The complex Hamiltonian systems and quasi‐periodic solutions in the derivative nonlinear Schrödinger equations

The complex Hamiltonian systems with real‐valued Hamiltonians are generalized to deduce quasi‐periodic solutions for a hierarchy of derivative nonlinear Schrödinger (DNLS) equations. The DNLS hierarchy is decomposed into a family of complex finite‐dimensional Hamiltonian systems by separating the temporal and spatial variables, and the complex Hamiltonian systems are then proved to be integrable in the Liouville sense. Due to the commutability of complex Hamiltonian flows, the relationship between the DNLS equations and the complex Hamiltonian systems is specified via the Bargmann map. The Abel‐Jacobi variable is elaborated to straighten out the DNLS flows as linear superpositions on the Jacobi variety of an invariant Riemann surface. Finally, by using the technique of Riemann‐Jacobi inversion, some quasi‐periodic solutions are obtained for the DNLS equations in view of the Riemann theorem and the trace formulas.


INTRODUCTION
The derivative nonlinear Schrödinger (DNLS) equation takes the form which models the propagation of circularly polarized nonlinear Alfvén waves in plasmas, 1 and the transmission of subpicosecond pulses in single mode optical fibers. 2 The DNLS equation is rich of many explicit solutions, as it appears to be the compatibility condition of Lax pair 3 and = (2) , where is the complex spectral parameter independent of variables and ,̄is the complex conjugate of , and Followed by the isospectral nature, 4 soliton equations always occur with a hierarchy of high-order candidates. In this paper, we dedicate to deduce quasi-periodic solutions simultaneously of the DNLS hierarchy stemmed from the Kaup-Newell (KN) spectral problem (under the reduction = −̄) characterized by the DNLS equation (1). 3 The quasi-periodic (finite-gap, or -phase) solutions are the natural analogues of multi-solitons in the sense that they can be constructed by the theory of finite-gap integration. [5][6][7] The significance of quasi-periodic solutions resides in the fact that they are related to the elliptic function solutions and multi-solitons; 5-7 also, the finite-gap method has been already used to study the rogue waves on the multi-phase solutions together with their magnification factors. 8,9 More recently, as was shown in the 1-genus case, [10][11][12] the implicit constants of integration of high-order stationary soliton (Novikov) equations 13 are established, so that parameters of periodic waves are related to eigenvalues in the Lax spectrum. The isolated rogue waves on the periodic background are then obtained at the branch point of instable spectral-bands.
After the advent of Its-Matveev formula, 14 Its, Kotlyarov, and Matveev first attained the finite-gap solutions for the NLS equation and its modified version, that is, the DNLS equation. 7,15,16 Note that the DNLS equation (1) corresponds to a nonself-adjoint operator. Kamchatnov gave the one-phase periodic solution of the DNLS equation in improving the effectiveness for applications. 17 Starting with a distinct version of the KN spectral problem, Geng et al arrived at all the quasi-periodic solutions to the KN hierarchy. Theoretically, the coupled DNLS equations can be reduced to the DNLS equation under the transformation =̄and the scaling → −2 (see eqs. 1.1 and 1.2 in Ref. 18), however, see their solutions (5.34) replacing with −2 , it cannot be confirmed that the second expression is the conjugate of the first formula multiplied by . This means that solutions of the DNLS hierarchy could not simply reduced from those solutions of the real KN hierarchy. Only recently, Wright deduced the squared amplitude of hyperelliptic solutions together with its upper bound for the standard DNLS equation (1); 19 Zhao and Fan retrieved finite-gap solutions for the Gerdjikov-Ivanov equation of DNLS type (see eq. 8 in Ref. 20) in view of the algebro-geometric and the Riemann-Hilbert method.
