Hybrid waves consisting of the solitons, breathers and lumps for a (2+1)-dimensional extended shallow water wave equation

Shallow water waves are studied for the applications in hydraulic engineering and environmental engineering. In this paper, a (2+1)-dimensional extended shallow water wave equation is investigated. Hybrid solutions consisting of H -soliton, M -breather and J lump solutions have been constructed via the modi(cid:12)ed Pfa(cid:14)an technique, where H , M and J are the positive integers. One-breather solutions with a real function ϕ ( y ) are derived, where y is the scaled space variable, we notice that ϕ ( y ) in(cid:13)uences the shapes of the background planes. Discussions on the hybrid waves consisting of one breather and one soliton indicate that the one breather is not a(cid:11)ected by one soliton after interaction. One-lump solutions with ϕ ( y ) are obtained with the condition k 21 R < k 21 I , where k 1 R and k 1 I are the real constants, we notice that the one lump consists of two low valleys and one high peak, as well as the amplitude and velocity keep invariant during its propagation. Hybrid waves consisting of the one lump and one soliton imply that the shape of the one soliton becomes periodic when ϕ ( y ) is changed from a linear function to a periodic function.


Introduction
Nonlinear evolution equations (NLEEs) have been used to describe the wave propagations in fluid mechanics, plasma physics and nonlinear optics [1][2][3][4][5][6][7]. Solitions have been found that their shapes and amplitudes unchanged during the propagations [8]. Breathers have depicted the growths of disturbances on the continuous backgrounds related to the modulation instabilities [9]. Lumps, localized in all the directions of the space and possessing the meromorphic structures, have been found to propagate stably and used to describe the nonlinear patterns in fluid mechanics, plasma physics and nonlinear optics [10]. To construct the nonlinear wave solutions of the NLEEs, methods have been proposed, e.g., the Wronskian technique has been used to construct the soliton solutions [11], Darboux transformation has been used to construct the breather solutions [12] and Kadomtsev-Petviashvili hierarchy reduction has been used to construct the lump solutions [13].
Shallow water waves, known as one type of water waves with small depth relative to the water wavelength, have been studied for the applications in hydraulic engineering and environmental engineering [14][15][16][17][18][19]. In this paper, we will investigate the following (2+1)dimensional extended shallow water wave equation [20][21][22][23][24][25][26], where u is a real function with respect to the scaled space variables x, y and time variable t, α is a real constant and the subscripts represent the partial derivatives. Multiple soliton solutions for Eq. (1) have been derived via the Hereman's simplified method and Cole-Hopf transformation method [20]. Travelling wave solutions for Eq. (1) have been derived via the (G ′ /G)-expansion method [22]. Periodic wave solutions for Eq. (1) have been derived via the Hirota direct method and Riemann theta function [21]. Lax pair, Bäcklund transformation and conservation laws for Eq. (1) have been investigated via the binary Bell polynomials method [23]. Lump solutions and interaction behaviors for Eq. (1) have been investigated via the ansätz technique [24]. Abundant wave solutions for Eq. (1) have been discussed via the (G ′ /G)-expansion method [25]. Ref. [26] has constructed the N -soliton solutions with the Pfaffian form for Eq. (1) as with ϕ(y) being a real function of y, f N being a real function and defined as in which whereĵ means that the element j is omitted, r, j and N are the positive integers, c r,j is a real constant satisfying the condition c r,j = −c j,r , φ r 's and φ j 's are the scalar functions of x, y, t and satisfying the following linear partial differential conditions, However, to our knowledge, hybrid solutions for Eq. (1) consisting of the soliton, breather and lump solutions have not been constructed via the modified Pfaffian technique. In Section 2, hybrid solutions consisting of the soliton and breather solutions for Eq. (1) will be constructed via the modified Pfaffian technique. In Section 3, based on the solutions obtained in Section 2, hybrid solutions consisting of the soliton, breather and lump solutions for Eq. (1) will be obtained. In Section 4, our conclusions will be presented.

