Soliton solutions and their degenerations in the (2+1)-dimensional Hirota–Satsuma–Ito equations with time-dependent linear phase speed

This paper focuses on the exact soliton solutions of the (2+1)-dimensional generalized Hirota–Satsuma–Ito equations with time-dependent linear phase speed. Based on the Painlevé integrability test of this equation, the condition of the integrability is determined. Then the general N-soliton solutions are constructed by Hirota bilinear method. Not only the expressions of exact solutions and their degenerations, but also the spatial structures are presented for different choices of the parameters, including the line soliton, periodic soliton, lump soliton and their interaction forms.

The (1+1)-dimensional Hirota-Satsuma (HS) model written as [21] was proposed by Hirota and Satsuma, which describes the propagation of unidirectional shallow water waves. The wave amplitude u = u(x, t) and v = v(x, t) are the function of the scaled spatial variables x and the temporal variable t. In [22], Ito gave the bilinear form of the HS equation (1) as an example of the modified Korteweg-de Vries (mKdV) type. KdV system explains the theory of weakly nonlinear long waves well in many fields of physics and engineering. The extended KdV equation including cubic nonlinearity [23] is a useful variant of the KdV equation and its linear phase speed of steady internal solitary waves is the eigenvalue of the Sturm-Liouville problem for the eigenmode.
Extending the HS equation (1) to the (2+1)-dimensional case, so-called Hirota-Satsuma-Ito (HSI) equation reads [24] u x x xt + 3(u x u t ) x + u yt + u x x = 0, where u = u(x, y, t) is a function of two scaled spatial variables x, y and the temporal variable t and represents the wave amplitude. The system (2) describes the propagation of small-amplitude surface waves with slow-changing depth and width and non-vanishing vorticity in straits or large channels. Later on, especially in the recent years, many scholars engaged in constructing the exact solutions of HSI equation by the Hirota bilinear method, which is a effective method to construct the N -soliton solution and is utilized widely for its directness, clarity and superposition. For examples, lump and lump-soliton solutions of the HSI equation were constructed of in [2]. The nonsingular complexiton solutions were obtained in [3]. The state transition from the breather to various nonlinear waves was demonstrated in [4]. A verified HSI equation was proposed in [5], and many types soliton solutions were constructed and their interactions were discussed. Some analytical lump solutions were determined in [7] and the physical behaviors involved in the solutions were also described.
The HSI equation was also generalized by adding the second-order derivative term(s) to Eq. (2) and adding constant coefficients to some or all terms in the resulting HSI equations [25][26][27]. In [25], lump solitons were presented by determining the conditions that the secondorder derivative coefficients should satisfy to ensure the existence of lump solitons. In [26], two-and three-lump solitons of the generalized equation were obtained and the hybrid solutions composing of the kink solitons, the lump wave and the breather wave. The authors of [27] focused on constructing lump and hybrid solutions of the generalized equation and analyzing the dynamical behaviors, such as the propagation orbits and velocities of lump waves and the collisions of hybrid solutions. The novelty of this paper lies in the fact that we consider the (2+1)-dimensional HSI equation with timedependent coefficients of the form where u, v and w are the analytic functions of spatial variables x, y and temporal variable t, and the coefficients α, β, ρ and γ are the functions of t. Here γ (t) is the linear phase speed and dependent on time t. It is obvious that Eq. (3) is just the HSI equation proposed in [5], as α(t) = β(t) = 1, ρ(t) = 3 and γ (t) = γ is a nonzero constant. In many branches of science, the equations with variable coefficients can give more physical information and are closer to the reality than the ones with constant coefficients. In addition, the structures of the solutions in the nonlinear equations with variable coef-ficients are more diverse. Many authors pay attention to a variety of variable-coefficient nonlinear equations, such as Hirota equation [28] and the nonlinear Schrödinger equation [29] in inhomogeneous optical fibers, the Kadomtsev-Petviashvili (KP) equation and the generalized Bogoyavlensky-Konopelchenko equation in a fluid [30,31], the KdV equations [32][33][34], and the Boiti-Leon-Manna-Pempinelli (BLMP) equations [10,35,36]. The bright one and two soliton solutions of Kundu-Eckhaus equation with variable coefficients were constructed by the bilinear method and the corresponding movements and collisions were illustrated in [37]. A fifth-order Sawada-Kotera equation with variable coefficient was investigated in [38], and the rational solutions were obtained by using the generalized unified method and many characters of solutions are also analyzed. In [36,[39][40][41], Wazwaz studied the BLMP equation, KP equation, high-dimensional KdV equation and high-dimensional sinh-Gordon equations by the same procedure. A constant-coefficient equation and a time-dependent-coefficient equation were developed first, the Painlevé integrability were confirmed next, and multiple real and multiple complex soliton solutions were obtained by complex forms of Hirota's method finally. Further, Kumar and Mohan in [42] gave many types of solutions and figures of KP equation developed in [39] by using the Hirota bilinear method. Currently, none of the research involves the variable coefficient HSI equation (3).
Motivated by the physical applications in explaining the local phenomena of nonlinear shallow water waves in ocean engineering and marine environment, we will do some working on the HSI equation with variable coefficients. In this paper, we will construct the Nsoliton solutions of the HSI equation (3) by the bilinear method. Compared with the constant coefficient HSI equation, the main difficulty is to determine whether the HSI equation with time-dependent coefficients can pass the Hirota three-soliton test [24] or possesses some kind of integrability. In the next section, the Painlevé integrability will be proved for the resulting HSI equation with constant and time-dependent coefficients (7). This is one point that the time-dependent coefficient HSI equation (7) to be studied is slightly different from other time-dependent coefficient shallow water wave equations [31,34,43]. The other different point is that the time-dependent coefficient γ (t) in the HSI equation (3) or (7) has its definite physical meaning, which is the linear phase speed.
Here is a brief overview in the next sections. In Sect. 2, we carry out the Painlevé test to an equation equivalent to the Eq. (3). Next, we give in Sect. 3 the bilinear form and the expression of N -soliton solution of the HSI equations with constant and time-dependent coefficients. Section 4 is devoted to present the different types of soliton solutions and their interaction solutions including formulas and figures for N from 1 to 4. We also do some asymptotic analyses on the exact solutions in this section. Finally, some conclusions and future research directions are proposed in Sect. 5.

