An Integrable Evolution System and Its Analytical Solutions with the Help of Mixed Spectral AKNS Matrix Problem

: In this work, a novel integrable evolution system in the sense of Lax’s scheme associated with a mixed spectral Ablowitz‒Kaup‒Newell‒Segur (AKNS) matrix problem is first derived. Then, the time dependences of scattering data corresponding to the mixed spectral AKNS matrix problem are given in the inverse scattering analysis. Based on the given time dependences of scattering data, the reconstruction of potentials is carried out, and finally analytical solutions with four arbitrary functions of the derived integrable evolution system are formulated. This study shows that some other systems of integrable evolution equations under the resolvable framework of the inverse scattering method with mixed spectral parameters can be constructed by embedding different spectral parameters and time-varying coefficient functions to the known AKNS matrix spectral problem.


Introduction
In nonlinear mathematical physics, the derivation, solution and integrability of equations are important topics [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Generally, an evolution equation is called integrable in the sense of Lax if it can be written as the compatibility condition between the related linear spectral problem and the adjoint time evolution equation [2]. For example [5], the well-known Korteweg-de Vries (KdV) equation 6 0 t x xxx u uu u    has the Lax integrability owing to the compatibility condition [8]: of a pair of given linear problems: where the eigenfunction  and the potential function u are dependent on the space variable x and the time variable t , and the spectral parameter  is a constant.
Since the isospectral AKNS matrix problem [2]: x t (4) and its adjoint time evolution equation: were proposed in 1974, a large number of important integrable equations [1][2][3][4][5][6][7][8][9] have been derived from the compatibility condition of Equations (4) and (5): such as the KdV equation, the modified KdV (mKdV) equation, the nonlinear Schrödinger (NLS) equation, and the sine-Gordon equation. In Equations (4) and (5), q and r are two smooth potential functions of x and t ; A , B and C are three undetermined functions of x , t , q , r and  ; and i is the imaginary unit. The findings of a large number of integrable equations are due to the pioneering work of Lax's scheme [1], including Equations (1) and (6) and their generalizations [5][6][7][8][9]. The generalizations of Equations (4) and (5) can be summarized as follows: (i) extending the isospectrum  , which is independent of t , to the nonisospectral case depending on t ; (ii) embedding some coefficient functions into the evolution equation satisfied by the nonisospectrum  and/or the function A for the derivation of time-varying nonisospectral equations or isospectral equations with time-varying coefficient functions; (iii) coupling isospectral equations and nonisospectral equations to mixed spectral equations; (iv) modifying local equations to nonlocal equations; (v) extension of the equations with integer-order derivatives to fractional-order equations. From the view of physics, the variable-coefficient equations and nonisospectral equations can be used to describe solitary waves in nonuniform media, and they have their own advantages [8] in being more suitable for approaching the essence of nonlinear phenomena than the constant-coefficient equations or isospectral equations. This work aims at generalizing Equations (4) and (5) to other different forms by proposing that the spectral parameter =ik  and the undetermined function A satisfy the following time evolution equation: and assumption: respectively. Here ( ) t are time-varying integrable functions, and B and C are supposed as: As a results, a novel system of integrable evolution equations: is derived in Section 2 for the first time. Equation (11) is a mixed spectral system and this, due to Equation (7), contains two kinds of spectra. One is isospectrum under the case of ( ) 2i ( ) 0 t k t     , and the other becomes nonisospectrum when ( ) 2i ( ) 0 t k t     . Thus, we call such a parameter ik in Equation (7) a mixed spectrum. Meanwhile, Equation (11) is called a mixed spectral system. Several special cases of Equation (11) and their corresponding simplified forms of Equations (7) and (8) can be found in Section 3. In Section 4, the inverse scattering method [2,3,9] combined with the mixed spectral parameter ik satisfying Equation (7) is established to solve Equation (11), and implicit solutions are obtained. Considering the reflectionless potential, the explicit unified formulae are reduced from the obtained implicit analytical solutions in Section 4. As a conclusion, we summarize the results of this article in Section 5.

Derivation of Equation (11) by Lax's Scheme
Substituting the matrices M and N in Equations (4) and (5) Then, the substitution of Equations (7) and (8) into Equations (12)- (14) shows that Equation (12) holds automatically, and Equations (13) and (14) are converted as follows: Further, we suppose that: where i b and i c are all undetermined functions of x and t . Substituting Equation (16) into Equation (15) and comparing the coefficients of the same powers of 2ik yields: 4 (2i ) k : Using Equations (18)-(21) we have: and then Equation (17) gives: Employing Equation (10), we easily find: and finally arrive at Equation (11) by means of Equations (25)-(27).
It should be noted that Equation (11) or Equation (25) cannot be included in the known mixed spectral AKNS hierarchy [7]: In fact, Equation (25) contains one sum of two nonisospectral terms: which cannot occur simultaneously in Equation (28). Similarly, Equation (28) cannot contain the other sum of two isospectral terms: In addition, all the four time-varying coefficient functions

Special Cases of Equation (11)
Proper selections of ( ) t can give some special cases of Equation (11), including the known equations.
qrq q xq xq q q r qrr r xr xr r r associated with: Special case 2. Constant-coefficient isospectral AKNS equations [5] under the case of ( ) 1 t associated with: Special case 3. Constant-coefficient nonisospectral AKNS equations [5] under the case of associated with: associated with Equation (7) and: Special case 5. Constant-coefficient isospectral KdV equation [5] under the case of associated with Equation (36) and: Special case 6. Constant-coefficient isospectral mKdV equation [5] under the case of q v  associated with Equation (36) and: Special case 7. Constant-coefficient isospectral sine-Gordon equation [5] under the case of / 2 associated with Equation (36) and: Special case 8. Variable-coefficient nonisospectral mKdV equation [5] under the case of associated with Equations (39) and (40).

Implicit Solutions of Equation (11)
In what follows, we assume that the potentials q , r and their derivatives of each order with respect to x and t are smooth functions, and when | | x   , they all tend to 0 . Theorem 1. Let us assume that ( , ) q x t and ( , ) r x t evolve according to Equation (11).