The dressing method and dynamics of soliton solutions for the Kundu–Eckhaus equation

The boundary value problem for the focusing Kundu–Eckhaus equation with nonzero boundary conditions is studied by the Dbar dressing method in this work. A Dbar problem with non-canonical normalization condition at infinity is introduced to investigate the soliton solution. The eigenfunction of Dbar problem is meromorphic outside annulus with center 0, which is used to construct the Lax pair of the Kundu–Eckhaus equation with nonzero boundary conditions, which is a crucial step to further search for the soliton solution. Furthermore, the original nonlinear evolution equation and conservation law are obtained by means of choosing a special distribution matrix. Moreover, the N-soliton solutions of the focusing Kundu–Eckhaus equation with nonzero boundary conditions are discussed based on the symmetries and distribution. As concrete examples, the dynamic behaviors of the one-breather solution and the two-breather solution are analyzed graphically by considering different parameters.


Introduction
The famous focusing Kundu-Eckhaus (KE) equation admits whereq(x, t) is the complex smooth envelop function with respect to x and t, β is a constant, β 2 is the quintic nonlinear coefficient. The last term represents the Raman effect, which is responsible for the selffrequency shift. This equation was first proposed by Kundu when he studied the gauge connections between the Landau-Lifshitz equation [1] and derivative nonlinear Schrödinger (NLS) type equations [2][3][4]. It adequately describes the propagation process of ultrashort optical pulses in nonlinear optics [5] and examines the stability of the Stokes wave in weakly nonlinear dispersive matter waves [6]. The focusing KE equation can be reduced to nonlinear Schrödinger equation when β = 0. Also Eq.(1) is completely integrable with Lax pair [7,8], soliton collisions [9], soliton solutions [10], rogue wave solutions [11][12][13][14][15], Darboux transformation [16] and long time asymptotic [17].
Although the Riemann-Hilbert (RH) approach or the inverse scattering transformation could be applied as well for this aim but, in our opinion, the Dbar dressing method is the most transparent and leads directly to the final results instead of analyzing the analytic regions. Recently, there has been a boom in the use of the RH approach to solve nonlinear equations with nonzero boundary conditions. However, few scholars take advantage of the Dbar method to study the soliton solution of nonlinear equations under nonzero boundary conditions.
In addition, even though there are many similarities between the Dbar method and the RH method, there are also many differences. For (1+1)-dimensional equations, the RH approach generally starts from the Lax pair to study analytic regions and symmetries so as to analyze the dynamical or long-time asymptotic behavior of the solution. However, the Dbar method is a direct method which can construct Lax pair instead of analyzing the analytic regions, nonlinear equations can be derived and then a series of properties such as conservation laws and their soliton solutions can be studied further. For (2+1)-dimensional equations, the Dbar method is more advantageous. Many problems that cannot be solved by RH approach can be transformed into the Dbar equation to study the solutions.
In this manuscript, the aim is to establish the nonlinear KE equation from the Dbar equation, together with its Lax pair, conservation law and exact soliton solution. It is worth noting that the two asymptotic expansions Eq.(23) and Eq.(24) derived from the Dbar equation are the basis for deriving the Lax pair and the KE equation. As examples, we also plotted the one-soliton solution and the two-soliton solution and analyzed their dynamic behaviors. The essentials of this Dbar dressing method have been explored and developed in the papers of Zakharov and Manakov [26,38] and books [39,40]. The Dbar dressing method is a powerful tool for studying the nonlinear integrable equations including one-dimensional equations [28,41,42] and twodimensional equations [27,[43][44][45][46][47][48]. Many nonlinear integrable equations with zero and nonzero boundary conditions have been well studied by the inverse scat-tering transformation or the Riemann-Hilbert approach [49][50][51][52][53][54][55][56]. Nevertheless, to our knowledge, few integrable equations with nonzero boundary are studied including the Kundu-Eckhaus equation by the Dbar dressing method. Consequently, as for the issue mentioned above, the KE equation is taken as a model to be investigated in detail and some new and interesting dynamical phenomena are demonstrated. The successful application of the Dbar method for the KE equations with nonzero boundary conditions illustrates that the analytical domains of eigenfunctions appear to be unimportant compared to the RH approach.
