{'fraction_non_alphanumeric': 0.06109474337106528, 'fraction_numerical': 0.027756241277717475, 'mean_word_length': 4.317394888705689, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A (2+1)-dimensional sine-Gordon and sinh-Gordon equations with symmetries and kink wave solutions

Abstract In this paper, a (2+1)-dimensional sine-Gordon equation and a sinh-Gordon equation are derived from the well-known AKNS system. Based on the Hirota bilinear method and Lie symmetry analysis, kink wave solutions and traveling wave solutions of the (2+1)-dimensional sine-Gordon equation are constructed. The traveling wave solutions of the (2+1)-dimensional sinh-Gordon equation can also be provided in a similar manner. Meanwhile, conservation laws are derived.


Introduction
It is well-known that the classical (1 + 1)-dimensional sine-Gordon (sG) equation u tt = u xx + sin u (1) or equivalent form u xt = sin u (2) appears in many scientific fields [1,2,12,[5][6][7]34], such as quantum-field and differential geometry theory [1,2,12,10,24,[5][6][7]. Many mathematicians and physicists studied this well-known equation from different aspects. The authors in [2] discussed the sG equation through using the inverse scattering method. Leibbrandt [12] studied solutions of the sine-Gordon equation in higher dimensions. Klein [10] considered geometric interpretation as surfaces of constant negative curvature. Rubinstein [24] presented a model of field theory and studied in detail. Gu and Hu [5] provided explicit solutions to the intrinsic generalization for the wave and sine-Gordon equations, and Hu [7] investigated the relationship between soliton and differential Geometry through the sG equation. In [25], the authors studied symmetry groups of the intrinsic generalized wave and sine-Gordon equations. A quantum-mechanical system is constructed over a Fock space of particles in [31] based on N-soliton solutions. In paper [35], one of the authors derived a (2+1)-dimensional KdV and mKdV equation from positive case. This paper is the continuation of that one. In this paper, from the extend AKNS system, we derive a (2+1)-dimensional sine-Gordon equation as well as a (2+1)-dimensional sinh-Gordon equation. Kink wave solutions and their interactional wave propagation are constructed for the (2+1)-dimensional sine-Gordon equation. Furthermore, Lie symmetries approach is employed to reduce the (2+1)-dimensional sine-Gordon and sinh-Gordon equations so that their traveling wave solutions are obtained.

Derivation of (2+1)-dimensional sine-Gordon and sinh-Gordon equations
It is well-known that the AKNS system [1] is one of the classical well-known integrable systems from which a great many of nonlinear evolution equations can be derived, such as the famous KdV equation, the MKdV equation, the nonlinear Schrödinger equation (NLS), the Burgers equation, the (1+1)-dimensional sine-Gordon equation, etc. Based on the AKNS system, let us consider the following (2+1)-dimensional zero curvature equation [1,33,11,34,35], where [X, T ] = XT − T X, Here in Eq. (4), i 2 = −1, ζ is an eigenparameter independent of time t (i.e. ζ t = 0), q, r are two potential functions of x, t, and A, B, C, D are the functions to be determined. Substituting Eq. (4) into Eq. (3) leads to the following equations From the first and the last equations, we can choose D = −A. Hence, Eq. (5) becomes In order to solve for A, B, C, let us target at expanding A, B, C in the form of truncated power series with regard to the eigenvalue ζ . Since the positive cases have already been studied in the literature, here in our paper we do the negative case. The negative order of integrable equations originated from the work in [15,16], and thereafter some interesting integrable equations with the properties of generalized Lax representations and algebraic structure were developed from the negative hierarchy [32,17,19,21,22]. The negative case may generate some new equations which have different physical meanings [18,20,23]. Therefore let us try employing the following expansions Substituting Eq. (7) into Eq. (6), it immediately generates In the special case n = 1, we have and Eq. (10) admits the following special solutions: and subsequently yields the following (2+1)-dimensional sine-Gordon equation In a similar way, we may select the following special solutions of Eq. (10) to get the (2+1)-dimensional sinh-Gordon equation below Remark.
In order to get some exact solutions, let us set up with the following constraint condition Then, we have which gives us the following single kink wave solution Because of its linearity, Eq. (20) admits the following solutions where c j , k j , l j (j = 1, 2) are constants. Apparently, substituting Eq. (27) into Eq. (21) produces where That is to say, Therefore, two kink wave solution is given by Adopting the same procedure shown above, we could obtain a 3-kink wave solution u = 2i arctan e ξ 1 + e ξ 2 + e ξ 3 − e ξ 1 +ξ 2 +ξ 3 +A 12 +A 13 +A 23 1 − (e ξ 1 +ξ 2 +A 12 + e ξ 1 +ξ 3 +A 13 + e ξ 2 +ξ 3 +A 23 ) , where Repeated the similar procedure N times, we can construct N -kink wave solution:

Determinant representation of the N -kink wave solution
The N -kink wave solution could be represented in the terms of determinants. Let us consider the following determinant associated with a parameter λ where δ ij is a characteristic function and where a 1 , a 2 , ..., a N are N coefficients. Obviously, Repeating the above procedure, we shall obtain where stands for dp (λ) dλ . Comparing all the coefficients (46) with the numerator of (34), we can readily find that and u = 2i ln where δ ij is defined in (36).
Adopting the similar procedure as above and playing the same scenario in the obit of constant ξ 2 , we then know that ξ 1 can be rewritten by ξ 2 as Therefore, in the obit of constant ξ 2 , when t → −∞ and t → ∞, we obtain the following two asymptotic formulations u ∼ 2i ln e ξ 1 +ξ 2 +A 12 − ie ξ 2 e ξ 1 +ξ 2 +A 12 + ie ξ 2 = 4 arctan e −(ξ 2 +A 12 ) and respectively. In light of the above asymptotic analysis, we can conclude that when the kink waves travel alone with the x-axis, the one on the left travels faster and interacts with the one on the right. After their interactions, two kink waves interchange their position.

Conservation laws
Below, we present the multipliers Q with the corresponding conserved forms T t dxdy + T y dxdt + T x dydt (where (T t , T x , −T y ) is the conserved vector). The computed multipliers are up to first order in derivatives of u. Given here, there are infinitely many as all functions f i (x + y + t), i = 1, 2, 3, of Eq. (12) viz.,