{'fraction_non_alphanumeric': 0.04635673371519915, 'fraction_numerical': 0.022778739842813374, 'mean_word_length': 4.170454545454546, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
General bright and dark soliton solutions to the massive Thirring model via KP hierarchy reductions

In the present paper, we are concerned with the tau function and its connection with the Kadomtsev-Petviashvili (KP) theory for the massive Thirring (MT) model. First, we bilinearize the massive Thirring model under both the vanishing and nonvanishing boundary conditions. Starting from a set of bilinear equations of two-component KP-Toda hierarchy, we derive the multi-bright solution to the MT model by the KP hierarchy reductions. Then, we show that the discrete KP equation can generate a set of bilinear equations of a deformed KP-Toda hierarchy through Miwa transformation. By imposing constraints on the parameters of the tau function, the general dark soliton solution to the MT model is constructed from the tau function of the discrete KP equation. Finally, the dynamics and properties of one- and two-soliton for both the bright and dark cases are analyzed in details.


Introduction
The massive Thirring (MT) model iv t + u + σv|u| 2 = 0 , with σ = ±1, was derived by Thirring in 1958 [1] in the context of general relativity. It represents a relativistically invariant nonlinear Dirac equation in the space of one dimension. It is one of the most remarkable solvable field theory models. Its complete integrability was firstly approved by Mikhailov [2] and Orfanidis [3] independently. The inverse scattering transform for the MT model was studied by Kuznetsov and Mikhailov [4] and many others [5,6,7,8,9]. The Darboux transformation, Bäcklund transformation of the MT model and its connection with other integrable systems have been investigated by Kaup and Newell [10], Lee [11,12], Prikarpatskii [13,14] , Franca et al. [15] and Degasperis [16]. Since the pioneer work by Date [17], the soliton solutions to the MT model including the ones with nonvanishing background were constructed by many authors [18,19,20,21,22,23,24]. In addition, the algebro-geometric solutions to the MT model has attracted attention and has been studied from late 1970s [25,26] to 1980s [27,28] until recently [29,30]. It should be pointed out that the rogue wave solutions to the MT model were recently investigated in [31,32,33] by the Darboux transformation.
The MT model admits the following Lax pair where in the sense that the compatibility condition U t − V x + [U, V ] = 0 gives the MT model (1)- (2). It is noted that the MT model can also be written in the laboratory coordinates as follows via the following transformation The MT model in the laboratory coordinates has application in nonlinear optics to describe pulse propagation in Bragg gratings [34]. Alexeeva et al. studied the PT-symmetry extensions of the MT model [35]. Regarding the integrable discretization, Nijhoff et al. gave the integrable discretization of the MT model in light cone coordinates [36,37] in 1980s. Most recently, Pelinovsky et al. proposed a semi-discrete integrable MT model in laboratory coordinates [38] and studied its solution solution via Darboux transformation [39]. Surprisingly, as far as we are aware, the bilinear formulation is missing in the literature. As far as we are aware, no paper published either in finding solutions via Hirota's bilinear method [40] or revealing the connection of the MT model to the KP theory by Kyoto school [41]. By combining the Hirota's bilinear method and the KP hierarchy reduction method, we have constructed general soliton solutions to many soliton equations such as the vector nonlinear Schrödiner equation [42] and the complex short pulse equation [43,44] for both the vanishing boundary condition (VBC) and nonvanishing boundary condition (NVBC). Therefore, the motivation and the goal of the present study is to bilinearize the MT model and find its soliton solutions under VBC and NVBC.
The remainder of the present paper in organized as follow. In section 2, we first bilinearize the MT model into a set of four equations corresponding to the bright soliton solution. Starting from a set of bilinear equations satisfied by the tau functions of two-component KP hierarchy, we arrive at above four bilinear equations satisfied by the bright soliton solutions by a series of reductions such as dimension and complex conjugate reductions. On the other hand, in section 3, we show that the discrete KP equation can generate a single-component KP-Toda hierarchy with four flows: one in positive flow and three in negative flows, along with its tau-function and a set of bilinear equations. By imposing one constraint on parameters, three reduction relations are achieved simultaneously, by which we are finally able to construct general N-dark soliton solution to the MT model. The dynamical behaviors and properties of one-and two-soliton solutions for both the bright and dark ones are analyed in section 4. The paper is concluded in section 5 by some comments and further topics.