Painlevé analysis and condition of integrability
The Hirota bilinear method is efficient to construct the solutions of the nonlinear equations, but the integrability may not be satisfy to the variable-coefficient equations. In this section, the main task is to conduct Painlevé test for integrability to Eq. (3). There are several approaches to test whether the given systems have Painlevé integrability, such as ARS approach [44], Kruskal's simplified method [45], WTC algorithm [46] and Lou's truncated expansion method [47].
Before Painlevé test, the transformations [48] convert Eq. (3) into the equivalent form where m is the function of x, y and t. The three-step WTC algorithm [36,39,46,49] will be used first to test the integrability of Eq. (4).
Suppose that α, β, ρ and γ are nonzero constants and the constant-coefficient (2+1)-dimensional HSI equation reads Firstly, analyze the leading order to determine the balance constant. Assume that the solutions of Eq. (5) has the Laurent expansion where m j (x, y, t) (j = 0, 1, 2, . . .) and ψ = ψ(x, y, t) are the analytic functions in some neighborhood of the singular manifold (x, y, t) = 0, and λ is an positive integer. Substituting the solution (6) into Eq. (5), we get the leading index λ = 1, the first term m 0 = 6βϕ x ρ by balancing the highest-order nonlinear and dispersive terms. Secondly, find the resonance points. Collecting the coefficients of m j (x, y, t) and factoring the efficient collected can lead to the resonance points j = −1, 1, 4, 6. The resonance point j = −1 is a general one, which corresponds to an arbitrary singular manifold. Lastly, check the compatibility at the resonances. Direct calculations not only present the explicit expressions of m 2 , m 3 and m 5 , but also prove the arbitrariness for m 1 , m 4 and m 6 . All these imply that Eq. (5) is Painlevé integrable.
Now we aim at the compatibility condition which can guarantee the integrability of Eq. (4). Repeating the above procedures, it is easy to find that Eq. (4) has the same resonances as Eq. (5). Furthermore, we get the compatibility conditions that α(t), β(t) and ρ(t) are nonzero constants and γ (t) can be any differentiable function of t. Under these conditions, Eq. (4) can pass the Painlevé test for integrability.

Bilinear form and N-soliton solution
On the basis of the results from Painlevé analysis, we turn back to the (2+1)-dimensional HSI equations with time-dependent linear phase speed given below where α, β and ρ are nonzero constants and γ (t) is a differentiable function of t. Here γ (t) represents the linear phase speed that depends on the time t, which can describe a variety of important physical phenomena. Combining the bilinear transformations and the bilinear differential operator D defined [49] as the bilinear formulism of Eq. (7) can be obtained It is easy to verify that Eq. (9) is equivalent to To maintain consistency with [5], we assume that the constant coefficients α = β = 1, ρ = 3 in (10). Applying the Hirota bilinear method, the N -soliton solution of Eq. (10) has the following same form as [5] In consideration of the time-dependent coefficient of Eq. (7), we take the phase variables as where c i (t) is the dispersive term and its derivative with respect to t satisfies and a i , b i and η 0i are the parameters to be determined. Hence, the phase shift where i, j = 1, 2, · · · , N and i < j. It can be verified that all these parameters A i j (i, j = 1, 2, · · · , N ) are constants.