For some other relevant works, the inverse scattering transformations with zero and nonzero boundary conditions were studied for the DNLS equation in Refs. 3 and 21. The multi-solitons and rogue wave solutions were presented by using the Bäcklund and Darboux transformation. 22,23 In the context of KN spectral problem, 3 Cao proposed a (2+1)-dimensional derivative Toda equation and gave its finite genus solution. 24 Pelinovsky confirmed the existence of global solutions to the DNLS equation without the small-norm assumption in the framework of direct and inverse scattering transformations. 25 Different from the above-mentioned treatments, we would like to set up the link between the complex finitedimensional Hamiltonian systems (FDHSs) and the DNLS hierarchy for getting their quasi-periodic solutions in terms of Riemann theta functions.
The separation of variables is one of the most universal methods in solving soliton equations. As soliton equations can be represented as the compatibility condition of two linear spectral problems (Lax pair), 4 the nonlinearization of Lax pair makes it possible to decompose soliton equations into FDHSs of solvable ordinary differential equations. 26 The real FDHSs have been extensively exploited to obtain solitons, quasi-periodic solutions, as well as the rogue periodic waves for a number of completely integrable models. [10][11][12][27][28][29][30][31] However, the complex FDHSs are not well studied, and are in the phase of collecting and classifying examples due to the complexity involving independent conjugates of eigenfunctions. Motivated by the above analysis, we develop a lucid algorithm not only to obtain a new formula of hyperelliptic solution of the DNLS equation (1) (compared with eqs. 98 and A13 in Ref. 19), but also to deliver all the quasi-periodic solutions for the DNLS hierarchy.
The main purpose of this work is to apply complex FDHSs for a simultaneous construction of quasiperiodic solutions of the DNLS hierarchy. Using the nonlinearization of Lax pair, the DNLS hierarchy is reduced to a family of complex FDHSs, so that simplifies the procedure for getting its explicit solutions. It follows from the Lax representations of DNLS hierarchy that a Lax matrix satisfied by the Lax equation is figured out, whose determinant gives rise to integrals of motion and a hyperelliptic curve for the complex FDHSs. With a set of quasi-Abel-Jacobi variables, the Liouville integrability of complex FDHSs is completed, which in a sense enriches the content of finite-dimensional integrable systems. Moreover, it turns out that involutive solutions of the complex FDHSs exactly yield finite parametric solutions of the DNLS hierarchy, and the used Bargmann map specifies a finitedimensional invariant subspace to the DNLS flows. The Abel-Jacobi (or angle) variables are suitably elaborated to linearize the DNLS flows, which display the evolution behavior of associated flows on the Jacobi variety of an invariant Riemann surface. Resorting to the Riemann theorem, 32,33 we apply the Riemann-Jacobi inversion to the Abel-Jacobi solutions of DNLS flows, and eventually arrive at quasi-periodic solutions of the DNLS hierarchy without any assumption on the periodic condition of potential.
The outline of this paper is as follows. In Section 2, the DNLS hierarchy is formulated into the zerocurvature pattern and further decomposed into a family of complex FDHSs. Section 3 focuses on the Liouville integrability of complex FDHSs, and the relationship between the DNLS hierarchy and the complex FDHSs is established in Section 4. Section 5 is devoted to the straightening out of the DNLS flows. In the last section, we briefly present the algebro-geometric construction of exact solutions for the DNLS hierarchy.

