. (3.5) When m = 1, setting ε = 1, we obtain pt = αpx + αhq − 1 2 αq(p2 + q2), qt = αqx − αhp+ 1 2 αp(p2 + q2). (3.6) When m = 2, setting ε = 1, we obtain pt = αqxx − αhxq − 2αhpx − αh2q + αq(p2 + q2) + αq((qpx − pqx) + 2αh(p2 + q2)), qt = −αpxx − αhxp− 2αhpx + αh2p− αp(p2 + q2) − αp((qpx − pqx) + 2αh(p2 + q2)). (3.7) We can construct bi-Hamiltonian structures for (3.5) by using the variational- trace identity δ δū ∫ 〈 V, ∂U(ū, λ) ∂λ 〉 dx = λ−γ ∂ ∂λ λγ 〈 V, ∂U(ū, λ) ∂ū 〉 , (3.8) where ū stands for (p, q)T , and 〈a, b〉 stands for the trace of the matrix a · b; here a · b stands for the matrix a times the matrix b. Computations yield ∂U ∂λ =  0 0 1 0 0 0 −1 0 0  , ∂U ∂p =  0 −1 2εp 1 0 −1 −2εp 1 0  , ∂U ∂q =  0 1 2εq −1 0 −1 −2εq 1 0  , EJDE-2022/71 INTEGRABLE NONLINEAR PERTURBED HIERARCHIES 17 and so we can obtain〈 V, ∂U ∂λ 〉 = −2a, 〈 V, ∂U ∂p 〉 = −4c− 4εpa, 〈 V, ∂U ∂q 〉 = −4b− 4εqa. (3.9) Now the corresponding trace identity (3.8) becomes δ δū ∫ −2a dx = λ−γ ∂ ∂λ λγ [ −4c− 4εpa −4b− 4εqa ] . (3.10) Balancing coefficients of each power of λ in the above equality, we have δ δū ∫ −2am+1 dx = (γ −m) [ −4cm − 4εpam −4bm − 4εqam ] . (3.11) The case of m = 1 tells γ = 0, and thus we have δ δū ∫ 2am+2 m+ 1 dx = [ −4cm+1 − 4εpam+1 −4bm+1 − 4εqam+1 ] . (3.12) Consequently, we obtain the following Hamiltonian structures for (3.5), ut = [ −4bm+1 + εqam+1 −4cm+1 − εpam+1 ] = J δHm δu , (3.13) with the Hamiltonian operator J = [ εq∂−1q 1 4 − εq∂ −1p − 1 4 − εp∂ −1q εp∂−1p ] , (3.14) and the Hamilton functionals Hm = ∫ − 2am+2 m+1 dx. Thus (3.5) has the following bi-Hamiltonian structures Ut = J δHm δu = JL δHm−1 δu , m ≥ 0. (3.15) Through a series of calculations, we obtain L = [ L11 L12 L21 L22 ] , (3.16) where L11 = −2p∂−1q − h− 2εp∂−1p∂ + 2εq∂−1q − 8ε∂p∂−1hq, L12 = ∂ + 2p∂−1p− 2εp∂−1q∂ − 2εq∂−1p+ 8ε∂p∂−1hp, L21 = −∂ − 2q∂−1q − 2εq∂−1p∂ − 2εp∂−1q + 8ε∂q∂−1hq, L22 = 2q∂−1p− h− 2εq∂−1q∂ + 2εp∂−1p− 8ε∂q∂−1hp. (3.17) So far, we are ready to see that (3.5) is intregrable in the sense of Liouville. 3.2. N-fold Darboux transformation of (3.6) and its explicit solution. In what follows, we search for a Darboux transformation of (3.6), which is the first equation in the hierarchy (3.5). The Darboux transformation is a special gauge transformation Φ̃ = T2Φ, (3.18) with T2 = A11 A12 A13 A21 A22 A23 A31 A32 A33  , (3.19) 18 Q. ZHAO, H. CHENG, X. LI, C. LI EJDE-2022/71 where A11 = A33 = 0, A12 = −A21 = N−1∑ i=0 A (i) 12λ i, A13 = −A31 = λN + N−1∑ i=0 A (i) 13λ i, A22 = λN + N−1∑ i=0 A (i) 22λ i, A23 = −A32 = λN + N−1∑ i=0 A (i) 23λ i. Then the Lax pair becomes Φ̃x = Ũ2Φ̃, Φ̃t = Ṽ2Φ̃, (3.20) and Ũ2, Ṽ2 satisfy T2,x + T2U2 = Ũ2T2, T2,t + T2V2 = Ṽ2T2. (3.21) Now we define Φ = ϕ1 ψ1 ϕ2 ψ2 ϕ3 ψ3  . (3.22) Using (3.22) we define the linear algebraic system N−1∑ i=0 A (i) 11λ i j + ω (1) j N−1∑ i=0 A (i) 12λ i j + ω (2) j N−1∑ i=0 A (i) 13λ i j = −ω(2) j λNj , − N−1∑ i=0 A (i) 12λ i j + ω (1) j N−1∑ i=0 A (i) 22λ i j − ω (2) j N−1∑ i=0 A (i) 32λ i j = −ω(1) j λNj , − N−1∑ i=0 A (i) 13λ i j + ω (1) j N−1∑ i=0 A (i) 32λ i j + ω (2) j N−1∑ i=0 A (i) 33λ i j = λNj , (3.23) with ω (1) j = γ (1) j ϕ2 + γ (2) j ψ2 γ (1) j ϕ1 + γ (2) j ψ1 , ω (2) j = γ (1) j ϕ3 + γ (2) j