Hybrid solutions consisting of the soliton and breather solutions for Eq. (1)
To construct the breather solutions for Eq. (1), we redefine (r, j) in Eq. (4) as where c 2r−1,2r = 1 (r = 1, 2, ..., N ), otherwise c r,j = 0, k r 's and k j 's are the complex constants, φ r 's and φ j 's are the complex functions. Introducing the definitions in the form of where * denotes the complex conjugate, r and j denote the complex conjugate of the elements of r and j, respectively, we have (r, j) = ( r, j) * and ( r, j) = (r, j) * .
Assuming that f N in Eq. (3) has the following form and where M and H are the positive integers, we can obtain Eq. (10) indicates that f 2M +H is a real function. Motivated by Ref. [26], we can set φ r 's and φ j 's in Conditions (5) as where Ω r 's and Ω j 's are the complex constants. Then we can derive the hybrid solutions consisting of the M -breather and H-soliton solutions for Eq. (1) as Hereby, taking M = 1 and H = 0 in Solutions (12), and noting k r 's and Ω r 's as the forms, where k 1R , k 1I , k 2R , k 2I , Ω 1R , Ω 1I , Ω 2R and Ω 2I are the real constants, the subscripts R and I indicate the real and imaginary parts, respectively, we can derive the one-breather solutions for Eq. (1) as with ) ϕ(y) ) ϕ(y) To construct the hybrid solutions consisting of onebreather and one-soliton solutions for Eq. (1), we can set M = H = 1 in Solutions (12) as = 1, 2, . . . , 6) , Solutions (14) indicate that the one breather is adjusted by k 1 , k 2 and ϕ(y), as well as the one breather is localized along the curve Figs. 1 and 2 present the propagations of the one breather via Solutions (14) with different backgrounds. Comparing Figs. 1 with 2, we notice that the background of the one breather exhibits periodic property along the y direction when ϕ(y) is changed from a linear function to a periodic function. Figs. 3 and 4 illustrate the hybrid waves consisting of one breather and one soliton via Solutions (16). Similar to Figs. 1 and 2, the backgrounds of the hybrid waves exhibit periodic property along the y direction when ϕ(y) is changed from a linear function to a periodic function. Furthermore, comparing Figs. 1 with 3, we find that the one breather is not affected by one soliton after interaction.

Hybrid solutions consisting of the soliton, breather and lump solutions for Eq. (1)
To construct the lump solutions for Eq. (1), we define the set J = {ζ 1 , ζ 2 , . . . , ζ J } ⊆ {1, 2, . . . , M, M + 1, . . . , M + J} with ζ 1 < ζ 2 < . . . < ζ J , and let with i = √ −1, and J is the positive integer. Based on Conditions (9) and taking k 4ζ1−2 → −k 4ζ1−3 , we can Hence, hybrid solutions consisting of the H-soliton, M -breather and J-lump solutions for Eq. (1) can be derived as where the vectors k 1 = (k 4ζ1−2 , k 4ζ2−2 , . . . , k 4ζJ −2 ) and k 2 = (k 4ζ1−3 , k 4ζ2−3 , . . . , k 4ζJ −3 ). Let J = ζ 1 = 1 and M = H = 0 in Hybrid Solutions (19), we can derive the one-lump solutions for Eq. (1) as with Let J = H = ζ 1 = 1 and M = 0 in Hybrid Solutions (19), we can derive the hybrid solutions consisting of the one-lump and one-soliton solutions for Eq. (1) as (r, j = 1, 2, . . . , 6) , From Solutions (20), we find that the one lump is adjusted by k 1 and ϕ(y), and the one lump needs to satisfy the condition k 2 1R < k 2 1I . Figs. 5 present the one lump whose background is a slope plane when ϕ(y) = 0.15y, we notice that the one lump consists of two low valleys and one high peak, as well as its amplitude and velocity keep invariant during the propagation. Figs. 6 present the one lump whose background is periodic when ϕ(y) = sin(0.3y). Comparing Figs. 5 with 6, we find that the background of the one lump is affected by ϕ(y). Figs. 7 and 8 present the hybrid waves consisting of the one lump and one soliton via Solutions (22), we notice that the shape of the one soliton becomes periodic when ϕ(y) is changed from a linear function to a periodic function.

Conclusions
Shallow water waves, which refer to the water waves with small depth relative to the water wavelength, have From Solutions (14), we have noticed that the one breather is adjusted by k 1 , k 2 and ϕ(y), as well as the one breather is localized along the curve (k 1R +k 2R )x+ ) ϕ(y) + (k 1I + k 2I )x + w 2 t + Ω 1I + Ω 2I = 0. Background of the one breather exhibits periodic property along the y direction when ϕ(y) is changed from a linear function to a periodic function, as seen in Figs Solutions (20) have indicated that the one lump is adjusted by k 1 and ϕ(y), and the one lump needs to satisfy the condition k 2 1R < k 2 1I . One lump, whose background is a slope plane when ϕ(y) is a linear function, has been presented in Figs. 5. We have noticed that the one lump consists of two low valleys and one high peak, as well as the amplitude and velocity keep invariant during the propagation. One lump, whose background is periodic when ϕ(y) is a periodic function, has been presented in Figs. 6. Comparing Figs. 5 with 6, we have found that the background of the one lump is affected by ϕ(y). Based on Solutions (22), hybrid waves consisting of the one lump and one soliton have been presented in Figs. 7 and 8, we have noticed that the shape of the one soliton becomes periodic when ϕ(y) is changed from a linear function to a periodic function.