Resonance Y-type soliton, hybrid and quasi-periodic wave solutions of a generalized (2+1)-dimensional nonlinear wave equation


 In this paper, we consider a generalized (2+1)-dimensional nonlinear wave equation. Based on the bilinear, the N-soliton solutions are obtained. The resonance Y-type soliton and the interaction solutions between M-resonance Y-type solitons and P-resonance Y-type solitons are constructed by adding some new constraints to the parameters of the N-soliton solutions. The new type of two-opening resonance Y-type soliton solutions are presented by choosing some appropriate parameters in 3-soliton solutions. The hybrid solutions consisting of resonance Y-type solitons, breathers and lumps are investigated. The trajectories of the lump waves before and after the collision with the Y-type solitons are analyzed from the perspective of mathematical mechanism. Furthermore, the multi-dimensional Riemann-theta function is employed to investigate the quasi-periodic wave solutions. The one-periodic and two-periodic wave solutions are obtained. The asymptotic properties are systematically analyzed, which establish the relations between the quasi-periodic wave solutions and the soliton solutions. The results may be helpful to provide some effective information to analyze the dynamical behaviors of solitons, fluid mechanics, shallow water waves and optical solitons.


Introduction
The construction of nonlinear localized waves of the integrable systems is one of the most important topics of the nonlinear sciences. The nonlinear localized waves appear in many fields of sciences and technology, such as fluids, Bose-Einstein condensation, shallow waver waves and nonlinear optics. Solitons, lumps, breathers and rogues waves, well known to us, are all the localized waves [1][2][3][4][5][6][7][8][9]. Lump solution is a rational solution localized in all directions in space, which can be seen limit of the infinite period of the breather wave [10][11][12][13][14]. The long wave limit method is one of the effective methods to construct the multiple lump solutions and the hybrid solutions of the integrable systems that can be transformed into bilinear equations [15][16][17]. The resonant solitary wave is a special kind of soliton. The resonant phenomena exist in many integrable systems. The fission and fusion of the solitary waves are all the resonance phenomena [18][19][20]. Soliton molecules are the bound states of solitons which have been experimentally discovered on optical systems [21][22][23]. Soliton molecule is essentially a velocity resonance soliton [24][25][26]. The linear superposition principle can be used for constructing the resonance solutions [27][28][29]. The hybrid solutions consisting of soliton molecules and lump waves were investigated by partial velocity resonance and partial long wave limits [30,31]. It is interesting that Li et al. considered a more generalized constraint of the parameters in N -solitons to construct the resonance Y -type soliton solutions and the hybrid solutions among the resonance Y -type solitons, breathers, soliton molecules and lumps [32,33].
In the last decade, the quasi-periodic wave solutions have attracted the attention of many scholars. Fan et al. developed the multi-dimensional Riemann-theta functions to the bilinear forms to construct the quasi-periodic wave solutions [34][35][36][37]. The main inspiration for their work comes from Nakamura's work. In 1980s, Nakamura first proposed a convenient way to construct the quasi-periodic wave solution with the aid of Hirota's bilinear method [38,39]. It is convenient to analyze the frequencies, wave numbers, phase shifts and amplitudes and all the parameters in the Riemann matrix are free. Then Tian et al. presented the Riemann-Hirota method to investigate the solvability of the quasi-periodic wave solutions of many integrable systems [40,41]. Chen et al. investigated the quasi-periodic wave solutions and discussed their asymptotic behaviors of some high-dimensional nonlinear equations [42,43]. The one-periodic wave solutions of the modified generalised Vakhnenko equation and a higher-order KdV-type equation were derived by Wang and Chen. [44,45]. Furthermore, the Bäcklund transformations were employed to establish a unifying scheme to construct the quasi-periodic wave solutions of the integrable equations which possess two or more equations in bilinear forms [46]. This scheme is called Riemann-Bäcklund method. It proved that the Riemann-Bäcklund method is an effective method to construct the quasi-periodic wave solutions of some integrable systems with constant and variable coefficients [47,48].
In this paper, we consider a generalized (2+1)-dimensional nonlinear wave (2DNW) equation where c 1 , c 2 and c 3 are arbitrary constants. Eq.(1) is a generalized model to investigate nonlinear dynamical phenomena in shallow water, plasma and nonlinear optics [49]. We investigated the M -lump, high-order breather and hybrid solutions of the 2DNW equation [50]. Zhao et al. studied the integrability and some mixed solutions of Eq.(1) [51]. This paper focuses on investigating the resonance Y -type soliton solutions, some new types of hybrid solutions and the quasi-periodic wave solutions of the 2DNW equation and analyzing dynamical characteristics of each kind of solutions.
The organization of the paper is as follows. In section 2, the N -soliton solutions of Eq.(1) are obtained by means of the bilinear method. In section 3, resonance Y -type soliton solutions and interaction solutions between M -resonance Y -type solitons and P -resonance Y -type solitions are obtained by taking some constraints to the parameters of the N -soliton solutions. A new type of two-opening resonance Y -type soliton is presented. The interaction between breathers and resonance Y -type solitons is investigated. In section 4, the hybrid solutions consisting of the resonance Y -type solitons and the lumps are constructed. The trajectories of the multiple lump waves before and after the interaction are analyzed from the perspective of mathematical mechanism. In section 5, the multi-dimensional Riemann theta function is used to construct the quasi-periodic wave solutions. The asymptotic properties of the one and two-periodic wave are studied. We show the quasi-periodic wave solutions convergent to the soliton solutions under a long time limit. Finally some conclusions are given in the last section.