ψ3 γ (1) j ϕ1 + γ (2) j ψ1 . Proposition 3.1. Noting T−12 = T ∗2 / detT2, we have T2,t + T2U2 = R(λ)T2, (3.24) with R(λ) =  0 R (0) 12 R (1) 13 λ+R (0) 13 −R(0) 12 0 −R(0) 32 −R(1) 13 λ−R13(0) R (0) 32 0  . (3.25) Proof. By (3.24), equating the coefficients of λN+i (i = 1, 2), we obtain R (1) 13 = 1, R (0) 32 = p− q +A (N−1) 32 = p̃+ q̃, R (0) 12 = p+ q +A (N−1) 12 = −p̃+ q̃, R (0) 13 = (−q + 1 2 A (N−1) 32 − 1 2 A (N−1) 12 )2 + (p+ 1 2 A (N−1) 32 + 1 2 A (N−1) 12 )2 = p̃2 + q̃2. We see that Ũ2 = R(λ), which completes the proof. � EJDE-2022/71 INTEGRABLE NONLINEAR PERTURBED HIERARCHIES 19 Proposition 3.2. Noting T−12 = T ∗2 /detT2, we have T2,t + T2V (1) 2 = S(λ)T2, (3.26) with S(λ) =  0 S (0) 12 S (1) 13 λ+ S (0) 13 −S(0) 12 0 −S(0) 32 −S(1) 13 λ− S13(0) S (0) 32 0  . (3.27) Proof. By (3.27), comparing the coefficients of λ on both sides, we have S (1) 13 = α, S (0) 13 = δ1, S (0) 32 = αp− αq + αA (N−1) 32 = αp̃+ αq̃, S (0) 12 = αp+ αq + αA (N−1) 12 = −αp̃+ αq̃. We see that Ṽ (1) 2 = S(λ). The Proof is complete. � Next we discuss the explicit solutions of (3.6). Firstly, we give a seed solutions p = q = 0 of (3.6), the spectral problems are Φx = U2Φ =  0 0 λ 0 0 0 −λ 0 0 Φ, (3.28) and Φt = V (1) 2 Φ =  0 0 αλ 0 0 0 −αλ 0 0 Φ. (3.29) Solving the above two equations, we have ϕ = ϕ1 ϕ2 ϕ3  = sin(λt)sin(λx)− cos(λt) cos(λx) 1 sin(λx) cos(λt) + sin(λt) cos(λx)  , ψ = ψ1 ψ2 ψ3  = cos(λx) sin(λt) + cos(λt) sin(λx) 1 cos(λt) cos(λx)− sin(λt) sin(λx)  . (3.30) In particular, when N = 1, T2 =  0 A (0) 12 λ+A (0) 13 −A(0) 12 λ+A (0) 22 −A(0) 32 −λ−A(0) 13 A (0) 32 0  , (3.31) and p̃ = 1 2 A (0) 12 − 1 2 A (0) 32 , q̃ = 1 2 A (0) 12 + 1 2 A (0) 32 . (3.32) If we set ω (1) j = ϕ2 + γjψ2 ϕ1 + γjψ1 , ω (2) j = ϕ3 + γjψ3 ϕ1 + γjψ1 , we have ω (1) j = 1 + γj sin(λt) sin(λx)− cos(λt) cos(λx) + γj(cos(λx)sin(λt) + cos(λt) sin(λx)) , ω (1) j = sin(λx) cos(λt) + sin(λt) cos(λx) + γj(cos(λt) cos(λx)− sin(λt) sin(λx)) sin(λt) sin(λx)− cos(λt) cos(λx) + γj(cos(λx) sin(λt) + cos(λt) sin(λx)) . 20 Q. ZHAO, H. CHENG, X. LI, C. LI EJDE-2022/71 By Φ̃ = T2Φ, we obtain A (0) 11 +A (0) 12 ω (1) j +A (0) 13 ω (2) j = −λjω(2) j , −A(0) 12 +A (0) 22 ω (1) j −A (0) 32 ω (2) j = −λjω(1) j , −A(0) 13 +A (0) 32 ω (1) j +A (0) 33 ω (2) j = λj . (3.33) Solving the above equations by using of Cramer’s rule, we can obtain A (0) 12 = ∆A (0) 12 ∆A12 , A (0) 32 = ∆A (0) 32 ∆A32 , (3.34) and ∆A12 = 1 ω (1) 1 ω (2) 1 1 ω (1) 2 ω (2) 2 1 ω (1) 3 ω (2) 3  , ∆A (0) 12 = 1 −λ1ω(1) 2 ω (2) 1 1 −λ2ω(2) 2 ω (2) 2 1 −λ3ω(3) 2 ω (2) 3  , ∆A32 = −1 ω (1) 1 ω (2) 1 −1 ω (1) 2 ω (2) 2 −1 ω (1) 3 ω (2) 3  , ∆A (0) 32 = −1 λ1 ω (2) 1 −1 λ2 ω (2) 2 −1 λ3 ω (2) 3  . (3.35) Remark 3.3. In Figure 3, we can see the different pulse propagation patterns. Figure 3(a, b), the solitons keep at a certain interval and have no interaction, we can find that they have certain symmetry. Figure 3(c, d) shows that the bright soliton structures of solutions. Therefore, the difference between Figure 3(a, b) and Figure 3(c, d) is obvious. (a) p (b) p (c) q (d) q Figure 3. Three-dimensional structure figures of explicit solu- tions of (3.6) and the density plots with α = 1; λ1 = −0.5, λ2 = 0.2, λ3 = −0.01, γ1 = −0.2, γ2 = 0.2, γ3 = −0.1. 