The Lax pair of the focusing KE equation iŝ then the Lax pair can be transformed into the following Lax pair with where q =qe −iβ |q| 2 dx . By setting the following transformation where ρ is a constant, and |ρ| = q 0 = 0, the equivalent Lax pair is denoted as the linear system where X = −ikσ 3 + Q, It is easy to calculate that the eigenvalues of the matrix X 0 = X (q = ρ) are ±iλ. Since these eigenvalues are doubly branched, the two-sheeted Riemann surface defined by λ 2 = k 2 + q 2 0 is introduced, which is glued two copies of the complex plane S 1 and S 2 along the segment [−iq 0 , iq 0 ], the branch points are k = ±iq 0 . Therefore, λ is a single-valued function of k on the Riemann surface which consists of two single-valued analytic branches with the distinction of function value by a negative sign. Thus, the local polar coordinates on S 1 are defined as the two single-valued analytic functions on the Riemann surface can be written as To ensure that the eigenvalue λ is single-valued, let us fix a uniformization variable to pledge two corresponding single-value functions Hence, the eigenfunction of the spectra problem (10) when q = ρ can be expressed as This work is organized as follows. In Sec.II, the asymptotic expansions of ψ at z → ∞ and z → 0 are obtained by the Dbar problem with non-canonical normalization conditions. In Sec.III, the Lax pair of the KE equation with nonzero boundary conditions is established and the symmetry relation is calculated via the Dbar dressing method. In Sec.IV, based on the distribution and spectra problem, the deformation of nonlinear KE equation and conservation law are constructed. In Sec.V, the N -soliton solution with nonzero boundary conditions is acquired by the Dbar equation. As application of the N -soliton solution, the figures of one-soliton and two-soliton solution with different parameters are plotted. The conclusion is given in the end.

Dbar dressing for KE equation with nonzero boundary condition
Consider a matrix Dbar problem where ψ(x, t, z) and r (x, t, z) are 2 × 2 matrix, the distribution r (z) is independent of x and t. From the expression of the eigenfunction, the following noncanonical normalization conditions are described as Define a new function then it satisfies the boundary conditionŝ and implies that the generalized Cauchy integral formula iŝ where R and ε are oriented circle with center at origin of z plane and radius R and ε, respectively, i.e., R = {z | |z| = R} and ε = {z | |z| = ε}, where − ε denotes a counter taking the negative direction of the circle ε . For simplicity, let us denote the first term of Eq. (20) as N (ψ), that iŝ where as R → ∞ and ε → 0. Equation (21) reduces to the following asymptotic behaviors about ψ at z → ∞ and z → 0 It is worth noting that the two coefficients a l (x, t) and b m (x, t) are not independent about the symmetry condition then we have for the circle R = {z | |z| = R} and ε = {z | |z| = ε}, where R → ∞ and ε → 0.
Proof There exist ε and R which satisfy 0 < |z| < 2ε and |z| > R/2 in Eq. (29) and Eq. (30), such that the function f (z) can be expressed as Let a m− j z j and H 1 (z) = ∞ l=0 a l z l represent the nonanalytic part and the analytic part of the function f (z) with |z| → 0. In a similar way, b l z l denote the corresponding nonanalytic and the analytic part of the function f (z) with |z| → ∞, respectively. According to the Cauchy formula, there is and Therefore, it is easy to find that the relation Eq.(31) holds.

