General soliton and (semi-)rational solutions of the partial reverse space y-non-local Mel’nikov equation with non-zero boundary conditions

General soliton and (semi-)rational solutions to the y-non-local Mel’nikov equation with non-zero boundary conditions are derived by the Kadomtsev–Petviashvili (KP) hierarchy reduction method. The solutions are expressed in N × N Gram-type determinants with an arbitrary positive integer N. A possible new feature of our results compared to previous studies of non-local equations using the KP reduction method is that there are two families of constraints among the parameters appearing in the solutions, which display significant discrepancies. For even N, one of them only generates pairs of solitons or lumps while the other one can give rise to odd numbers of solitons or lumps; the interactions between lumps and solitons are always inelastic for one family whereas the other family may lead to semi-rational solutions with elastic collisions between lumps and solitons. These differences are illustrated by a thorough study of the solution dynamics for N = 1, 2, 3. Besides, regularities of solutions are discussed under proper choices of parameters.


Introduction
In the past two decades, the studies on parity-time (PT )-symmetric systems have grown significantly. A seminal work by Bender & Boettcher [1] revealed that a large class of non-Hermintian Hamiltonians exhibiting PT -symmetry can still possess entirely real spectra. Soon afterwards, PT -symmetry spread out to various physical fields, such as optics [2,3], mechanical systems [4], quantum field theory [5], electric circuits [6] and many others [7]. A comprehensive review of the developments of PT -symmetry is provided in [8].
In 2013, the concept of PT -symmetry was introduced to integrable systems by Ablowitz & Musslimani [9]. By considering a novel non-local reduction of the Ablowitz-Kaup-Newell-Segur (AKNS) scattering problem, they proposed the non-local nonlinear Schrödinger (NLS) equation iq t (x, t) ¼ q xx (x, t) þ 2sq(x, t) 2 q Ã ( Àx, t), s ¼ +1, (1:1) where the asterisk Ã represents complex conjugation. Remarkably, this equation is PT -symmetric as it can be viewed as a linear Schrödinger equation where the self-induced potential V(x, t) ≡ 2σq(x, t)q Ã (−x, t) admits the condition of PT -symmetry V(x, t) = V Ã (−x, t). The non-locality of equation (1.1) stems from the fact that the solution's evolution at x depends not only on its property at x, but also on its behaviour at −x. Subsequently, this equation has been extensively studied. Soliton solutions of equation (1.1) have been derived using various methods [10,11] and several types of rogue wave solutions of equation (1.1) were obtained via Darboux transformation [12]. With different symmetry reductions from the AKNS hierarchy and other integrable hierarchies, many new non-local equations were proposed, and some of them include the non-local complex/real sine-Gordon equation [13,14], the non-local complex/real reverse space-time modified Korteweg-de Vries (mKdV) equation [13], the non-local Davey-Stewartson (DS) equation [15][16][17], to name a few. The semi-discrete version [18,19] and multi-component generalizations [13,20] of the non-local NLS have been reported as well. From these studies, several distinctive features of solutions to non-local equations compared to their local counterparts were revealed, such as finite-time blow-up [9], the simultaneous existence of bright and dark solitons [21], and coexistence of solitons and kinks [22]. It should also be pointed out that nonlocal integrable equations may produce new physical effects and thus trigger novel physical applications. For instance, the non-local NLS (1.1) has been clarified to be related to an unconventional magnetic system [7]. Various methods of constructing exact solutions to the integrable equations have been developed, such as the Darboux transformation [23], the method of inverse scattering transformation [10], the Kadomtsev-Petviashvili (KP) hierarchy reduction method and so on [24,25]. Among them, the KP hierarchy reduction method is very powerful in deriving soliton solutions of integrable equations. This method was developed by the Kyoto school [26] and has been applied to construct soliton and breather solutions of many equations, including the NLS equation, the modified KdV equation, the Davey-Stewartson (DS) equation and the derivative Yajima-Oikawa system [27][28][29][30]. This method was also improved later to derive rogue wave and semi-rational solutions of various integrable equations [31][32][33] as well as their discretization [34,35]. Nevertheless, applications of this technique to non-local equations are not as successful as expected to local equations. The main obstacle is the simultaneous reductions of both the non-locality and complex conjugacy. Only very recently was this difficulty overcome by Feng et al. [11] in the study of soliton solutions to equation (1.1). They started with tau functions of the KP hierarchy expressed in Gram-type determinants of size 2J, where J is a positive integer. The reductions of the non-locality and complex conjugacy can be realized simultaneously by dividing the corresponding matrices into four J × J submatrices and imposing certain symmetry relations on the parameters in each sub-matrix. Subsequent to this, by making use of similar arguments, Rao et al. [36,37] obtained various solutions to the DS I equation, which contain 2J soliton/lump solutions and semi-rational solutions consisting of 2J solitons and 2J lumps either on the constant background or on the periodic background, where J is a positive integer.
Despite the successful extension of the reduction method to non-local equations, there are still some unsolved problems. On the one hand, solitons or lumps derived in both of the non-local NLS equation [11] and non-local DS I equation [36,37] always appear in pairs. On the other, the collisions between lumps and solitons that correspond to semi-rational solutions of the non-local DS I equation are inelastic. Therefore, it motivates the present work. We will solve these problems by investigating the partial reverse space y-non-local Mel'nikov equation 3u yy À u xt À (3u 2 þ u xx þ kC(x, y, t)C(x, À y, t)) xx ¼ 0 where κ = ±1, u depicts the long wave amplitude, and Ψ is the complex short wave envelope. Mel'nikov introduced the local counterpart of this equation [38,39] (Ψ(x, −y, t) replaced by Ψ Ã (x, y, t)) to model the royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 8: 201910 interaction of long waves with short wave packets. Recently, studies of the partial reverse space-time (x,t)non-local Mel'nikov equation have been carried out in [40,41], where solutions containing even numbers of solitons or lumps were derived. Compared to them, the main contributions of this paper are listed as follows: (a) For the reduction from the tau functions of the KP hierarchy to the bilinear equations of equation (1.3), two families of parameter relations in the N × N Gram-type determinants are found. When N is even, while one of them is similar to that in [11,36,37,40], which only generates pairs of solitons or lumps, the new one can give rise to odd numbers of solitons or lumps. (b) For even N, the interactions between lumps and solitons are always inelastic for the old family (similar to [36,37]) of parameter relations, whereas the new family may lead to semi-rational solutions with elastic collisions between lumps and solitons.
The rest of this paper is organized as follows. In §2, general soliton and (semi-)rational solutions of equation (1.3) are presented in theorem 2.1 and the regularity of solutions is explained in proposition 2.5. Then the proofs are provided. Sections 3 and 4 are, respectively, devoted to the discussions of soliton and (semi-)rational solutions on both constant and periodic backgrounds. We will summarize this paper in §5.

