A study on soliton, lump solutions to a generalized (3+1)-dimensional Hirota--Satsuma--Ito equation

Feng-Hua Qi; Zhen-Huan Li; Shuang Li; Pan Wang

doi:10.1515/phys-2023-0272

Article Open Access

A study on soliton, lump solutions to a generalized (3+1)-dimensional Hirota--Satsuma--Ito equation

Feng-Hua Qi , Zhen-Huan Li , Shuang Li and Pan Wang

Published/Copyright: July 29, 2023

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From the journal Open Physics Volume 21 Issue 1

Abstract

In this article, through the Hirota bilinear method and long wave limit method, based on the N-solitons, we construct the multiple lump solutions of the generalized (3+1)-dimensional Hirota–Satsuma–Ito equation. Furthermore, to enhance our understanding of the solutions obtained, we further elucidate the physical implications of these solutions with three-dimensional and two-dimensional graphs. The solutions obtained might have practical applications in elucidating the dynamic behaviors of higher-dimensional systems, particularly in the study area of waves in shallow water and the study of nonlinear optics.

Keywords: Hirota bilinear; long wave limit; generalized (3+1)-dimensional Hirota–Satsuma–Ito equation; multiple lump solutions

1 Introduction

Over the past decades, nonlinear evolution equations (NLEEs) have proven to be effective in describing numerous nonlinear phenomena observed in nature, including fluid mechanics, optical fibers, solid-state physics, and plasma physics [1–5]. Researchers have tried to find exact solutions for NLEEs by introducing different methods such as Hirota bilinear method [6–9], Darboux transformations [10,11], Bäcklund transformations [12,13], Painlevé analysis [14,15], and Lie symmetry analysis [16,17]. This work contributes to a more comprehensive comprehension of both qualitative and quantitative aspects of NLEEs [18–21]. Especially, lump wave in recent years has captured considerable attention in the field of the nonlinear science, and it belongs to the category of rational function waves that display energy concentration and space localization properties [22–26]. The concept of lump waves was initially introduced by Manakov et al. in their study [27]. Later, the long wave limit method was proposed by Satsuma and Ablowitz as an efficient technique to construct the multiple lump (M-lump) solutions [28]. This method has been shown to be effective in studying lump solutions of integrable systems that can be transformed into bilinear forms. Typical examples include the Sawada–Kotera equation [29,30], Korteweg–de Vries equation [31,32], Caudrey–Dodd–Gibbon–Kotera–Sawada equation [33], and shallow water wave equation [34–37]. Inspired by the significance of lump waves, the focus of our study is the generalized (3+1)-dimensional Hirota–Satsuma–Ito (HSI) equation proposed by Singh et al. and it provides a means of describing the dynamic behavior of wave motion in fluid dynamics and shallow waters, given as [38]:

(1) Γ 1 [ 3 ( u x u t ) x + u x x x t ] + Γ 2 [ 3 ( u x u y ) x + u x x x y ] + Γ 3 u y t + Γ 4 u x x + Γ 5 u x y + Γ 6 u x t + Γ 7 u y y + Γ 8 u z z = 0 ,

where u = u ( x , y , z , t ) . Γ i ( i = 1 , 2 , 3 , … , 8 ) are the arbitrary constants, and the integrability nature and higher-order rogue wave solutions of Eq. (1) have been discussed in the study of Singh et al. [38]. It is a novel nonlinear model that has been proposed in recent years. It comprises nine distinct nonlinear soliton equations, such as (2+1)-dimensional HSI equations, (3+1)-dimensional generalized Kadomtsev–Petviashvili (KP) equations, dimensionally reduced Jimbo–Miwa equation, generalized Calogero–Bogoyavlenskii–Schiff equation, and other substantial nonlinear wave equations. Particular instances of Eq. (1) have been studied in the following manner:

(1) When Γ 2 = Γ 5 = Γ 7 = Γ 8 = 0 , Γ 1 , Γ 3 , Γ 4 , and Γ 6 are the arbitrary constants, Eq. (1) can be reduced to a generalized Hirota–Satsuma–Ito (GHSI) equation in (2+1) dimensions [36]:

(2) c 1 [ 3 ( u x u t ) x + u x x x t ] + c 2 u y t + c 3 u x x + c 4 u x t = 0 ,

the physical significance of nonlinear shallow waves, as well as the hybrid and M-lump solutions, have been examined in the study about Eq. (2) to understand the dynamic behaviors of each type of solution.