Bright solitons in the MT model 2.1 Bilinearization of the MT model under VBC
The bilinearization of the MT system is established by the following proposition.

the MT model (1)-(2) is transformed into the following bilinear equations
where D is the Hirota D-operator defined by [40] D n s D m

Reductions to the multi-bright solution from the two-component KP-Toda hierarchy
Define the following tau functions of two-component KP-Toda hierarchy, where A n and B m are N × N matrices whose elements are where Φ n , Ψ n ,Φ n andΨ n are N-component row vectors Then we have the following lemma:

Lemma 1. The above tau functions of two-component KP-Toda hierarchy satisfy the following bilinear equations
Proof. Let us take the following notations: then the above matrix elements possess the differential and difference rules: By using the differential formula of determinant, it can be checked that the derivatives and shifts of the tau functions are expressed by the bordered determinants as follows: By using the Jacobi identity of determinant, we can obtain the following relations which completes the proof of bilinear equations (12)- (15).
In what follows, we proceed to the reduction procedure. To this end, we impose the reduction condition then one can easily show which leads to from Eqs. (12)-Eqs. (13). On the other hand, by performing row and column operations, we rewrite the tau function f nm as with the elements From the condition (16), one can deduce which implies Thus Eqs. (14)- (15) become Next, we consider complex conjugate condition by giving the following lemma: and drop the dummy variables y 1 and y −1 , then which implies f 00 = f * 10 ,ḡ 00 = −g * 00 ,ḡ 01 = g * 0,−1 .
Proof. Let C = (α 1 , α 2 , · · · , α N ) be a row vector. Direct calculations give Therefore, we arrive at exactly the same set of bilinear equations (7)-(10) by setting In summary, we can give the N-bright soliton solution to the MT model by the following theorem:

Theorem 1. The MT model (1)-(2) admits the multi-bright soliton solution
where I is an N × N identity matrix, A, A ′ , and B are N × N matrices whose entries are Here p i , α i and ξ i0 are arbitrary complex parameters for i = 1, · · · , N.
3 Dark solitons in the MT model 3

.1 Bilinearization of the MT model under NVBC
The bilinearization of the MT model (1)-(2) under NVBC is established by the following proposition.

Proposition 2. By means of the dependent variable transformations
where ρ 1 and ρ 2 are real constants, the MT model (1)-(2) is transformed into the following bilinear equations Proof. By rewriting the dependent variable transformations (34) Bilinear equations (35) and (36) are deduced by taking zero for each group inside bracket. Similarly, we can drive bilinear equations (37) and (38) by substituting (34) into Eq.(2).

Discrete KP equation and bilinear equations for the KP-Toda hierarchy
Let us start with a concrete form of the Gram determinant expression of the tau functions for the extended KP hierarchy with negative flows where Here p i ,p j , ξ i0 ,ξ j0 , a, b are constants. We have the following lemma regarding the bilinear equations satisfied by above tau function:

Lemma 3. The discrete KP equation generates a set of bilinear equations
satisfied by above tau function (40).
Applying Miwa transformation by taking b → 0 and x 1 = −k 2 b, i.e., τ n,k (k 2 + 1) → τ n,k − b∂ x 1 τ n,k one obtains which is equivalent to (41) by taking p i → p i , q i →p i , c ij = δ ij , d ij = ip i , a → −a and adding l to each tau function.
In what follows, we further show that bilinear equation (41) can generate (42)-(44) by a dual relation between positive flow and negative flow. To be specific, we notice that Therefore, x 1 and x −1 is exchangeable by redefining p −1 i → p i ,p −1 j →p j . Furthermore, by redefining index n + k → −n, k → l and a −1 = b, (41) is converted into which is nothing but Eq. (42).
On the other hand, by reparametrizing which makes positive flow (x 1 ) and negative flow (x −1 ) exchangeable. Thus, by taking x 1 → t a , and a −1 → −a, one obtains which is exactly Eq. (43) by dropping the prime. Eq. (44) is just a parallel copy of Eq. (43) from (a, t a , k) → (b, t b , l).
Remark. It is very interesting to observe that the discrete KP equation can generate KP-Toda hierarchy with asymmetric positive flow and negative flow. Furthermore, the positive flow and negative flow are exchangeable by reparametrizing the wave numbers in the tau function. Remark. It is noted that some of the bilinear equations derived above are also bilinear equations of the Fokas-Lenells equation [49], the complex short pulse equation [44] and the modified Camassa-Holm equation [50,51]. In other words, the tau function behind these equations is the same before the reductions. Remark. We can also give proof of above bilinear equations via determinant identities such as the Jacobi identity. However, the proof we give here is more systematic instead of technical. Moreover, in scrutinizing the connection between the discrete KP equation and the nonlinear PDEs, it makes possible for us to construct the integrable discrete analogues of the nonlinear PDEs.