N -soliton solutions
Eq.(1) can be transformed into a bilinear equation with D t , D x and D y are the bilinear derivative operators, which can be defined by , by means of the variable transformation where the f (x, y, t) is a function about variables x, y and t. It is a fact that u = u (x, y, t) is a solution of Eq.(1) if and only if f is a solution of the bilinear equation (2). In order to construct the N -soliton solutions, the auxiliary function f can be taken as where with k j , l j and η 0 j are arbitrary constants, is the summation of possible combinations of µ j = 0, 1 (j = 1, 2, · · · , N ). It is interesting that the resonance Y -type soliton solutions of Eq.(1) can be obtained by removing some items from the expression of formula (4). The exp (x) = 0 is true if and only if x = ln (0). If one takes all the A ij to 0, the N -soliton solutions can be reduced to the resonance Y -type soliton solutions with the form Since the structure of this resonance soliton is simple, it is difficult to further construct the hybrid solutions between Y -type soliton and other types of solitons. Chen et al. considered a generalized form of the auxiliary function  [32,33]. Inspired by their work, we shall study the constraint of the parameters of function f (4) to investigate the resonance Y -type solitons and the hybrid solutions between resonance Y -type solitons and other type of localized waves. where (7) Theorem 1 provides a generalized form of resonance solutions of Eq.(1). Because the formula (7) contains the symbol "±", the resonance Y -type solitons not only simply refer the the fissionable Y -type solitons, but the fission or fusion Y -type solitons generated by stripe solitons. In order to distinguish the two different physical phenomena, we call the fusion type wave which tends to fuse along the y-axis. If one takes M = 2, P = 2 in Theorem 1, the interaction solutions between two 2-resonance Y -type solitons can be given as where the relevant parameters η j and A js are determined by (5), (6) and (7).
To describe the evolutionary dynamic behaviors of the three different types of interaction between 2-resonance Y -type solitons, three different figures are plotted by choosing appropriate parameters. (I) Fig.1 shows an interaction between two fission resonance waves. (II) An interaction between two fusion resonance Y -type solitary wave are shown in Fig.2. (III) A mixed solutions including a fusion and a fission resonance Y -type solitary wave presented in Fig.3. By observing Fig.1, 2 and 3, we can conclude that the interaction among the resonance Y -type solitary waves is elastic.
If the function f is selected as f = 1 + e η 1 + e η 2 + e η 3 + e η 2 +η 3 +A 23 , and the parameters l 2 and l 3 satisfy the condition (7), and A 23 is determined by Eq.(5), we can obtain a new type of two-opening resonance Y -type soliton by means of the transformation (3). From Fig.4, we can see that the wave fuses first and then fissions. Due to the structure of this resonance soliton is similar to the capital letter X, it can also be called X-type resonance soliton. Furthermore, if we choose the parameters in the N -soliton solutions as the interaction solutions between a M -resonance Y -type soliton and P -order breather wave can be constructed. Fig.5 is presented to show the interaction between a fusion 2-resonance Y -type soliton and a breather wave.
parameters of N -soliton solutions as where the trajectories of the lump wave determined by the parameters K 2m−1 , K 2m , L 2m−1 and L 2m , and the central coordinates of the lump wave before and after the interaction with the P -resonance Y -type soliton are The height of the wave remains constant, which have the fixed values , before and after the interaction. If we take M = 1, P = 2 in formula (9), we gain a hybrid wave consisting of a 2-resonance Y -type soliton and single lump wave. Under this condition, the auxiliary function f can be expanded as t. By choosing the parameters as K 1 =   Fig.(7)) before the interaction. Then the trajectory of the lump wave changes into line y = − 1 2 x + 294 293 (red line in Fig.(7)) after the interaction. It is notable that the center of the lump wave is located at the bifurcation point of the resonance solitary wave as t = 0. Based on the results of the theorem 2, we can further investigate more complex hybrid solutions of the resonance Y -type solitons and high-order lump waves.