General soliton and (semi-)rational solutions of the y-non-local Mel'nikov equation
In this section, we present the general soliton and (semi-)rational solutions of equation (1.3).

General soliton and (semi-)rational solutions
Through the independent variable transformation 3) can be transformed into the bilinear form where c is an arbitrary constant. Here f, g and h are functions in x, y and t that satisfy and D is Hirota's bilinear differential operator [42] defined by royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 8: 201910 Remark 2.2. In this paper, we focus on the dynamics of solutions with n i ¼ 0 or 1, i ¼ 1, 2, . . . , N: Besides, without loss of generality, we may set a i0 = b i0 = 1, i = 1, 2, …, N. Remark 2.3. Three types of solutions (2.4) in theorem 2.1 will be discussed under different parameter restrictions: soliton solutions (c i ≠ 0, n i = 0, i = 1, …, N), rational solutions (c i ¼ 0, P N k¼1 n k ! 1) and semirational solutions ( P N i¼1 jc i j . 0, P N k¼1 n k ! 1).
Remark 2.4. For n 1 = n 2 = · · · = n N = 1, compared with Case I, the solutions (2.4) corresponding to Case II contain more free parameters c [N/2]+1 , …, c 2[N/2] , where bkc refers to the largest integer that is less than or equal to k. It is also noted that for even N = 2J, in Case I, solitons or lumps always appear in pairs, whereas Case II can give rise to odd numbers of solitons or lumps. For the dynamics of semi-rational solutions with N = 2J, the collisions between soliton and lump in Case I are always inelastic, while elastic collisions between them may appear for Case II. For example, if we choose ( 2 :11) where all the =c i are positive (or negative).

Proofs of theorem 2.1 and proposition 2.5
We first recall a lemma that will be needed.
Lemma 2.6. [31,43] The bilinear equations in the KP hierarchy (2:12) have the Gram-type determinant solutions Here, m (n) ij are functions in x −1 , x 1 , x 2 and x 3 defined by and where δ ij is the Kronecker delta, n i , m j are non-negative integers and p i , q j , a ik , b jl , c i , ξ i0 , η j0 are arbitrary complex constants, i, j = 1, 2, …, N. Now, we consider the reductions of the bilinear equations (2.12) in the KP hierarchy to the bilinear equations (2.2), by which the soliton and (semi-)rational solutions of (1.3) can be derived. Therefore, we define and take the variable transformations With these conditions, it will be shown that the functions in (2.1) satisfy equation (1.3) in theorem 2.1 as long as proper parameter constraints are imposed.
Further, we impose parameter conditions from Case I on (2.5) for even and odd N.
which implies the first term in (2.41) is real. As a consequence, if all the =c i are positive (or negative), then vMv = 0 for any v ≠ 0, which gives τ 0 ≠ 0 and thus f ≠ 0. The proof is completed. ▪

Dynamics of the soliton solutions
In this section, we analyse the dynamics of the soliton solutions of equation (1.3) on both constant and periodic backgrounds.