THE DNLS HIERARCHY AND THE COMPLEX FDHSS
The Lax representations of soliton equations contain most information of exact solutions, and in particular the machinery of finite-gap integration may be used as the Lax representations are available. Let us first specify the Lax representation for each equation in the DNLS hierarchy.
It follows from the recursive formulas (5) that the Lenard gradients { } and the Lenard operator pair and are defined by with a supplementary definition −1 = (0, 0) T , and are two skew-symmetric operators, and −1 is to denote the inverse operator of = ∕ with the condition −1 = −1 = 1, and 2 = 2 ∕ 2 , 2 = 2 ∕ , etc. Clearly, it is seen from (6) and (7) that Let us introduce an auxiliary spectral problem in terms of the Lenard gradients { } where = ( (1) , (2) The zero-curvature equation − ( ) + [ , ( ) ] = 0 of spectral problems (2) and (10) gives rise to the DNLS hierarchy together with a fundamental identity where It is easy to check that the first nontrivial member in (11) is the DNLS equation (1) with replacing the notation 2 = . For the concreteness, the second equation of (11) reads which is the compatibility condition of spectral problems (2) and where ] .
In general, the th DNLS equation (11) allows the Lax representations (2) and = ( ) . Remark: To fix = 1 in (10), we know from the spectral matrix (1) = that the variable 1 is in fact the spacial variable .
Due to the symmetry of (2), (̄, −̄) T corresponds to the eigenvaluē. Recalling the nonlinearization of Lax pair, 26 we take copies of spectral problem (2) together with their complex conjugates written as From Refs. 26 and 34, a direct computation yields the functional gradients of and̄on and̄, which is a special solution to the Lenard eigenvalue equations Consider the Bargmann (symmetric) constraint in the process of nonlinearization of Lax pair 26 which leads to a Bargmann map between the potential and the eigenfunctions ( , ) To progress further, on ℂ 2 we bring in the symplectic structure 2 = ∧ +̄∧̄and define the Poisson bracket 35 Simply substituting (19) back into the Lax representations (2), (3), (14), and using (15) leads to three complex FDHSs with real-valued Hamiltonians As the DNLS equations (1) and (13) appear to be the compatibility condition of spectral problems (2), (3), and (14), on (ℂ 2 , 2 ) they are indeed reduced to three complex FDHSs (20), (22), and (24) via separating the temporal and spatial variables. Introduce a bilinear generating function which satisfies the Lenard eigenvalue equation corresponding to the spectral parameter where which satisfies the Lax equation in view of (12) and (27). From the Lax equation (29), with | | > max{| 1 |, | 2 |, … , | |} we attain a generating function det of integrals of motion for the complex FDHSs (20) 36 where It is observed that We introduce an infinite sequence of complex FDHSs described as where the real-valued Hamiltonian is given by the recursive formula which together with (35) can be put into the unified form In what follows, we exhibit that the complex FDHSs (20), (22), (24), and (36) constitute the decompositions of the DNLS hierarchy (11), whose involutive solutions exactly generates finite parametric solutions to the DNLS hierarchy.

THE LIOUVILLE INTEGRABILIT Y
The integrability of a given complex FDHS is the existence of a set of 2 smooth functions, which is termed by a Liouville set if they are involutive in pairs and functionally independent on (ℂ 2 , 2 ). 35 Recalling (31)-(34), we have a sequence of integrals of motion to the complex FDHSs ( 0 , 2 , ℂ 2 ). Actually, they are integrals of motion for the complex FDHSs (22), (24), and (36) and which signifies that { } ( ≥ 0) are involutive integrals of motion for all the complex FDHSs (20), (22), (24), and (36).
Conforming to the Liouville theorem, 35 the other essential element of integrability is the functional independence of conserved quantities. The rest of this section is thus instructed to the functional independence of { } and { } ( ≥ 0), which ensures that the complex FDHSs could be integrated completely.

THE FINITE PARAMETRIC SOLUTIONS
The real FDHSs have been adapted to deduce exact solutions in quite a few cases. 10,[27][28][29][30][31] The key point lies in the fact that the relationship between real FDHSs and soliton equations is established, in which involutive solutions of real FDHSs naturally produce finite parametric solutions of soliton equations and the used symmetric constraint specifies a finite-dimensional invariant subspace to soliton equations. Based on the above observation, in this section we show that the Bargmann map (19) gives not only finite parametric solutions of the DNLS hierarchy, but also the finite-gap potential of the complex Novikov equation.
A solution is said to be a finite-gap potential if it satisfies a high-order stationary soliton (Novikov) equation. 13 To solve the DNLS hierarchy, we propose the conception of complex Novikov equation, which specifies a finite-dimensional invariant subset for the DNLS flows. (20). Then