(2) When Γ 1 = Γ 4 = Γ 5 = Γ 7 = 0 , Γ 2 = Γ 3 = Γ 6 = 1 , and Γ 8 = − 1 , Eq. (1) can be transformed into a generalized KP equation in (3+1) dimensions:

(3) u x x x y + 3 ( u x u y ) x + u t x + u t y − u z z = 0 ,

which was investigated to find general phase shifts and wave frequencies by multiple exp-function algorithms in the study of Ma and Zhu [39], with the assistance of Maple.

(3) When Γ 1 = Γ 4 = Γ 5 = Γ 6 = Γ 7 = Γ 8 = 0 and Γ 2 = Γ 3 = 1 , Eq. (1) can simplify to a Korteweg–de Vries equation in (2+1) dimensions:

(4) u t y + u x x x y + 3 ( u x u y ) x = 0 ,

which was initially derived by Boiti et al. [40] using the concept of the weak Lax pair. It has been demonstrated to possess several notable properties, including a Lax pair, integrability characteristics, multiple soliton solutions, and an infinite number of conservation laws.

(4) When Γ 1 = Γ 4 = Γ 5 = Γ 6 = Γ 7 = 0 , Γ 2 = 1 , Γ 3 = 2 , and Γ 8 = − 3 , Eq. (1) can be simplified as a Jimbo–Miwa equation in (3+1) dimensions [41]:

(5) u x x x y + 3 ( u x u y ) x + 2 u y t − 3 u z z = 0 ,

which has been analyzed through the principle of linear superposition of exponential traveling waves [42], and by combining exponential traveling waves linearly, a specific sub-class of N-soliton solutions has been constructed.

(5) When Γ 1 = Γ 5 = Γ 6 = Γ 7 = Γ 8 = 0 , Γ 2 = − 1 , Γ 3 = 1 , and Γ 4 = 3 , Eq. (1) can be transformed into a Bogoyavlenskii–Kadomtsev–Petviashvili equation in (2+1) dimensions [43]:

(6) u y t − u x x x y − 3 ( u x u y ) x + 3 u x x = 0 ,

which has been studied using the method of soliton equations, based on the Clifford algebra of free fermions, vertex operators, and τ functions (or Hirota’s dependent variables).

In order to better solve and understand the physical performance of Eq. (1) in nonlinear shallow waves, we take Γ 1 = Γ 2 = α , Γ i ( i = 3 , 4 , 5 , 6 , 7 ) = β and Γ 8 = γ of Eq. (1), where α , β , and γ are constants, then obtain [38]

(7) α [ 3 ( u x u t ) x + u x x x t + 3 ( u x u y ) x + u x x x y ] + β [ u y t + u x x + u x y + u x t + u y y ] + γ u z z = 0 .

As far as we know, Singh et al. [38] have successfully derived the explicit 1-, 2-, and 3-order rogue wave solutions of Eq. (7). However, the investigation into M-lump solutions of Eq. (7) has not yet been explored. In general, Eq. (7) can be regarded as a simplified model of Eq. (1). If we could solve Eq. (7), Eq. (1) could also be solved by the same method. Therefore, we will solve Eq. (7) and aim to obtain new types of solutions in this article.

This article is structured as follows. In Section 2, we use the Hirota bilinear method to construct N-soliton solutions of Eq. (7). In Section 3, we obtain M-lump solutions by applying the long wave limit to the previously derived solutions in Section 2. Section 4 deals with conclusions.

2 N-soliton solutions of Eq. (7)

The construction of M-lump solutions for Eq. (7) involves a preliminary investigation of the N-soliton solutions, so we first study the N-soliton solutions. By the transformation u = 2 ( ln f ) x , Eq. (7) can be rephrased as a bilinear equation, given as:

(8) [ α ( D x 3 D t + D x 3 D y ) + β ( D y D t + D x 2 + D x D y + D x D t + D y 2 ) + γ D z 2 ] f ⋅ f = 0 ,

where D x , D y , D z , and D t are the bilinear derivative operators [44], which can be defined by:

(9) D x m D y n D z l D t k ( f ⋅ g ) = ∂ ∂ x − ∂ ∂ x ′ m ∂ ∂ y − ∂ ∂ y ′ n × ∂ ∂ z − ∂ ∂ z ′ l ∂ ∂ t − ∂ ∂ t ′ k f ( x , y , z , t ) × g ( x ′ , y ′ , z ′ , t ′ ) ∣ x ′ = x , y ′ = y , z ′ = z , t ′ = t .