Lax pair of KE equation with nonzero boundary conditions
In order to construct the relation between the potential function of KE equation and the solution of the Dbar problem, the Lax pair of the KE equation is necessary to discuss. For this purpose, it is important to search for two sets of operator which have same normalization conditions. The Dbar dressing method is based on the assumption that the homogeneous equation of Eq. (21) only has the zero solution, that is to say For convenience, the solution space of the Dbar problem (16) is denoted as follows In fact, the result can be used to construct the Lax pair of KE equation. There is an indisputable fact that and C are some 2 × 2 matrices. In order to acquire the spatial linear spectra problem, let us start from Eq.(23) at z → ∞, then where [σ 3 , a 1 ] = σ 3 a 1 −a 1 σ 3 . It is easy to observe that which implies that ψ x and −ikσ 3 ψ + i 2 [σ 3 , a 1 ]ψ share the same principal part at z → ∞. Now, in a similar way, there is an asymptotic expression at z → 0, which demonstrates that (45) which suggests that ψ x and −ikσ 3 ψ + i 2 [σ 3 , a 1 ]ψ also share the same principal part at z → 0.
Furthermore, the equality − i is received in virtue of the Proposition 1 and definition of N (ψ), which implies the spatial linear spectral problem In what follows, the temporal linear spectra problem is deduced in similar to the process of the spatial linear spectra problem. Firstly, the asymptotic behavior at z → ∞ is discussed by means of Eq.(23), that is which hints ψ t and −ik 2 σ 3 ψ + k Qψ + i 2 (Q x − Q 2 − q 2 0 )σ 3 ψ share the same principal part at z → ∞.
Similar to the process of spatial and temporal spectral problems, the asymptotic behavior at z → 0 is calculated from Eq. (24), It is easy to find that ψ t ∈ F and obtain the relation on account of Eq. (47) and Eq. (52). Analogously, it can be concluded that according to Eq. (50) and Eq. (53). There is no doubt that the temporal linear spectra problem is simplified as Moreover, for the sake of the solution of KE equation with nonzero boundary condition, the symmetry condition of the matrix Q is investigated to prove σ Q The eigenfunction ψ(x, t, z) satisfies the symmetry Eq. (27) and

KE equation and conservation law
In this subsection, the KE equation and its conservation law are constructed from the Dbar problem. Firstly, suppose that the matrix R(x, t, z) has the zero diagonal part and From the Dbar problem (16) and (18), the new Dbar problem is determined where R = e iθσ 3 re −iθσ 3 , and the associated solution space is defined asF. It can be easily verified that Then the matrix functionψ is a solution of Eq.(58) and possesses the asymptotic behaviorŝ where the coefficient a l and b m have been shown in Eq. (25) and Eq.(26), respectively. The linear system aboutψ t can be written aŝ where 3 . In fact, the Abel formula can be used to achievē which implies that the detψ(x, t, z) is analytic in z ∈ C \ {0}. The Cauchy integral formula and the asymptotic behaviors aboutψ can be used to receive The relation γψ −1 = σψ T σ is acquired for z = ±iq 0 . Furthermore, Eq. (61) can be expressed aŝ where =ψσ 1ψ T and σ 1 = 0 1 1 0 . Note that has the asymptotic behavior as z → ∞ When z → 0, the asymptotic behavior of can also be calculated, For the purpose of searching for the Kundu-Eckhaus equation, substituting Eq.(61) and Eq.(65) into Eq.(64) and comparing the term of O(z −1 ) yields, Making use of the spatial linear spectral problem Eq.(46) and the relation between ψ andψ, there is also a linear expression which implies that (x, t, z) admits the linear formula From the expression Eq. (71), it is obvious to see that and [o] where [d] and [o] denote the diagonal and off-diagonal matrix of . Then let us substitute the expansion Eq.(65) into the above two expressions at z → ∞, 1 (x, t, z) and Consequently, the first few formulas of n (n = 1, 2, · · · ) can be written as 3 (x, t, z) = 2Q xx σ − 4(Q 2 + q 2 0 )Qσ, The nonlinear equation is given by the off-diagonal part of Eq.(69) which is the deformed form of the KE equation. The original KE equation (1) is derived by means of the transformation q =qe −iβ |q| 2 dx . Moreover, the conservation law can be calculated as The nonlinear evolution equation and the conservation law can also be deduced from Eq. (64) in virtue of the expansion Eq.(67) and the symmetry condition Eq. (27) when z → 0.

