Continuum Soliton Chain Analog to Heisenberg Spin Chain System. Modulation Stability and Spectral Characteristics

The one-dimensional (1D) Heisenberg spin chain system (HSCS) allows to investigate anomalous features originating from strong quantum fluctuations, which become more significant than those in higher dimensions. A continuum model equation of the HSCS, based on the discrete model, was constructed in the literature. It is nonlinear Schrodinger equation (NLSE) with biquadratic dispersion and fifth degree nonlinearity. Rare research works were done in this area. Notably for deriving the exact solutions and investigate the physical phenomena produced. Our objective, here, is to obtain these solutions, which we think they are new. Further, an analog of the different geometric solutions structures to the known characteristics of HSCS is performed. The unified method is implemented to find the exact solutions of the continuum model equation. A variety of solutions are obtained where they are evaluated numerically and represented in graphs. In these graphs, it is remarked that the solutions exhibit soliton chain (or dense soliton chain).in an analog to spin chain. In the contour plots, they show different shapes of super lattices. Furthermore, complex chirped waves are observed. A significant result is that these solutions are bounded by − 1/4 and 1/4, which can be relevant to the spins− 1/2 and 1/2. The analysis of modulation stability is carried and it is found that there is a critical value for the dominant parameters, where below this value, modulation instability holds otherwise modulation stability occurs.For the spectral characteristics, it is shown that the wave number increases abruptly and decreases to an asymptotic state, while the frequency is monotonic increasing. The spectrum is periodic wave away from the origin, but near the origin it is soliton.

The method used here is compared with the known methods, known in the in the literature.
Indeed, the UM [36] prevails the all known methods such as, the tanh, modified, and extended versions, the F-expansion, the exponential, the G'/G expansion method, the Kerdyashov methods, as it is of low time cost in symbolic computation.Further, It provides solutions which cannot be obtained by the other methods.
The outlines of the paper is in what follows. Section 2 is devoted to mathematical equations and outlines of the UM and GUM. the solutions in the polynomial form are presented in Section 3. While the solutions in the rational forms are carried in Section 4. Section 5 is devoted to modulation stability analysis. Conclusions are given in Section 6. (1) where S n := S n (t) and α is the biquadratic exchange parameter, which is considered in a dominant parameter. By assuming that the lattice side, a, is small, in the continuum analog, we write x = na and S n+1 = S(x, t) + aS x (x, t) + a 2 S xx (x,t)/2!+... Up to the order of a 4 , the equation of motion takes the form,
The spectral characteristics are, here, introduced. To this issue, we write, where K is the w2ave number and is the frequency; and the spectrum content is, For the objective of finding the solutions of (3), we introduce a transformation for w(x, t) with complex amplitude, This transformation allows to inspect the effect of soliton-periodic wave collision, which is elastic or inelastic depending on the waves solutions if they are smooth or not smooth. When inserting (7) into (3) gives rise, for the real and imaginary parts respectively, to, (9) We search for traveling waves solutions. To this end, we use the transformations u(x, t) = U(z), v(x, t) = V (z) and z = ax + bt. Under these transformations (8) and (9) become, respectively, Here, the solutions of (10) and (11) are found by using the UM and GUM. The Um asserts that the solutions of NLPDEs (NLODEs) are formulated in polynomial and rational forms, in auxiliary functions that satisfy appropriate auxiliary equations AE).

Polynomial forms
By considering (10) and (11), the polynomial forms are, To determine m i , i = 1, 2 and r, we use the balance and compatibility conditions. First, we consider the case when p = 1.The balance condition is determined by balancing the highest order derivative and highest nonlinearity terms. In the present case, the balance condition reads m 1 = m 2 = r − 1. To determine the consistency condition, we require the following (a) the number of equations that result from inserting (12) into (10) and (11) and by setting the coefficients of g(z) i ., i = 0, 1, 2, ..., say h(k).
(b) the number of arbitrary parameters a i , b i , c i , say f(k). For integrable equations the condition is h(k) − f (k) ≤ s,where s is the highest order derivative (here s = 4). In the present case the consistency condition reads 1 ≤ r ≤ 3.
The case case when p = 2 can be analyzed by the same way. We mention that when p = 1, the solutions of the AE are elementary functions, while they are periodic or ellptic whenp = 2.