Quasi-periodic waves and asymptotic properties
In this section, we shall use the Riemann-theta function to construct the quasiperiodic wave solutions and analyze the dynamical behaviors of the quasi-periodic waves. The multi-dimensional Riemann-theta function of genus can be written as where the integer value vector n = (n 1 , n 2 , · · · , n N ) T , and the complex phase variables ξ = (ξ 1 , ξ 2 , · · · , ξ N ) T ∈ C N . The inner product of two vectors is defined as ⟨f, g⟩ = f 1 g 1 + f 2 g 2 + · · · + f N g N .
The periodic N × N matrix τ is positive and real-valued symmetric, in which the entries τ ij can be considered as free parameters of the theta function (11). For an arbitrary vector ξ ∈ C N , the series (11) converges to a real-valued function.
In order to find the Riemann-theta function quasi-periodic wave solutions, we consider the solution of Eq.(1) in the form where u 0 is a free constant and ξ j = α j x+ρ j y+δ j t+σ j , j = 1, · · · , N. Substituting (12) into (1) and integrating with respect to x, then we obtain the following bilinear form where c is an integral constant.
Prof. Based on the theory of the quasi-periodic wave solutions in [34], in order to make the theta function (15) satisfies the bilinear form (13), the parameters of the one-periodic wave solution (14) should satisfy the following constraint equations To write above constraint equations into a linear system, we introduce the notations as the formula (16). Then liner system about parameters δ and c can be obtained ( a 11 a 12 a 21 a 22 By solving the linear equations, the theorem of the one-periodic wave can be established. Fig.8 shows the structure of the space for the one-periodic wave solution (14). As we can see, one-periodic wave is a parallel superposition of single solitons. The one-period wave has strict periodicity in each direction of the coordinate axis.
It is interesting that one can consider the asymptotic properties of the oneperiodic wave solution. The relation between the one-periodic wave solution and one-soliton solution can be established as follows. Based on the form of the Nsoliton solutions (5), the one-soliton solution can be given as For the case u 0 = 0, we write the coefficient matrix and the vector (b 1 , b 2 ) T into power series of λ as ( a 11 a 12 a 21 a 22 Substituting (20) into the liner system (18), we obtain the solution (δ, c) T with the form of series Thus, it concludes that as λ → 0. From this fact, we know that the phase variable 2πiξ tends to η − η 0 under the assumption In addition, if the parameters σ and η 0 satisfy the relation σ = η 0 + πτ 2πi (22) the theta function can be reduced into ) λ 6 + · · · → 1 + eξ, as λ → 0, whereξ = 2πiξ − πτ. According to the formulas (21) and (22), one concludes that ξ → η, ϑ (ξ) → 1 + e η , as λ → 0.
From above analysis, we can conclude that the two-periodic wave solution (23) tends to the two-soliton solution (28) under the condition λ 1 , λ 2 → 0.

Conclusions
This paper mainly focuses on investigating the localized wave solutions of the 2D-NW equation. Based on the bilinear method, the N -soliton solutions are obtained. By considering a more generalized constraints of the parameters, the N -soliton solutions are reduced to the resonance Y -type solitons. We study three different interactions between the fusion-type and fission-type resonance Y -type solitons through numerical simulation. The new kind of two-opening resonance Y -type solitons that can be regarded as resonance X-type solitons are presented. In order to study more complex structure of the solutions, we investigate the hybrid solutions consisting resonance Y -type solitons, breather waves and high-order lump waves. The trajectories of the multiple lump waves before and after the interaction with the resonance Y -type solitons are explicitly given. Furthermore, on the basis of the bilinear form of the 2DNW equation, the multi-dimensional Riemann-theta function is employed to construct the quasi-periodic wave solutions. A limiting procedure is presented to analyze the asymptotic behaviors of the one-periodic and two-periodic wave solutions. It is shown that the quasi-periodic wave solutions tend to the corresponding soliton solutions under a limit of the small amplitude. The dynamical characteristics of the quasi-periodic waves are analyzed be means of comparing the different parameters in Riemann-theta function. The results may be helpful to provide some useful information to study the dynamical behaviors of solitons, fluid mechanics, shallow water waves and optical solitons.