Soliton solutions of KE equation
where the functionψ(z) has been defined in Eq.(18) and satisfies the integral equation Eq. (21). Notice that the distribution r (z) in C\0 has the symmetries in view of Eq. (27) and Eq. (57).
In the following discussion, the propagation characteristics of the soliton solutions for KE equation will be analyzed. As a result of the above description, an expression for the N -soliton solution Eq. (83) is obtained. Next the dynamic behaviors of the solution will be figured by choosing the appropriate parameters. In particular, for N = 1, the one-breather solution is where Therefore, the figures of N = 1 are shown as Fig.1-4 by means of selecting the different parameters.
As can be seen from Figs. 1 and 2, the soliton solutions are the one-breather solutions which show periodic changes over t-axis. The propagation of the solutions for these illustrations is parallel to the t-axis when the discrete spectrum ξ 1 is pure imaginary, in other words, the solutions of Figs. 1(a) and 2(a) are called the stationary breather solutions. It is obvious to observe that the top of Fig.2 is parallel to the time axis as the boundary values get smaller and smaller, that is to say, the phenomenon of the breather will tend to disappear as the boundary value is very small.
It can be seen from Figs. 3 and 4 that the propagation of the solutions is neither parallel to the t-axis nor parallel to the x-axis except that they have similar propagating properties to Figs. 1 and 2. Moreover, the solutions of Figs. 3(a) and 4(a) are called the non-stationary breather solutions. This is because the discrete spectra are no longer purely imaginary, which results in different propagation behaviors of the breather solutions. Therefore, it is obvious that the value of the discrete spectrum ξ determines whether the breather solution is stationary or non-stationary.
Analogously, the dynamic behaviors of the two-breather solutions are analyzed when N = 2. Figure 5 reveals an interesting phenomenon that there are two columns of breather solutions in the process of propagation when the discrete spectra are purely imaginary. The propagation characteristic of the solutions in Fig. 5-7 is similar to Fig. 1-2, and Fig. 5-7 is illustrated by changing the boundary value ρ. These illustrations show that the breathing phenomenon gradually vanishes and only a sharp soliton exists when the boundary value ρ becomes smaller and smaller gradually. The soliton solutions also change periodically over time and the top of the breather will get steeper when the boundary value is smaller.
In addition, the images of the soliton solutions are symmetric with respect to the x-axis when the discrete spectra present opposite numbers by comparing Figs. 5-7 and Figs. 8-10 in the case where the discrete spectra are purely imaginary. Figures 8-10 show the same propagation patterns and structures as Figs. 5-7. The breathing phenomenon is weakening as the initial value gets smaller and smaller, and there s a decrease in the number of breathers. Figure 11 illustrates an intriguing phenomenon that there are two columns of breather solutions interacting during propagation. Figures 11-12 are described by changing the boundary value ρ which can change the propagation path of the soliton solution. When the discrete spectrum is a complex number whose both the real and imaginary parts are not zero, the change in the boundary value by a factor of 10 has led to a significant change in the picture of the soliton solution, so let us just draw two graphs Figs. 11-12 to see how the boundary value changes. By observing Figs. 11(a)-12(a), these illustrations show that the breathing phenomenon gradually vanishes and only a sharp soliton exists when the boundary value ρ becomes smaller and smaller gradually. On the contrary, when the discrete spectrum is opposite, Figs. 13-14 exhibit the same propagation characteristics as Figs. 11-12 except that the paths of the propagated images are symmetric about the x-axis. There is only a sharp soliton generated in the place where the two columns of waves interact during the propagation process as the boundary value ρ gradually becomes smaller.  Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.