Novel solutions of the generalized mixed nonlinear Schrödinger equation with nonzero boundary condition

We present a determinant representation of generalized Darboux transformation for a generalized mixed nonlinear Schrödinger equation, and obtain several novel solutions with nonzero boundary condition. A complete classification of first-order solutions with a nonzero boundary condition is considered, and several second-order solutions, including some interesting structures, are discussed. Furthermore, by selecting special parameters in the equation, several novel kinds of solutions for the equation are displayed. Finally, we discuss the effects of parameter β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}, representing quintic nonlinearity term, on the breather solutions.

(GVD) and the self-phase modulation (SPM) reach a good balance. The SPM, caused by the Kerr effect, is a significant nonlinear effect, whereas the high-bit-rate transmission of solitary waves of ultrashort pulses is compulsory nowadays, due to the high-speed development of laser technology. For instance, the few-cycle pulses at attosecond scale could be produced in [1,2], in which the standard NLS was inadequate to model the evolution of the ultra-short pulse [3]. Therefore, it is compulsory to provide new models or generalized equations to simulate the ultra-short pulse. To deal with this issue, several nonlinear terms should be considered, for example, the self-steepening, self-frequency shift, non-Kerr nonlinearity, and so on. As a consequence, several new nonlinear equations were proposed [4][5][6][7][8][9][10], including several high-dimensional equations [11][12][13].
In this paper, we will study an integrable generalized mixed nonlinear Schrödinger (GMNLS) equation as follows: with a, b, β ∈ R. It was first proposed by Kundu and also known as Kundu equation [14]. The GMNLS equation can yield many significant nonlinear integrable equations, by setting suitable values of a, b [14], so it has attracted a lot of attention. Due to the integrability of the GMNLS equation, its Painlevé property [15], infinitely many conservation laws [16], Darboux transformation [17,18] were studied, and its multi-soliton [17,19,20] and rogue wave solutions were obtained [21,22]. The Darboux transformation was applied to Korteweg-de Vries (KdV) equation by Matveev [23], and the 'positon' solutions of KdV were obtained by considering the degenerated case. Recently, Guo, Ling, and Liu [24] developed the theory of Matveev, proposed a generalized Darboux transformation, and applied it to the celebrated NLS equation. The 'degenerated' solutions of NLS equation were obtained in [24], including 'positon' and rogue wave solutions. The generalized Darboux transformation in [24] was applied to several other integrable equations [25][26][27][28][29][30][31][32][33]. Motivated by the works in [18,24], we will consider the nonvanishing boundary solution of the GMNLS equation in the manuscript from the following aspects.
-We will consider the generalized Darboux transformation for the GMNLS equation, with each entry of Darboux matrices represented by a ratio of determinants, which are useful to construct a lot of solutions. -As shown above, many kinds of solutions of the GMNLS equation have been obtained. However, to our best knowledge, the soliton and breather solutions with a nonzero boundary condition have never been considered by Darboux transformation. Furthermore, it is noted that the GMNLS equation is a generalization of the derivative type of nonlinear Schrödinger equation, and the classification of the first-order solution of derivative nonlinear Schrödinger equation was discussed in [34]. Thus, it is natural to ask the question that can we give a classification for the first-order solution of the GMNLS equation according to related parameters. -Based on the classification of the first-order solutions of GMNLS equation, we will further continue to study the abundant second-order solutions with special parameters for the GMNLS equation according to three cases of two-fold generalized Darboux transformation. We conclude that there are eight kinds of solutions: periodic wave, dark soliton interacting with a periodic wave, bright soliton interacting with a periodic wave, two-bright soliton, two-dark soliton, bright-dark soliton I, brightdark soliton II, and breather solution. In particular, among them the generation mechanisms of two-bright soliton, two-dark soliton, bright-dark soliton I, and bright-dark soliton II are very interesting.
This paper is organized as follows: We review the main results of the generalized Darboux transformation of the GMNLS equation in Sect. 2. From the plane wave solution, we construct the first-order solution of the GMNLS equation and discuss the classification of this solution according to the special eigenvalue in Sect. 3. In Sect. 4, we construct the second-order solutions of the GMNLS equation and discuss the classification of these solutions based on the results in Sect. 3. The discussion and conclusion are given in the final section.