Soliton solutions and their degenerations
As to the soliton solution of the HSI equation with all constant coefficients (namely α, β, ρ and γ ) have been studied in [5], we will present the soliton solutions of the HSI equation with constant and time-dependent coefficients (7) in this section.

One soliton
For N = 1, from the formula (11), the auxiliary function has the following form Hence, one-soliton solutions of Eq. (7) are Choosing the different function γ (t) will lead to the different line soliton solutions of Eq. (7). Here are three examples. For the case of the time-dependent coefficient γ (t) = t, the dispersion c 1 (t) is a function of t 2 , and the corresponding line soliton is parabolic, whose opening direction and size can be controlled by the parameters a 1 and b 1 . We name this kind of parabolic soliton as "U-shape" soliton (shown in Fig. 1). For the case of γ (t) = t 2 , c 1 (t) is a function of t 3 , and the corresponding line soliton has the shape of cubic parabola, which can be named as "S-shape" soliton (shown in Fig. 2). If the time-dependent coefficient γ (t) is a trigonometric function, the corresponding line soliton will present periodicity, shown in Fig. 3 with γ (t) = sin t. Such periodicity usually appears in at least two soliton for the constant-coefficient equations.

Two solitons
For different N , according to the formula (11), the auxiliary function f has different forms. Substituting the function f into the transformations (8) gives the N soliton solutions of Eq. (7) just as one soliton solutions. The formulas of u, v and w will be omitted for the sake of obviousness and lengthiness. But all the results of u, v and w will be demonstrated by figures. The same goes in the following parts. The parameters of a 1 , a 2 , b 1 , b 2 , η 01 and η 02 in (11)-(15) are assumed different values and the different types of soliton solutions of Eq. (7) will be obtained, such as line soliton, periodic soliton and lump soliton.
Case 1. Two-line soliton Let a 1 , a 2 , b 1 , b 2 be real number and η 01 = η 02 = 0, from the formula (11), we get where η i (i = 1, 2) and A 12 are defined as (12) and (13), respectively. The solutions u, v and w from (14) and (8) are the interaction solutions between two soliton waves. For γ (t) = t, c i (t) can be deduced as .
], (i = 1, 2).  (11) It means that the velocities of two lines [37] along The values a 1 = 1, a 2 = −1, b 1 = 2 and b 2 = 2 lead to the different propagating direction along x-axis at the time t and two back-to-back U-shape lines in Fig. 4. Either wave keeps its own amplitude and propagating direction after elastic collision at time t = 0. On one hand, we suppose that the soliton 1 whose velocity is V 1x is motionless and rewrite η 2 as Because a 2 < 0 and 1 , as t → ∞ (both +∞ and −∞), η 2 → −∞ and e η 2 → 0. The fact that f ∼ 1+e η 1 implies that two solitons approach single soliton (soliton 1) asymptotically. On the other hand, we suppose that the soliton 2 whose velocity is V 2x is motionless and rewrite η 1 as Repeating the above analysis, we get the similar results, namely, for a 1 > 0 and 1 a 2 2 +δ 2 < 1 a 2 1 +δ 1 , as t → ∞, e η 1 → 0 and two solitons approach the soliton 2 asymptotically.
For different values of a 1 , a 2 , b 1 and b 2 , the asymptotic behaviors of two line solitons can also be analyzed similarly. The results are not listed here.
Case 2. One-breather soliton Letting a 1 and a 2 , b 1 and b 2 be the conjugate complex numbers, respectively, and η 01 = η 02 = 0, we get the periodic soliton solutions of the Eq. (7). Setting where i stands for the imaginary unit, the auxiliary function in (11) is where ζ 1 and θ 1 are defined as In addition, with For γ (t) = t, assuming μ 1R = 1, μ 1I = 2, ν 1R = 2 and ν 1I = 4, the U-shape periodic soliton solutions u, v and w of Eq. (7) are presented in Fig. 5. As y = 0, the  (11) soliton wave propagates along the parabola on the x-t plane and its period is π . The parameters have effects on opening direction and size of parabola besides the period.
Case 3. One-lump soliton Taking b 1 = δ 1 a 1 , b 2 = δ 2 a 2 , a 1 = k 1 ε, a 2 = k 2 ε, and η 01 = η 02 = iπ , by the means of Taylor expansion and long wave limit, f can be rewritten as where and The expression (20) will appear again in the cases involving the lump solitons. Assuming that δ 1 = δ * 2 = δ R + iδ I and substituting (18) into (8), we obtain the rational solutions of Eq. (7) where Notice that δ R > 0 can ensure that u is nonsingular. As |x| → ∞, |y| → ∞ (namely |X | → ∞, |Y | → ∞), u will decay as O(x −2 , y −2 ) and its velocities at time t along x and y axis are respectively. The analyses of v and w can be made analogously.