The presence of solution u = u ( x , y , z , t ) for Eq. (7) is dependent on whether f satisfies the bilinear Eq. (8). In other words, there is a one-to-one correspondence between the solutions of Eqs. (7) and (8). The N-soliton solutions of Eq. (8) can be obtained using the Hirota bilinear method, which is given as follows [44]:

(10) f = f N = ∑ μ = 0 , 1 exp ∑ i < j N μ i μ j A i j + ∑ i = 1 N μ i η i ,

where

(11) η i = k i ( x + p i y + z + ω i t ) + η i 0 , ω i = − k i 2 p i α − β − p i β − p i 2 β − λ k i 2 α + β + p i β ,

and

(12) e A i j = − λ ( k i − k j ) 2 + α ( k i − k j ) 3 [ k i ( p i + w i ) − k j ( p j + w j ) ] + β ( − l + h + g ) λ ( k i + k j ) 2 + α ( k i + k j ) 3 [ k i ( p i + w i ) + k j ( p j + w j ) ] + β ( l + h + g ) ,

(13) l = k i k j [ ( w i + 1 ) p j + p i ( 2 p j + w j + 1 ) + w i + w j + 2 ] ,

(14) h = k i 2 [ p i ( w i + 1 ) + p i 2 + w i + 1 ] ,

(15) g = k j 2 [ p j ( w j + 1 ) + p j 2 + w j + 1 ] ,

with k i , p i , η i 0 and α , β , λ are all real constants. The notation ∑ μ = 0 , 1 refers to the sum over all possible combinations of μ i = 0 , 1 ( i = 1 , 2 , … , N ) , and the expression ∑ i < j ( N ) denotes the sum over every possible way of selecting a pair of distinct elements from a set of N elements, with the restriction i < j .

3 M-lump solutions

The focus of this section is to use the long wave limit method to generate M-lump solutions for Eq. (7) based on N-soliton solutions.

Taking η i 0 = − 1 , k i → 0 ( i = 1 , 2 , … , N ) , we have discovered that the N-soliton solutions of Eq. (7) can be expressed in a new form:

(16) f N = ∏ i = 1 N θ i + 1 2 ∑ i , j N B i j ∏ s ≠ i , j N θ s + … + 1 M ! 2 M × ∑ i , j , … , p , q N B i j B k l … B p q ︷ M ∏ m ≠ i , j , k , l , … , p , q N θ m + … ,

where

(17) θ i = x + p i y + z + − β − p i β − p i 2 β − λ β + p i β t , B i j = − 6 α ( 2 + p i + p j ) β ( p i − p j ) 2 ,

while ∑ i , j , … , p , q N represents a summation taken over every conceivable combination of i , j , … , p , q , which are taken from distinct integers ranging from 1 to N . The general M-lump solutions can be derived by applying the condition: p M + i = p i ∗ ( i = 1 , 2 , … , M ) and N = 2 M , where “ ∗ ” represents complex conjugation to Eq. (16).

To generate one-lump solution, we consider the case where N is 2, Eq. (16) becomes

(18) f 2 = θ 1 θ 2 + B 12 ,

where θ 1 = x + p 1 y + z + − β − p 1 β − p 1 2 β − λ β + p 1 β t , θ 2 = x + p 2 y + z + − β − p 2 β − p 2 2 β − λ β + p 2 β t , B 12 = − 6 α ( 2 + p 1 + p 2 ) β ( p 1 − p 2 ) 2 . If we take p 2 = p 1 ∗ = a − b I , where I is an imaginary number unit, and a and b are the real constants, then Eq. (18) is modified to

(19) f 2 = ( x ′ + a y ′ + z ) 2 + b 2 y ′ 2 + 3 ( 1 + a ) α b 2 β ,

where

(20) x ′ = x − ( 1 + 2 a ) ( β + λ ) ( 1 + 2 a + a 2 + b 2 ) β t , y ′ = y − 2 a β + a 2 β + b 2 β − λ ( 1 + 2 a + a 2 + b 2 ) β t .

By replacing the expression given in Eq. (19) into the transformation of u = 2 ( ln f ) x , we can generate a rational solution for Eq. (7).