Rational forms
In the UM the rational solutions are written, 3 Polynomial solutions of (10) and (11) Here, we consider the following cases.

When p = 1and r = 2
In this case (12) reduces to When inserting (14) into (10) and (11), and by setting the coefficients of g(z) i = 0, i = 0, 1, .., we have, (for linearity dependent solutions By solving the AE in (14), the solutions of (8) and (9) are, The solutions in (16) are evaluated numerical and the results are used to display Rew in Fig. 1(i)-(v). Figures 4 and 5. In (iv), the contour plots is displayed, while in (v) the variation of Re wagainst x for different values of tis done. Figure 1(i) shows "continuum" soliton chain with trapping, while Fig. 1(ii) and (iii) show complex soliton chain. Fig. 1(iv) shows super lattices and Fig. 1(v) shows "continuum" solton chain.
The Spectral characteristics which are given in (5) and (6) are shown in Fig. 2(i)-(iii), for the wave number, the frequency and spectrum respectively. Figure 2(i) shows the there a critical value of νwhere it incre4ases and decreases abruptly. Figure 2(ii) shows that the frequency increases with ν. Figure 2(iii) shows soliton chain with small amplitude apart near x = 0.

When p = 2and r = 2
In this case consider the solution in (14), but the AE is,  . 2 The wave number, the frequency and the spectrum are shown for the same caption as in Fig. 1 (i)-(v), but in (iii)ν = 0.5 By substituting from (14) and (17) into (10) and (11), by the same way as in Section 3.1, we get, Finally, the solutions of (8) and (9) are,.
The solutions in (20) are used to display Rew in Fig. 3 Fig. 3(i) and , the 3D plot , contour plots are displayed for Re w, while the variation againstx for different values oft is displayed in Fig. 3(iii)

Rational solutions of (10) and (11)
We consider the solution in (13), together with AE, By using (13) and (23) into (10) and (11), we get, Fig. 4 In Fig. 4(i) and , the 3D plot , contour plots are displayed for | w |, while the variation againstx for different values of tis displayed in Fig. 4(iii) The solutions of (8) and (9) are, The results in (25) are used to display Rew in Fig. 5 (i)-(iii). In Fig. 5(i) and (ii0, the 3D and contour plots of Rew are displayed, while the variation of Rewagainst x fo0r different values of t is displayed. Figure 5(i) shows "-continuum" soliton chain with trap, while )ii) shows mixed latticesolitons.

Modulation stability analysis
To study the modulation instability of a system, it should exhibit a normal mode. That is a periodic standing waves. Here, (3) has a solution of the form.
By inserting (24) into (3), we get We write the solution expansion near w m , in (3) and Calculations give rise to, The solution of (26) is detH = 0, which yields, We solve the eigenvalue problem in (27) subjected to the boundary conditions | U(±∞) |≤ U 0 and | V (±∞) |≤ V 0 . Thus the eigenfunctions take the form.

Conclusions
The 1D Heisenberg spin chain system is considered. In continuum analog a o equation wad derived in the literature. Which is a nonlinear Schrodinger equation with bi-quadratic dispersion and fifth degree nonlinearity. Here, this equation is studied for the objectives of finding the exact solutions and investigating the relevant phenomena vis-a-vis with spin chain.The exact solutions are obtained by using the unified method in polynomial and rational forms. A transformation that enables us to inspect the effects of soliton-periodic wave collision is proposed.The collision can be elastic or inelastic according to the waves solutions are smooth or not smooth. Numerical evaluation of the solutions are carried and displayed in figures. It is found that the solutions exhibit "continuum" soliton chain, while the contour plots show super lattices or lattices with trapping. It is remarked the the solutions are bounded by −1/4 and 1/4 which may be relevant with the spin −1/2 and 1/2.The modulation stability analysis is carried and it is shown that there is a critical value of the dominant parameters that separates stability and instability. It is worth noticing that the waves solutions found here are smooth, so waves collision is elastic. article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.