3.3. N-fold Darboux transformation of (3.7) and its explicit solutions. In this subsection, we apply the same Darboux matrix (3.19) to construct the Darboux transformation of (3.7). Proposition 3.4. Noting T−12 = T ∗2 /detT2, we have T2,t + T2V (2) 2 = W (λ)T2, (3.36) EJDE-2022/71 INTEGRABLE NONLINEAR PERTURBED HIERARCHIES 21 with W (λ) =  0 W (1) 12 +W (0) 12 W (2) 13 λ 2 +W (1) 13 λ+W (0) 13 −W (1) 12 −W (0) 12 0 −W (1) 32 −W (0) 32 −W (2) 13 λ 2 −W (1) 13 λ−W13(0) W (1) 32 +W (0) 32 0  . (3.37) Proof. By (3.37), comparing the coefficients of λ on both sides, we have W (1) 32 = αp− αq + αA (N−1) 32 = αp̃+ αq̃, W (1) 12 = αp+ αq + αA (N−1) 12 = −αp̃+ αq̃, W (2) 13 = α, W (0) 13 = −α[(−q + 1 2 A (N−1) 32 − 1 2 A (N−1) 12 )2 + (p+ 1 2 A (N−1) 32 + 1 2 A (N−1) 12 )2] = −α(p̃2 + q̃2), W (1) 13 = δ1, W (0) 12 = −α(−qx + 1 2 A (N−1) 32,x − 1 2 A (N−1) 21,x )− αh(p+ 1 2 A (N−1) 32 + 1 2 A (N−1) 12 ) = −αp̃x − αhq̃. We see that Ṽ (2) 2 = S(λ). The Proof is complete. � Next we will apply Darboux transformation (3.19) to give explicit solution of (3.7). We choose the seed solution p = q = 0, then the spectral problems become Φx = U2Φ, U2 =  0 0 λ 0 0 0 −λ 0 0  , (3.38) and Φt = V (2) 2 Φ, V (2) 2 =  0 0 αλ2 0 0 0 −αλ2 0 0  . (3.39) Solving the above two equations, we have ϕ = ϕ1 ϕ2 ϕ3  = sin(λ2t) sin(λx)− cos(λ2t) cos(λx) 1 sin(λx) cos(λ2t) + sin(λ2t) cos(λx)  , ψ = ψ1 ψ2 ψ3  = cos(λx) sin(λt) + cos(λt) sin(λx) 1 cos(λt) cos(λx)− sin(λt) sin(λx)  . (3.40) Ultimately, we obtain explicit solution of (3.7), p̃ = 1 2 A (0) 32 − 1 2 A (0) 12 , q̃ = 1 2 A (0) 32 + 1 2 A (0) 12 . (3.41) Remark 3.5. In Figure 4, one can see that all of these solitary waves move from right to left. During the propagation, their shapes nearly keep invariant but am- plitudes changes. It is worth pointing out that these one solitary waves display oscillating nonlinear waves with multiple peaks. Compared with Figure 4(a, b), the amplitude change of Figure 4(c, d) are more obvious. 22 Q. ZHAO, H. CHENG, X. LI, C. LI EJDE-2022/71 (a) p (b) p (c) q (d) q Figure 4. Three-dimensional structure figures of explicit solu- tions of (3.7) and the density plots with α = 1, λ1 = −0.2, λ2 = −0.1, λ3 = 0.3, γ1 = −0.5, γ2 = 0.2, γ3 = 0.3. 4. Sixth-order nonlinear integrable perturbed hierarchy of NLS-mKdV equation and their explicit solutions 4.1. Sixth-order nonlinear integrable perturbed hierarchy of NLS-mKdV equation. To obtain a sixth-order nonlinear integrable perturbed hierarchy of NLS-mKdV equation, we assume that the V3 has the form: Φt = V3Φ and V3 = [ V2 V4 0 V2 ] =  0 b− c a 0 f − g d −b+ c 0 −b− c −f + g 0 −f − g −a b+ c 0 −d f + g 0 0 0 0 0 b− c a 0 0 0 −b+ c 0 −b− c 0 0 0 −a b+ c 0  . (4.1) Now, we set V3 = ∞∑ i=0  0 bi − ci ai 0 fi − gi di −bi + ci 0 −bi − ci −fi + gi 0 −fi − gi −ai bi + ci 0 −di bi + ci 0 0 0 0 0 bi − ci ai 0 0 0 −bi + ci 0 −bi − ci 0 0 0 −ai bi + ci 0 λ −i. (4.2) Solving the stationary zero curvature equation (2.1) we obtain am+1,x = 2pbm+1 − 2qcm+1, bm+1 = −cm,x + qam − hbm, cm+1 = bm,x + pam − hcm, , dm+1,x = 2pfm+1 − 2qgm+1 + 2rbm+1 − 2scm+1, fm+1 = −gm,x − 1 2 cm,x + (s− p)am + hcm + qdm − hfm, gm+1 = fm,x − 1 2 bm,x + (r − q)am + hbm + pdm − hgm. (4.3) Substituting (1.7) and (4.2) into (2.1), and comparing the powers of coefficient of λ, we obtain initial values a0 = α, b0 = 0, c0 = 0, e0 = β, f0 = 0, g0 = 0, with α, β are arbitrary constants. The values of first few terms are calculated as follows a1 = 0, b1 = αq, c1 = αp, d1 = 0, f1 = α(s− q) + βq, EJDE-2022/71 INTEGRABLE NONLINEAR PERTURBED HIERARCHIES 23 g1 = α(r − p) + βp, a2 = α(−p2 − q2), b2 = −px − αhq, c2 = αqx − αhp, d2 = −2αpr − 2αqs+ 2α(q2 + p2) + β(−p2 − q2), f2 = −αrx + 2αpx − βpx + 2αhq − αhs− βhq, g2 = αsx − 2αqx + βqx + 2αhp− αhr − βhp, . . . We choose the auxiliary problem Φt = V3Φ. Noting that V3 = V3,m + ∆3,m, where the correction term is ∆∆3,m =  0 0 δm 0 0 0 0 0 0 0 0 0 −δm 0 0 0 0 0 0 0 0 0 0 δm 0 0 0 0 0 0 0 0 0 −δm 0 0  , where δm = −2ε(am+1 + dm+1). Next, we set V3,m = m∑ i=0  0 bi − ci ai 0 fi − gi di −bi + ci 0 −bi − ci −fi + gi 0 −fi − gi −ai bi + ci 0 −di bi + ci 0 0 0 0 0 bi − ci ai 0 0 0 −bi + ci 0 −bi − ci 0 0 0 −ai bi + ci 0 λ m−i. Then the corresponding zero curvature equation (2.6) give rise to the integrable coupling of NLS-mKdV system pt = −bm+1 + 1 2 εq(am+1 + em+1), qt = cm+1 − 1 2 εp(am+1 + em+1), rt = −bm+1 − fm+1 + 1 2 εs(am+1 + em+1), st = cm+1 + gm+1 − 1 2 εr(am+1 + em+1). (4.4) When m = 1, setting ε = 1 in (4.4) gives rise to pt = αpx + αhq − 1 2 αq2s− 1 2 αpqr − 1 2 (α+ β)q(p2 + q2), qt = αqx + αhp+ 1 2 αqs+ 1 2 αp2r + 1 2 (α+ β)p(p2 + q2), rt = αrx − αpx + βpx − αhq + αhs+ βhq − 1 2 αqs2 − 2αpsr − 1 2 (α+ β)s(p2 + q2), rt = αsx − αqx + βqx + αhq − αhs− βhq + 1 2 αqsr − 2αpr2 + 1 2 (α+ β)r(p2 + q2). (4.5) 24 Q. ZHAO, H. CHENG, X. LI, C. LI EJDE-2022/71 When m = 2, setting ε = 1 in (4.4) yields pt = αqxx − 2αhpx + αphx + 1 2 q(6α(qpx − pqx) + 2α(spx − psx) + β(pqx − qpx) + 2α(qrx − rqx) + 4αh(p2 + q2)− βh(q2 − p2) + 2h2), qt = −αpxx − 2αhqx + αh2p− αh2p+ αp(p2 + q2)− αqhx − 1 2 p(6α(qpx − pqx) + 2α(spx − psx) + β(pqx − qpx) + 2α(qrx − rqx) + 4αh(p2 + q2)− βh(q2 − p2) + 2h2), rt = αsxx − αqxx + βqxx + 2αphx + 4αhpx − αrhx − αhrβphx − 2βhpx − αhrx − 4αh2q + 2αh2s+ 2βh2q + 2αpqr − 2αq2s+ 2αq(p2 − q2) + βq(p2 − q2) + αs(p2 − q2) + 1 2 s(6α(qpx − pqx) + 2α(spx − psx) + β(pqx − qpx) + 2α(qrx − rqx) + 4αh(p2 + q2)− βh(q2 − p2) + 2h2), st = −αrxx − αpxx + βpxx + 2αqhx + 4αhqx − αshx − αhsβqhx − 2βhqx − αhsx − 4αh2p+ 2αh2r + 2βh2p+ 2αp2r − 2αpqs+ 2αp(p2 − q2) + βp(p2 − q2) + αr(p2 − q2)− 1 2 r(6α(qpx − pqx)2α(spx − psx) + β(pqx − qpx) + 2α(qrx − rqx) + 4αh(p2 + q2)− βh(q2 − p2) + 2h2). (4.6) We are ready to show that a sixth-order nonlinear integrable perturbed hierarchy of NLS-mKdV equation (4.4) is Lax integrable. 4.2. N-fold Darboux transformation of (4.5) and its explicit solutions. In this section, we investigate the Darboux transformation of (4.5). To construct the Darboux transformation of (4.5), we consider a gauge transformation Φ̃ = T3Φ, (4.7) where T3 satisfies Ũ3 = (T3,x + T3U3)T−13 , Ṽ3 = (T3,x + T3V3)T−13 . (4.8) Suppose the Darboux matrix T3 is of the form T3 =  A11 A12 A13 A14 A15 A16 A21 A22 A23 A24 A25 A26 A31 A32 A33 A34 A35 A36 0 0 0 A11 A12 A13 0 0 0 A21 A22 A23 0 0 0 A31 A32 A33  , (4.9) with A11 = A33 = A16 = A34 = 0, A12 = −A21 = N−1∑ i=0 A (i) 12λ i, A13 = −A31 = λN + N−1∑ i=0 A (i) 13λ i, A16 = λN + N−1∑ i=0 A (i) 16λ i, A22 = λN + N−1∑ i=0 A (i) 22λ i, EJDE-2022/71 INTEGRABLE NONLINEAR PERTURBED HIERARCHIES 25 A23 = −A32 = λN + N−1∑ i=0 A (i) 23λ i, A25 = λN + N−1∑ i=0 A (i) 25λ i, A35 = −A26 = N−1∑ i=0 A (i) 26λ i, A15 = −A24 = N−1∑ i=0 A (i) 15λ i. We define Φ =  ϕ1 ψ1 ϕ2 ψ2 ϕ3 ψ3 ϕ4 ψ4 ϕ5 ψ5 ϕ6 ψ6  . (4.10) By (4.9) and (4.10), we have Φ̃ = T3Φ =  A11 A12 A13 A14 A15 A16 A21 A22 A23 A24 A25 A26 A31 A32 A33 A34 A35 A36 0 0 0 A11 A12 A13 0 0 0 A21 A22 A23 0 0 0 A31 A32 A33   ϕ1 ψ1 ϕ2 ψ2 ϕ3 ψ3 ϕ4 ψ4 ϕ5 ψ5 ϕ6 ψ6  , (4.11) which equals a matrix or two Column 1 is A11ϕ1 +A12ϕ2 +A13ϕ3 +A14ϕ4 +A15ϕ5 +A16ϕ6 A21ϕ1 +A22ϕ2 +A23ϕ3 +A24ϕ4 +A25ϕ5 +A26ϕ6 A31ϕ1 +A32ϕ2 +A33ϕ3 +A34ϕ4 +A35ϕ5 +A36ϕ6 A11ϕ4 +A12ϕ5 +A13ϕ6 A21ϕ4 +A22ϕ5 +A23ϕ6 A31ϕ4 +A32ϕ5 +A33ϕ6  and the column 2 is A11ψ1 +A12ψ2 +A13ψ3 +A14ψ4 +A15ϕ5 +A16ϕ6 A21ψ1 +A22ψ2 +A23ψ3 +A24ψ4 +A25ϕ5 +A26ϕ6 A31ψ1 +A32ψ2 +A33ψ3 +A34ψ4 +A35ϕ5 +A36ϕ6 A11ψ4 +A12ψ5 +A13ϕ6 A21ψ4 +A22ψ5 +A23ψ5 A31ψ4 +A32ψ5 +A33ψ5  . So there exists γj (j = 1, 2) satisfying γ (1) j ϕ̃+ γ (2) j ψ̃ = 0. (4.12) Substituting (4.12) into (4.11), we obtain γ (1) j (A11ϕ1 +A12ϕ2 +A13ϕ3 +A14ϕ4) + γ (2) j (A11ψ1 +A12ψ2 +A13ψ3 +A14ψ4) = 0, γ (1) j (A21ϕ1 +A22ϕ2 +A23ϕ3 +A24ϕ4) + γ (2) j (A21ψ1 +A22ψ2 +A23ψ3 +A24ψ4) = 0, 26 Q. ZHAO, H. CHENG, X. LI, C. LI EJDE-2022/71 γ (1) j (A11ϕ3 +A12ϕ4) + γ (2) j (A11ψ3 +A12ψ4) = 0, γ (1) j (A21ϕ3 +A22ϕ4) + γ (2) j (A23ψ3 +A24ψ4) = 0. Proposition 4.1. The matrix Ũ3 has the same form as U3 determined by (1.7), that is U3 =  0 −p̃+ q̃ λ+ q̃s̃+ p̃r̃ 0 −r̃ + s̃ λ p̃− q̃ 0 −p̃− q̃ r̃ − s̃ 0 −r̃ − s̃ −λ− q̃s̃− p̃r̃ p̃+ q̃ 0 −λ r̃ + s̃ 0 0 0 0 0 −p̃+ q̃ λ+ q̃s̃+ p̃r̃ 0 0 0 p̃− q̃ 0 −p̃− q̃ 0 0 0 −λ− q̃s̃− p̃r̃ p̃+ q̃ 0  , in which the relations between old potentials and new ones are p̃ = −q + 1 2 A (N−1) 32 − 1 2 A (N−1) 12 , q̃ = p+ 1 2 A (N−1) 32 + 1 2 A (N−1) 12 , r̃ = −s+ 1 2 A (N−1) 35 − 1 2 A (N−1) 15 , s̃ = r + 1 2 A (N−1) 35 + 1 2 A (N−1) 15 . (4.13) Proof. Setting (T3,t + T3V3)T ∗3 =  Γ11 Γ12 Γ13 Γ14 Γ15 Γ16 Γ21 Γ22 Γ23 Γ24 Γ25 Γ26 Γ31 Γ32 Γ33 Γ34 Γ35 Γ36 0 0 0 Γ11 Γ12 Γ13 0 0 0 Γ21 Γ22 Γ23 0 0 0 Γ31 Γ32 Γ33  , where T−13 = T ∗3 / detT3. Then, we deduce that T3,t + T3U3 = X(λ)T3, (4.14) with X(λ) =  0 X (0) 12 X (1) 13 λ+X (0) 13 0 X (0) 15 X (1) 16 λ −X(0) 12 0 −X(0) 32 −X(0) 15 0 −X(1) 35 λ −X(1) 13 λ−X (0) 13 −X(0) 32 0 −X(0) 16 λ −X(0) 35 0 0 0 0 0 X (0) 12 X (1) 13 λ+X (0) 13 0 0 0 −X(0) 12 0 −X(0) 32 0 0 0 −X(1) 13 λ−X (0) 13 −X(0) 32 0  . Equating the coefficients of λN+i (i = 0, 1) in the above expression, we obtain X (1) 13 = 1, X (0) 12 = p+ q +A (N−1) 12 = −p̃+ q̃, X (0) 15 = r + s+A (N−1) 15 = −r̃ + s̃, X (1) 16 = 1, X (0) 13 = qs+ pr = q̃s̃+ p̃r̃, X (0) 32 = p− q + 2A (N−1) 32 = p̃+ q̃, X (0) 35 = r − s+A (N−1) 35 = r̃ + s̃. The proof is complete. � Proposition 4.2. We have (T3,t + T3V (1) 3 )T ∗3 = (detT3)Y (λ), (4.15) EJDE-2022/71 INTEGRABLE NONLINEAR PERTURBED HIERARCHIES 27 where Y (λ) =  0 Y (0) 12 Y (1) 13 λ+ Y (0) 13 0 Y (0) 15 Y (1) 16 λ −Y (0) 12 0 −Y (0) 32 −Y (0) 15 0 −Y (1) 35 λ −Y (1) 13 λ− Y (0) 13 −Y (0) 32 0 −Y (0) 16 λ −Y (0) 35 0 0 0 0 0 Y (0) 12 Y (1) 13 λ+ Y (0) 13 0 0 0 −Y (0) 12 0 −Y (0) 32 0 0 0 −Y (1) 13 λ− Y (0) 13 −Y (0) 32 0  . Proof. Equation (4.15) can be rewritten as T3,t + T3V (1) 3 = Y (λ)T3, (4.16) by means of T−13 = T ∗3 / detT3. According to (4.16), equating the coefficients of λN+i (i = 1, 2), we obtain Y (1) 13 = α, Y (0) 12 = αp+ αq + αA (N−1) 12 = −αp̃+ αq̃, Y (0) 13 = qs+ pr, Y (0) 32 = p− q + 2A (N−1) 32 = p̃+ q̃, Y (1) 16 = β, Y (0) 15 = α(s+ r − q − p) + β(p+ q) + αA (N−1) 15 + (β − α)A (N−1) 12 = α(−s̃− q̃ − r̃ + p̃) + β(q̃ − p̃), Y (0) 35 = α(r − p− s+ q) + β(p− q) + αA (N−1) 35 + (β − α)A (N−1) 32 = α(s̃+ r̃ − q̃ − p̃) + β(q̃ + p̃). The proof is complete. � To obtain the explicit solutions of (4.5), we choose the seed solution p = q = 0, r = s = 1. The spectral problems become Φx = U3Φ, U3 =  0 0 λ 0 0 λ 0 0 0 0 0 −2 −λ 0 0 −λ 0 0 0 0 0 0 0 λ 0 0 0 0 0 0 0 0 0 −λ 0 0  , (4.17) and Φt = V (1) 3 Φ, V (1) 3 =  0 0 αλ 0 0 βλ 0 0 0 0 0 −2α −αλ 0 0 −βλ 0 0 0 0 0 0 0 αλ 0 0 0 0 0 0 0 0 0 −αλ 0 0  . (4.18) In particular, when N = 1, T3 =  A11 A12 A13 A14 A15 A16 A21 A22 A23 A24 A25 A26 A31 A32 A33 A34 A35 A36 0 0 0 A11 A12 A13 0 0 0 A21 A22 A23 0 0 0 A31 A32 A33  , where A11 = A33 = A16 = A34 = 0, A12 = −A21 = A (0) 12 , A13 = −A31 = λ+A (0) 31 , 28 Q. ZHAO, H. CHENG, X. LI, C. LI EJDE-2022/71 A15 = −A24 = A (0) 15 , A16 = λ+A (0) 16 , A22 = λ+A (0) 22 , A23 = −A32 = λ+A (0) 32 , A25 = λ+A (0) 25 , A35 = −A26 = A (0) 35 . From the above results, we obtain the explicit solutions of (4.5) as follows p̃ = 1 2 A (0) 32 − 1 2 A (0) 12 , q̃ = 1 2 A (0) 32 + 1 2 A (0) 12 , r̃ = −1 + 1 2 A (0) 35 − 1 2 A (0) 15 , s̃ = 1 + 1 2 A (0) 35 + 1 2 A (0) 15 . (4.19) (a) p = q (b) p = q (c) r = s (d) r = s Figure 5. Soliton solutions of (4.5) and the density plots with α = 1, β = 0, λ1 = −0.6, λ2 = −0.6, λ3 = −0.2, λ4 = 0.2, λ5 = −0.3, λ6 = −0.3, γ1 = −0.8, γ2 = 0.8, γ3 = −0.3, γ4 = 0.3, γ5 = −0.2, γ6 = 0.2. Remark 4.3. From Figure 5(a, b), it can be observed that the shapes and ampli- tudes of three single solitons hardly change during propagation, which means that these propagation process is elastic. Compared to Figure 5(a, b), Figure 5(c, d) shows that both the shapes and amplitudes of two single solitons change during propagation, thus the propagation is inelastic. In addition, the existence of elastic and inelastic in the same system is an interesting physical phenomenon. 4.3. N-fold Darboux transformation of (4.6) and its explicit solutions. Next we use the same Darboux matrix (4.9) to solve the (4.6). Proposition 4.4. Let (T3,t + T3V (2) 3 )T ∗3 =  Θ11 Θ12 Θ13 Θ14 Θ15 Θ16 Θ21 Θ22 Θ23 Θ24 Θ25 Θ26 Θ31 Θ32 Θ33 Θ34 Θ35 Θ36 0 0 0 Θ11 Θ12 Θ13 0 0 0 Θ21 Θ22 Θ23 0 0 0 Θ31 Θ32 Θ33  . Then (T3,t + T3V (2) 3 )T ∗3 = (detT3)Z(λ), (4.20) EJDE-2022/71 INTEGRABLE NONLINEAR PERTURBED HIERARCHIES 29 with Z(λ) =  0 Z12 Z13 0 Z15 Z16 Z21 0 Z23 Z24 0 Z26 Z31 Z32 0 Z34 Z35 0 0 0 0 0 Z12 Z13 0 0 0 Z21 0 Z23 0 0 0 Z31 Z32 0  , (4.21) where Z12 = Z (1) 12 + Z (0) 12 , Z13 = Z13λ (2) + Z (1) 13 λ+ Z (0) 13 , Z21 = −Z(1) 12 λ− Z (0) 12 , Z23 = −Z(1) 32 λ− Z (0) 32 , Z31 = −Z(1) 13 λ 2 − Z(1) 13 λ− Z (0) 13 , Z15 = Z (1) 15 λ+ Z (0) 15 , Z16 = Z (1) 16 λ 2 + Z (0) 16 , Z24 = Z (1) 15 λ− Z (0) 15 , Z26 = −Z(1) 35 λ− Z (0) 35 , Z35 = Z (1) 35 λ+ Z (0) 35 , Z32 = Z (1) 32 λ+ Z (0) 32 . Proof. From (4.20), we obtain T3,t + T3V (2) 3 = Z(λ)T3. (4.22) Comparing the coefficients of λ in (4.22), we have Z (2) 13 = α, Z (1) 12 = αp+ αq + αA (N−1) 12 = −αp̃+ αq̃, Z (1) 13 = qs+ pr, Z (1) 32 = p− q + 2A (N−1) 32 = p̃+ q̃, Z (2) 16 = β, Z (1) 15 = α(s+ r − q − p) + β(p+ q) + αA (N−1) 15 + (β − α)A (N−1) 12 = α(−s̃− q̃ − r̃ + p̃) + β(q̃ − p̃), Z (1) 35 = α(r − p− s+ q) + β(p− q) + αA (N−1) 35 + (β − α)A (N−1) 32 = α(s̃+ r̃ − q̃ − p̃) + β(q̃ + p̃), Z (0) 13 = −α[(−q + 1 2 A (N−1) 32 − 1 2 A (N−1) 12 )2 + (p+ 1 2 A (N−1) 32 + 1 2 A (N−1) 12 )2] = −α(p̃2 + q̃2), Z (0) 12 = −α(−qx + 1 2 A (N−1) 32,x − 1 2 A (N−1) 21,x )− αh(p+ 1 2 A (N−1) 32 + 1 2 A (N−1) 12 ) − α(px −A(N−1) 32,x +A (N−1) 12,x ) + αh(−qx + 1 2 A (N−1) 32,x +A (N−1) 21 ) = −αp̃x − αhq̃ − αq̃x + αhp̃. The proof is complete. � Next we apply Darboux transformation (4.9) to give explicit solution of (4.6). We choose the seed solution p = q = 0, r = s = 1. Then the spectral problems become Φx = U3Φ, U3 =  0 0 λ 0 0 λ 0 0 0 0 0 −2 −λ 0 0 −λ 0 0 0 0 0 0 0 λ 0 0 0 0 0 0 0 0 0 −λ 0 0  , (4.23) 30 Q. ZHAO, H. CHENG, X. LI, C. LI EJDE-2022/71 and Φt = V (2) 3 Φ, V (2) 3 =  0 0 αλ2 0 0 βλ2 0 0 0 0 0 −2αλ −αλ2 0 0 −βλ2 0 0 0 0 0 0 0 αλ2 0 0 0 0 0 0 0 0 0 −αλ2 0 0  . (4.24) The exact solution of (4.6) is p̃ = 1 2 A (0) 32 − 1 2 A (0) 12 , q̃ = 1 2 A (0) 32 + 1 2 A (0) 12 , r̃ = −1 + 1 2 A (0) 35 − 1 2 A (0) 15 , s̃ = 1 + 1 2 A (0) 35 + 1 2 A (0) 15 . (4.25) (a) p = q (b) p = q (c) r = s (d) r = s Figure 6. Soliton solutions of (4.6) and the density plots with α = 1, β = 0, λ1 = −0.2, λ2 = 0.2, λ3 = −0.3, λ4 = 0.3, λ5 = −0.3, λ6 = 0.3, γ1 = −0.2, γ2 = 0.2, γ3 = −0.5, γ4 = 0.3, γ5 = −0.2, γ6 = 0.2. Remark 4.5. Based on single soliton solution (4.6) under the suitable parameters, Figure 6 describes the soliton pulse propagation in (x, t) plane. In Figure 6, we find that the components p q and r s are composed of bright solitons and the amplitude of r s are significantly higher than that of p q. It should be pointed out that the image shapes of components in the same system are similar but the amplitudes are different, which is an interesting phenomenon that may explain some physical significance and worthy of further investigation. 5. Conclusions In this article, three integrable nonlinear perturbed hierarchies of NLS-mKdV equation associated with their corresponding spectral problem are proposed and their Lax integrability is proved. Although the three problems are integrable cou- pled in different ways and have different perturbation terms, they all have Lax integrability, which is the interesting part of this paper. The purpose of studying the three problems together is that they all belong to an integrable coupled system of NLS-mKdV equations, which are promising for classification and comparison. EJDE-2022/71 INTEGRABLE NONLINEAR PERTURBED HIERARCHIES 31 We found three different Darboux matrices to construct their Darboux transforma- tions and derive the relationship between old and new potentials. Explicit solutions of these equations are investigated by Darboux transformation. Three-dimensional plots and density