One-and two-soliton solutions on the periodic background
Previously, we have derived one-solitons Ψ s (3.6), two-solitons Ψ 2s (3.8) and the periodic background Ψ p (3.7), which naturally motivate us to obtain one-and two-soliton solutions to equation (1.3) on the periodic background.
One-soliton solutions. Take N = 2 of Case II in theorem 2.1 and we impose the parameter constraints: q 1 = p 1 , q 2 = p 2 . As we mentioned before, once p 2 is purely imaginary, one-soliton solutions on the periodic background can be obtained. By choosing proper parameter values, anti-dark soliton and dark soliton solutions on the periodic background are presented in figure 4.
Two-soliton solutions. Similarly, by taking N = 3 and p 3 to be purely imaginary in the basis of twosolitons Ψ 2s , five types of two-solitons can be observed on the periodic background (figure 5).
Two-and three-lump solutions. We construct the two-and three-lump solutions by taking N = 2 and N = 3 of Case I in theorem 2.1, respectively, with the same matrix entries According to the parameter constraints discussed in one-lump solutions, we can sort the two-lump solutions into three types, i.e. bright-bright lump solutions, four-petalled-four-petalled lump solutions and dark-dark lump solutions (see figure 7). Whereas the three-lump solutions have real parameters p 3 and q 3 leading to the three-lumps consisting of one bright lump and three types of two-lumps ( figure 8).

One-and two-lump solutions on the periodic background
Similar to §3, we will construct one-and two-lump solutions on the periodic background.
One-lump solutions. We take N = 2 and n 1 = 1 of Case II to solutions of equation (1.3) in theorem 2.1, hence with matrix entries  Based on the one-lump solutions, we set p 2 and q 2 to be purely imaginary. Naturally, we get the one bright lump on the periodic background ( figure 9). Two-lump solutions. Similarly, we take N = 3, n 1 = n 2 = 1 of Case I in theorem 2.1 and p 3 , q 3 to be purely imaginary numbers, then three types of two-lump solutions on the periodic background can be derived (figure 10).

Semi-rational solutions
In this section, we continue to study the dynamics of semi-rational solutions containing combinations of solitons and lumps. First, we consider solutions on the constant background.
One-soliton-two-lump solutions. To obtain an one-soliton-two-lump solution, we take parameter values (Case II in theorem 2.1) The corresponding solution |Ψ| is displayed in figure 13. It is seen that one bright lump moves toward one anti-dark soliton as t < 0. After they interact, another new larger bright lump splits from the soliton and these two lumps move away from the soliton with different velocities. Two-soliton-one-lump solutions. The dynamics of two-soliton-one-lump solutions with parameter values (Case II in theorem 2.1) is illustrated in figure 14. When t < 0 anti-dark and dark solitons move toward each other until they intersect at t = 0. Meanwhile, one bright lump separates from the interaction of these two solitons and gradually moves away from them. royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 8: 201910 Two-soliton-two-lump solutions. In this case, we take N = 2 and the parameter restrictions (Case I) listed in theorem 2.1 and proposition 2.5. Depending on the choices of parameter values, we discover nine types of semi-rational solutions consisting of two-solitons (three types) and two-lumps (three types). Since all models have similar dynamical behaviours, we just select one of them to illustrate in detail while the other eight types are shown at a specific time (t = 2) to demonstrate the various components ( figure 15).  royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 8: 201910 As shown in figure 16a-c, two bright lumps move toward two dark solitons when t < 0 and merge into them at t = 0 leaving just two solitons on the constant background. The opposite dynamical properties of these solutions are also described in figure 16d-f. When t < 0, two dark solitons appear and then two bright lumps gradually appear and move away from them as t > 0.
Finally, the dynamics of semi-rational solutions on the constant background discussed above can be extended to the periodic background (figures 17-19). The arguments are similar to those in soliton and rational solutions, and thus we omit the details.

Conclusion
In this paper, we have derived general soliton and (semi-)rational solutions of equation (1.3) with nonzero boundary conditions on both constant and periodic backgrounds, by using the KP hierarchy reduction method. These solutions are expressed in terms of N × N Gram-type determinants with an arbitrary positive integer N, from which N-soliton/lump solutions can be obtained. Regularities of solutions are given in proposition 2.5 under proper choices of parameters.
Two sets of parameter relations in the Gram-type determinants are found and they demonstrate several distinctive features. The solutions (2.4) corresponding to Case II in theorem 2.1 contain more free parameters than Case I. It is also noted that for even N = 2J, in Case I, solitons or lumps always appear in pairs, whereas Case II can give rise to any odd number of solitons or lumps. For the dynamics of semi-rational solutions with even N, the collisions between soliton and lump in Case I are always inelastic, where the fission or fusion of lumps can take place, while elastic collisions between them may appear for Case II. These differences are illustrated by a comprehensive study on the dynamics of solutions for N = 1, 2, 3. In conclusion, compared with earlier works on the non-local Mel'nikov equation, solutions to the y-non-local Mel'nikov equation under two different cases of parameter constrains have both even and odd numbers of solitons/lumps and richer dynamical behaviours.
To the best of our knowledge, the results obtained in this paper are entirely new and provide a further extension of the KP hierarchy reduction method to non-local equations. Finally, the physical implications of our results await future efforts of researchers.  royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 8: 201910  royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 8: 201910