Generalized Darboux transformation for the GMNLS equation
In this section, we consider the generalized Darboux transformation for the GMNLS equation (1.1). To this end, we first recall its Lax pair or the linear spectral problem, given by where and the symbol * represents complex conjugation. We use k = (φ k , ϕ k ) to denote an eigenfunction of Eq.
Moreover, the n-order new solution induced by the generalized n-fold Darboux transformation can be given by 3 First-order solution from a periodic seed by Darboux transformation In this case, α is expressed as Substituting (3.1) into the Lax pair (2.1), the eigenfunction corresponding to λ is solved as Due to the symmetry of Lax pair (2.1), we infer that is also a solution of the Lax pair (2.1) associated with λ. Using the principle of superposition, we may obtain a new solution of the Lax pair (2.1) with parameter λ j as follows: . (3.5) Next, we will apply this eigenfunction to construct new solutions for the GMNLS equation by using the formula (2.5). Firstly, we shall consider the simplest situation n = 1. In this case, we choose the parameter λ 1 = iη 1 , η 1 ∈ R, then the first-order solution is given by with u [1] gives a soliton solution, and its amplitude is Thus, one may conclude that under the condition R and |u [1] A 1 | < |u [1] A 2 |, it gives rise to a dark soliton. Otherwise, it generates a bright soliton with a non-vanishing boundary. Furthermore, when [1] presents a periodic wave. To discuss the classification of the solution conveniently corresponding to parameter η 1 , we hereafter restrict our discussion to parameters a = −1, b = 1 4 , β = 1 4 . Two curves related to functions Fig. 1 Intervals of η 1 in the onefold DT generate different kinds of solution (dark soliton, bright soliton and periodic solution) of u [1] with a = −1, b = 1 4 , β = 1 4 and Fig. 1. It is easy to calculate four roots for y 1 , which are Note that the (η 1 ) 1 , (η 1 ) 2 are also two roots for y 2 , so we may conclude that -For the first-order solution u [1] (3.6) displays a periodic wave. -For the solution u [1] presents a dark soliton. -For u [1] is a bright soliton solution.
The dynamical evolutions of these three kinds of solution of u [1] are plotted in Fig. 2. Above discussions of the classification of the solution u 2b are summarized in Table 1.

Second-order solutions from a periodic seed by Darboux transformation
For the twofold generalized Darboux transformation (2.4) (n = 2), there are three admissible cases for the reductions of spectral parameters and eigenfunctions if we ignore the order of different spectral parameters: For the case (2a), we set λ 1 = iη 1 , then the 2-order new solution u 2a generated by generalized Darboux Fig. 2 Profiles of |u [1] | with parameters a = −1, b = 1 4 , β = 1 4 . a A bright soliton with η 1 = 1 2 . b A dark soliton with η 1 = − 1 2 . c A periodic wave with η 1 = 1 transformation is given by -For the solution shows a dark soliton interacting with a periodic wave shown in Fig. 3a and e. -For η 1 ∈ I 1 \Î 1 , the profiles in Fig. 3b and f of |u 2c | display a soliton interacting with a periodic wave. -For η 1 ∈ I 2 and η 2 ∈ I 3 , although the figures in Fig. 3c and d both depict a dark soliton interacting with a bright soliton, they are different. Also, the difference can be observed obviously in Fig. 3g and h. Thus, we may call the solution in Fig. 3c and d For the case (2b), we set λ 1 = iη 1 , λ 2 = iη 2 , then the 2-order solution u 2b , containing two free parameters η 1 , η 2 (|η 1 | = |η 2 |), is given by Through a simple calculation, we list the dynamical evolutions in terms of different values of η 1 and η 2 in Figs. 4 and 5. We find that for η 1 ∈ I 1 , η 2 ∈ I 2 and η 1 ∈ I 1 , η 2 ∈ I 3 the profiles, respectively, correspond to a dark soliton interacting with a periodic wave and a bright soliton interacting with a periodic wave, which can be checked in Fig. 5a, c and b, d. When there will be a dark soliton interacting with a bright soliton, which is similar to the corresponding situations in case (2a) and the related dynamical evolutions are shown in Fig. 4c, g and d, h. It is remarked that there will be two interesting kinds of dynamical profiles, which are not found in case(2a). If we set there will be a dark two-soliton (or a bright two-soliton) for the situation |η i | > |η j |( or |η i | < |η j |). The corresponding profiles of these two cases are arranged in Fig. 4a, e and b, f. Furthermore, one may observe from    Fig. 4e that the amplitude of the dark two-soliton will be enhanced when two peaks collide at t = 0, while the figure in Fig. 4f shows that the amplitude of the bright two-soliton will be depressing at the collision time t = 0. For the situation η 1 , η 2 ∈ I 1 , there will be a periodic wave, and we omit the related profiles. The above discussions on the classification of solution u 2b are summarized in Table 3. We finally consider the solution of case (2c), and we will discuss the effect of β on the solution with a = −1, b = 1 4 , λ 1 = 1 + i. Similar to case (2a) and case (2b), we obtain the expression of the solution as follows The density plots of |u 2c | with five different values of β are shown in Fig. 6, which is a breather solution. Furthermore, we infer that parameter β in the GMNLS equation will have two effects on the solution |u 2c |. One is that as the value of β(< 0) increases, the profile of the breather will have a counterclockwise rotation, while the profile of the breather will experience a clockwise rotation for the case of parameter β(> 0) increasing. The other influence induced by β is that the period of the solution will be increasing as the value of |β| increases.

Conclusion
In this work, we studied a generalized mixed nonlinear Schrödinger equation (1.1), which is an important physical model and can be used to describe the effects of the quintic nonlinearity and the self-steepening and self-frequency shift for the ultra-short optical pulse propagation in non-Kerr media. The following interesting results are summarized:  Tables 1, 2 and 3. It is worth to mention that the formation of two-bright soliton, two-dark soliton, bright-dark soliton I, and bright-dark soliton II related to the selections of different values of the parameter are very interesting. Moreover, we have studied the effects of parameter β on the secondorder solution u 2c (see Fig. 6).
Our analysis of the classification of first-order and second-order solutions of the GMNLS equation related to parameter η 1 may supply a generation mechanism to obtain several interesting higher-order solutions. It is noted that the GMNLS equation can be also viewed as a special reduction in some physical models [9,10,[35][36][37], and we expect that our results could be observed in the experiment in shortly soon.