Three solitons
As N increases, the different types of soliton solutions will emerge simultaneously in the solutions u, v and w. In this and next parts, the typical combinations will be illustrated by figures.
For N = 3, from the formula (11), the auxiliary function has the following form are defined as (12) and (13), respectively, and A 123 = A 12 A 13 A 23 . Case 1. Three-line soliton The principle of parameter choice is consistent with the case 1 in two solitons. If the time-dependent coefficient γ (t) is a function of t 2 , we get three U-shape line solitons. In Fig. 7, a 1 = 1, a 2 = −1, a 3 = 1, b 1 = 2, b 2 = 2 and b 3 = 1 3 , these values influence on the opening directions and sizes. It is not difficult to find that the amplitudes of u, v and w are all invariant before and after collision at the time  1I , a 3 and b 3 real, and η 01 = η 02 = η 03 = 0, the auxiliary function in (11) is

Case 2. One-line and one-breather solitons Setting
where ζ 1 , θ 1 and A 12 can be seen in Eqs.(15)- (17). In order to express the similar formulas for N = 4 briefly, we note Hence, ζ 3 and A 13 are obvious for s = 3. As a 1 and a 2 , b 1 and b 2 are conjugative, respectively, A 13 and A 23 are also conjugative and can be simply denoted as Taking γ (t) = −2t, we obtain the interaction solutions between one U-shape line and one U-shape breather, shown in Fig. 8. The two soliton waves propagate along the parabolas on the x-t plane whose opening directions are same.
Case 3. One-line and one-lump solitons The parameter choices are that b s = δ s a s , (s = 1, 2, 3), a 1 = k 1 ε, a 2 = k 2 ε, and η 01 = η 02 = iπ , η 03 = 0. Expanding f in (21) and taking limit as ε → 0, f can be rewritten as where ξ 1 , ξ 2 and 12 obey the expressions in (19)- (20), ζ 3 is well defined in (22) for s = 3, and If taking δ 1 = τ R + iτ I = δ * 2 and γ (t) = t 2 , the solutions u, v and w from (25) and (8) will reduce to the interaction solutions between one S-shape line and one S-shape lump soliton, seen in Fig. 9. The propagation of the line and the movement of the lump are both along the cubic parabola on the x-t plane.

Case 1. Two-line and one-breather solitons Letting
where ζ 1 , θ 1 , A 12 and ζ 3 , A 13 , A 23 have been noted in Eqs. (15)- (17) and (22)- (24). Taking s = 4 in (22) and (23), ζ 4 and A 14 will be clear. A 14 is further denoted as A 14 = M 2 exp(iω 2 ) and A 24 = A * 14 = M 2 exp(−iω 2 ) according to the conjugative relations. In addition,  Figure 10 shows the interaction solution of u, v and w among two U-shape line solitons and one U-shape periodic soliton for γ (t) = t. Both the parabolas' opening size and the period of the breather are controlled by the parameter choices. The different opening sizes make the structure more clear.
(I) Two-line and one-lump solitons If only taking δ 1 conjugated with δ 2 and δ 3 , δ 4 real, we will get the interaction solution among two lines and one lump soliton. The figures of u, v and w from (28) and (8) present two S-shape lines and one S-shape lump in Fig. 11 for γ (t) = 3t 2 .
(II) One-breather and one-lump solitons If δ 1 is conjugated with δ 2 and δ 3 is conjugated with δ 4 , the interaction solution between one breather and one lump is constituted. For γ (t) = 2t, the structures of u, v and w from (28) and (8) can be seen Fig. 12. It is obvious that both the breather and the lump are U-shape and their movement loca are both parabolas on the x-t plane.
Conversely, if it is supposed that the other lump is fixed whose whose phase ξ 3 is constant, Similar argument can be done and the result is that So, it is concluded that two lumps approach single lump asymptotically as t → ±∞ and there is no phase difference after collision.

Conclusions
In this paper, we focused on a (2+1)-dimensional HSI equation with variable coefficients. We determined the condition that guarantees the Painlevé integrability of the system and obtained the HSI equation with constant and time-dependent coefficients (7). For different coefficient functions, such as power functions and trigonometric functions of t, we constructed various soliton solutions and their degenerations to the HSI equations by Hirota's bilinear method and presented the expressions of solutions in detail. We also analyzed the asymptotic behaviors of two line solitons for N = 2 and the asymptotic behaviors of two lump solitons for N = 4. The results obtained above may be helpful to explain the local phenomena of nonlinear shallow water waves and the ideas here may do some help to the research of equations with time-dependent coefficients and become a starting point for further studies. As for future researches on HSI equation with variable coefficients, neglecting the integrability temporarily, we will consider more than one coefficient function or random coefficients and attempt to obtain more diverse solutions closer to the realistic situations. We will also try applying the Hirota bilinear method to construct the exact solutions of the nonlinear evolution equations with variable coefficients.