Theorem 2. Let ( ( ), ( )) T be a solution of the complex Hamiltonian system
is a finite-gap solution of the complex Novikov (high-order stationary DNLS) equation and̂2,̂3, … ,̂2 +1 are some constants of integration with a supplementary definition̂1 = 1.
Proof. Expanding the polynomial ( ) in powers of gives rise to Resorting to (18) and (61), we have By applying the Lenard operator on (73), we derive the complex Novikov equation (72). ■

THE EVOLUTION OF DNLS FLOWS
The most attractive point of the Liouville-Arnold theory is to find action-angle variables for linearizing various flows on a complex invariant torus. 35 In fact, as for the classical integrable systems subjected to the inverse scattering transformation, the standard construction of action-angle variables using poles of the Baker-Akhiezer function is equivalent to the separation of variables. 37 Note that the DNLS hierarchy has been reduced to a family of complex integrable FDHSs in Section 4. In the next step, we present a systematical way of constructing the Abel-Jacobi (or angle) variable to straighten out the DNLS flows on the Jacobi variety of an invariant Riemann surface.

=1
( 2 ), we suitably introduce the Abel-Jacobi variable = ( 2 , ( 2 )), which will be used to straighten out the complex Hamiltonian flows and the DNLS flows. To progress further, it is assumed that By the expansion of power series, we arrive at where
Additionally, by combining the formulas (65), (74), (51), (53), (78), and (75), we get which signifies that the first formula of (76) holds true. From the definition of Poisson bracket and (38), we derive The comparison of (79) and (80) of the same powers in leads to the second formula of (76). ■ As a concrete application of Theorem 3, on the Jacobi variety (Γ) the Abel-Jacobi variable can be integrated by direct quadratures Let us restrict to be a finite number of terms. We achieve the Abel-Jacobi solutions described as linear superpositions of flow variables for the complex Hamiltonian flows and the DNLS flows X k − f low ∶ = 0 + Ω 0 + Ω +1 , ≥ 1.

THE RIEMANN-JACOBI INVERSION
Apart from solitons, quasi-periodic solutions are another class of interesting exact solutions, which enjoys a much richer structure due to its connection to algebraic geometry inherent in its construction. The Bargmann map (19) delivers both the finite parametric solutions of the DNLS hierarchy and the finite-gap potential of the high-order stationary DNLS (or complex Novikov) equation. The significance of finite-gap is to specify a finite-dimensional invariant subspace from the infinite-dimensional function space, similar to the sufficiently fast-decaying initial dada in the inverse scattering transformation. Moreover, the Lax matrix (28) determines a hyperelliptic curve of Riemann surface, whose genus coincides with the number of elliptic variables. We are now in a position to complete the Riemann-Jacobi inversion ⇐⇒ { 2 } 2 −1 =1 , such that the potential is represented by Riemann theta functions. To do so, first we represent the amplitude of potential as a symmetric function of elliptic variables Proof. Multiplied by ( ) on both sides of (48), the expansion in of the right-hand side (RHS) of (48) gives while the left-hand side (LHS) of (48) reads By comparing the coefficient of power 2 −2 in (85) and (86), we derive On the other hand, from (15) and (19), we have The combination of (87), (88), and their conjugates immediately gives rise to (84). ■ Followed by Lemma 1, we do not need to figure out each elliptic variable 2 (1 ≤ ≤ 2 − 1), but the symmetric function of { 2 } 2 −1 =1 . As a matter of the above fact, we turn to the Riemann theorem, 32,33 for the Abel-Jacobi variable defined by (74), there exists a vector of Riemann constant = ( 1 , 2 , … , 2 −1 ) T ∈ ℂ 2 −1 such that • ( ) =∶ (( ( )) − − ) has 2 − 1 simple zeros at { 2 1 , 2 2 , … , 2 2 −1 }.