profiles of these explicit solution behaviors are presented. These will help us understand the role of higher order nonlinearity or larger amplitude waves. The propagation of pulsed soliton in the integrable nonlinear perturbed hier- archies of the NLS-mKdV equation is systematically analyzed, three-dimensional plots and density profiles of soliton solutions are obtained by balancing nonlinear term and dispersion term by adjusting the parameters. In the perturbed system, the adjustment of parameter is more complicated, we can find the desired image by adjusting the parameters to achieve a balance between nonlinear term and dis- persion term. For unselected images, these images maybe irregular because the balance of nonlinear term and dispersion term is not achieved. If we take h = 0, the perturbed system becomes unperturbed. In other words, unperturbed system is a special case of perturbed system, and the image of the perturbed system we show already includes the image of the unperturbed system. Therefore, there is no need to discuss unperturbed systems separately. In fact, soliton images of unperturbed systems have been widely studied [28], but soliton images of perturbed systems have not been widely studied due to their com- plexity. It seems that perturbed systems are important to study than unperturbed systems. Acknowledgements. This work was supported by the National Nature Science Foundation of China (No. 11701334), and the Jingying Project of Shandong Uni- versity of Science and Technology. References [1] Ablowitz, M. J.; Clarkson, P. 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Qiulan Zhao (corresponding author) College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, Shandong, China Email address: qlzhao@sdust.edu.cn Hongbiao Cheng College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, Shandong, China Email address: chenghongbiao1997@163.com Xinyue Li College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, Shandong, China Email address: xyli@sdust.edu.cn Chuanzhong Li College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, Shandong, China Email address: lichuanzhong@sdust.edu.cn 1. Introduction 2. Fourth-order nonlinear integrable perturbed hierarchy of NLS-mKdV equation and their explicit solutions 2.1. Fourth-order nonlinear integrable perturbed hierarchy of NLS-mKdV equation 2.2. N-fold Darboux transformation of (2.10) and its explicit solutions 2.3. Darboux transformation of (2.11) and its explicit solutions 3. Third-order nonlinear integrable perturbed hierarchy of NLS-mKdV equation and their explicit solutions 3.1. Third-order nonlinear integrable perturbed hierarchy of NLS-mKdV equation 3.2. N-fold Darboux transformation of (3.6) and its explicit solution 3.3. N-fold Darboux transformation of (3.7) and its explicit solutions 4. Sixth-order nonlinear integrable perturbed hierarchy of NLS-mKdV equation and their explicit solutions 4.1. Sixth-order nonlinear integrable perturbed hierarchy of NLS-mKdV equation 4.2. N-fold Darboux transformation of (4.5) and its explicit solutions 4.3. N-fold Darboux transformation of (4.6) and its explicit solutions 5. Conclusions Acknowledgements References