(21) u = 2 ( ln f 2 ) x = 4 ( z + x ′ + a y ′ ) 3 ( 1 + a ) α b 2 β + b 2 y ′ 2 + ( z + x ′ + a y ′ ) 2 ,

where the parameter 3 ( 1 + a ) α b 2 β > 0 , Eq. (21) is the one-lump solution and it moves along a specific straight line y = 2 a β + a 2 β + b 2 β − λ ( β + λ ) ( 1 + 2 a ) x with the velocity v x = ( 1 + 2 a ) ( β + λ ) ( 1 + 2 a + a 2 + b 2 ) β and v y = 2 a β + a 2 β + b 2 β − λ ( 1 + 2 a + a 2 + b 2 ) β . From the observation of Figures 1 and 2, it is evident that the original shape, velocity, and amplitude of the one-lump wave without undergoing any modifications during propagation.

$Figure 1 The time evolution of the one-lump solution in setting z = 1 z=1 , a = 1 a=1 , b = 1 b=1 , and α = β = λ = 1 \alpha =\beta =\lambda =1 at (a) t = ‒ 5 t=‒5 , (b) t = 0 t=0 , and (c) t = 5 t=5 .$

Figure 1

The time evolution of the one-lump solution in setting z = 1 , a = 1 , b = 1 , and α = β = λ = 1 at (a) t = ‒ 5 , (b) t = 0 , and (c) t = 5 .

$Figure 2 Contour plots of the one-lump solution in setting z = 1 z=1 , a = 1 a=1 , b = 1 b=1 , and α = β = λ = 1 \alpha =\beta =\lambda =1 at (a) t = ‒ 5 t=‒5 , (b) t = 0 t=0 , and (c) t = 5 t=5 .$

Figure 2

Contour plots of the one-lump solution in setting z = 1 , a = 1 , b = 1 , and α = β = λ = 1 at (a) t = ‒ 5 , (b) t = 0 , and (c) t = 5 .

To generate two-lump solutions, we consider the case where N is 4, Eq. (16) becomes

(22) f 4 = θ 1 θ 2 θ 3 θ 4 + B 12 θ 3 θ 4 + B 13 θ 2 θ 4 + B 14 θ 2 θ 3 + B 23 θ 1 θ 4 + B 24 θ 1 θ 3 + B 34 θ 1 θ 2 + B 12 B 34 + B 13 B 24 + B 14 B 23 ,

where θ i = x + p i y + z + − β − p i β − p i 2 β − λ β + p i β t , B i j = − 6 α ( 2 + p i + p j ) β ( p i − p j ) 2 , ( 1 ≤ i ≤ 4 , 1 ≤ j ≤ 4 ) . By taking p 3 = p 1 ∗ = a 1 − b 1 I , p 4 = p 2 ∗ = a 2 − b 2 I and using the transformation u = 2 ( ln f 4 ) x , we derive the two-lump solutions of Eq. (7). The solutions consist of two lump waves, each of which moves along a different line, namely, y 1 = 2 a 1 β + a 1 2 β + b 1 2 β − λ ( β + λ ) ( 1 + 2 a 1 ) x and y 2 = 2 a 2 β + a 2 2 β + b 2 2 β − λ ( β + λ ) ( 1 + 2 a 2 ) x . The evolution of the two-lump waves is shown in Figures 3 and 4.

$Figure 3 Time evolution of the two-lump solutions in setting z = 1 z=1 , a 1 = 1 ∕ 2 {a}_{1}=1/2 , a 2 = 1 ∕ 4 {a}_{2}=1/4 , b 1 = b 2 = 1 {b}_{1}={b}_{2}=1 , and α = β = λ = 1 \alpha =\beta =\lambda =1 at (a) t = ‒ 5 t=‒5 , (b) t = 0 t=0 , and (c) t = 5 t=5 .$

Figure 3

Time evolution of the two-lump solutions in setting z = 1 , a 1 = 1 ∕ 2 , a 2 = 1 ∕ 4 , b 1 = b 2 = 1 , and α = β = λ = 1 at (a) t = ‒ 5 , (b) t = 0 , and (c) t = 5 .

$Figure 4 Contour plots of the two-lump solutions in setting z = 1 z=1 , a 1 = 1 ∕ 2 {a}_{1}=1/2 , a 2 = 1 ∕ 4 {a}_{2}=1/4 , b 1 = b 2 = 1 {b}_{1}={b}_{2}=1 , and α = β = λ = 1 \alpha =\beta =\lambda =1 at (a) t = ‒ 5 t=‒5 , (b) t = 0 t=0 , and (c) t = 5 t=5 .$

Figure 4

Contour plots of the two-lump solutions in setting z = 1 , a 1 = 1 ∕ 2 , a 2 = 1 ∕ 4 , b 1 = b 2 = 1 , and α = β = λ = 1 at (a) t = ‒ 5 , (b) t = 0 , and (c) t = 5 .