Reduction to the dark soliton of the MT model
In what follows, we briefly show the reduction processes of reducing bilinear equations of extended KP hierarchy (41)- (44) to the bilinear equation (35)- (38). Firstly, we start with dimension reduction by noting that the determinant expression of τ nkl , can be alternatively expressed by by dividing j-th column by ψ nkl j and i-th row by ϕ nkl i for 1 ≤ i, j ≤ N. By imposing the reduction condition or one can easily check the following relations hold which implies τ n−1,k,l+1 = τ n,k+1,l , τ n,k,l+1 = τ n+1,k+1,l .

Theorem 2. The MT system (1)-(2) admits the multi-dark soliton solution
where f , f * , g and h are the following N × N determinant with Here p i , ξ i0 are complex constants and α is a real constant which need to satisfy the constraint condition:

Dynamics of bright and dark soliton solutions 4.1 One-and two-bright solitons and their dynamics
Based on the N-bright soliton solution to the MT model, one-and twosoliton solutions are calculated as follows. For simplicity, we reparameterize ξ ′ i = ξ i +ξ ′ i with α i = eξ ′ i , and redefine the complex constants p i and ξ ′ i as where a i , b i , η i0 and θ i0 are real constants. Then the variables ξ ′ 1 and ξ ′ 2 are rewritten as (73) By taking N = 1 in (32)-(33), we obtain the tau functions for one-soliton solution This leads to the square of the modulus of one-soliton solution for the MT model (1) and (2) with e 2δ = √ a 2 1 +b 2 1 4a 2

1
. Then the amplitudes A u and A v are given by and decreases in the interval 1 Two cases of the amplitude-velocity relations are depicted in Fig. 1. Amplitude Besides, we need to point out that the amplitudes A u and A v remain finite in the limit of a 1 → 0. In this limit case, the bright one-soliton solution (75) degenerates to the following algebraic soliton solution where we take e η 10 +iθ 10 = ∓2a 1 σ b 1 e a 1 x 0 +iθ ′ 10 . The profiles of one-soliton are plotted in Fig. 2 with different values of wave number p 1 . The tau functions corresponding to two-bright soliton solution can be obtained by taking N = 2 in (32)- (33) f = 1 + c 11 * e ξ 1 +ξ * 1 + c 21 * e ξ 2 +ξ * 1 + c 12 * e ξ 1 +ξ * 2 +c 22 * e ξ 2 +ξ * 2 + c 121 * 2 * e ξ 1 +ξ 2 +ξ * where .
The interaction of two-bright solitons are illustrated in Fig.3 for different parameters. As shown in Fig. 3(a) and (b), two-bright solitons undergo the regular collisions. The bound soliton state is exhibited in Fig. 3(c), in which two solitons move with the same velocity 1

One-and two-dark solitons and their dynamics
In this subsection, we list one-and two-dark soliton solutions to the MT model. Same as the bright soliton case, we rewrite the complex constants p i and ξ i as where a i , b i , η i0 and θ i0 are real constants. Then the variables ξ 1 and ξ 2 are rewritten as By taking N = 1, we have tau functions for one-dark soliton solution which leads to the squares of the modulus of u and v: . This implies that u exhibits a dark soliton when αa 1 < 0 and an anti-dark soliton on the background ρ 1 when αa 1 > 0, while v represents a dark soliton when (α + σαρ 1 ρ 2 )a 1 < 0 and an anti-dark soliton on the background ρ 1 when (α + σαρ 1 ρ 2 )a 1 > 0.
Case 1: a 1 > 0. Two components exhibit as anti-dark soliton and the amplitude-velocity relations are given by

an increasing function whereas
A v is a decreasing function in the intervalk − 1 < κ 1 <k + 1 , which is shown in Fig.4(a).

Concluding Remarks
In this paper, we give bilinear formulation of the massive Thirring model which is missing in the literature. Based on the bilinear equations, we constructed both the bright and dark soliton solutions to the massive Thirring model under VBC and NVBC, respectively, via the KP hierarchy reduction technique. Especially, we have shown that the discrete KP equation can generate a set of four bilinear equations, which are reduced to bilinear equations corresponding to the dark soliton of the MT model. We have also listed oneand two-bright and dark soliton solutions and analyzed their propertied in details.
We should point out that, even if we give the bright and dark soliton solution to the MT model in the light cone coordinates, one can easily obtain the corresponding soliton solutions in the laboratory coordinates. Moreover, based on the bilinear formulation established in the present work, we can also construct general breather and rogue wave solutions of the MT model (1)-(2) via the KP hierarchy reduction method. We will report our results elsewhere in the near future.
Finally, we comment that, since we have established a link between the discrete KP equation and the MT model in terms of tau functions and bilinear equations, similar to our recent work on modified Camassa-Holm (mCH) equation [51], it paves a way for constructing integrable semi-discrete and fully discrete MT model, which definitely deserves immediate investigation.