In the case of N = 6 , the expansion of f 6 results in a polynomial function consisting of 76 individual terms, where θ i = x + p i y + z + − β − p i β − p i 2 β − λ β + p i β t , B i j = − 6 α ( 2 + p i + p j ) β ( p i − p j ) 2 , ( 1 ≤ i ≤ 6 , 1 ≤ j ≤ 6 ) . Taking p 4 = p 1 ∗ = a 1 − b 1 I , p 5 = p 2 ∗ = a 2 − b 2 I , p 6 = p 3 ∗ = a 3 − b 3 I , and substituting these polynomial functions into u = 2 ( ln f 6 ) x , we can obtain three-lump solutions. Three linear paths are traveled by the three lump waves, namely, y 1 = 2 a 1 β + a 1 2 β + b 1 2 β − λ ( β + λ ) ( 1 + 2 a 1 ) x , y 2 = 2 a 2 β + a 2 2 β + b 2 2 β − λ ( β + λ ) ( 1 + 2 a 2 ) x , and y 3 = 2 a 3 β + a 3 2 β + b 3 2 β − λ ( β + λ ) ( 1 + 2 a 3 ) x , respectively, which can be seen in Figures 5 and 6.

$Figure 5 Time evolution of the three-lump solutions in setting z = 1 z=1 , a 1 = 1 ∕ 2 {a}_{1}=1/2 , a 2 = 1 ∕ 4 {a}_{2}=1/4 , a 3 = 0 {a}_{3}=0 , b 1 = b 2 = b 3 = 1 {b}_{1}={b}_{2}={b}_{3}=1 , and α = β = λ = 1 \alpha =\beta =\lambda =1 at (a) t = ‒ 8 t=‒8 , (b) t = 0 t=0 , and (c) t = 8 t=8 .$

Figure 5

Time evolution of the three-lump solutions in setting z = 1 , a 1 = 1 ∕ 2 , a 2 = 1 ∕ 4 , a 3 = 0 , b 1 = b 2 = b 3 = 1 , and α = β = λ = 1 at (a) t = ‒ 8 , (b) t = 0 , and (c) t = 8 .

$Figure 6 Contour plots of the three-lump solutions in setting z = 1 z=1 , a 1 = 1 ∕ 2 {a}_{1}=1/2 , a 2 = 1 ∕ 4 {a}_{2}=1/4 , a 3 = 0 {a}_{3}=0 , b 1 = b 2 = b 3 = 1 {b}_{1}={b}_{2}={b}_{3}=1 , and α = β = λ = 1 \alpha =\beta =\lambda =1 at (a) t = ‒ 8 t=‒8 , (b) t = 0 t=0 , and (c) t = 8 t=8 .$

Figure 6

Contour plots of the three-lump solutions in setting z = 1 , a 1 = 1 ∕ 2 , a 2 = 1 ∕ 4 , a 3 = 0 , b 1 = b 2 = b 3 = 1 , and α = β = λ = 1 at (a) t = ‒ 8 , (b) t = 0 , and (c) t = 8 .

4 Conclusions and discussions

In this article, we study an extended form of the Hirota–Satsuma–Ito equation, which characterizes the behavior of shallow water waves within a (3+1)-dimensional environment. In order to generate new types of solutions for Eq. (7), we use the Hirota bilinear method to construct the N-soliton solutions Eq. (10) of Eq. (7). Moreover, the one-lump solution Eq. (21), two-lump solutions ( u = 2 ( ln f 4 ) x ), and three-lump solutions ( u = 2 ( ln f 6 ) x ) of Eq. (7) are constructed by applying the long wave limit to Eq. (10). Meanwhile, in order to ensure the rigor of our study, the existence of the gained solutions has been verified and drawn with three-dimensional and two-dimensional graphs (Figures 1–6) using specific values to better reveal the dynamical characteristics of solutions for the studied equation. From the aforementioned graphics, we can observe that the one-lump wave exhibits one positive and one negative peak. In the case of the two-lump and three-lump waves, they are composed of two and three independent one-lump waves, respectively. Additionally, all of the obtained lump waves have the characteristic of lim x → ± ∞ u ( x , y , z , t ) = 0 and lim y → ± ∞ u ( x , y , z , t ) = 0 . The outcomes of this study expand the range of solution types for the (3+1)-dimensional HSI equation. The solutions obtained might have practical applications in elucidating the dynamic behaviors of higher-dimensional systems, particularly in the study area of waves in shallow water and the study of nonlinear optics.

Funding information: This study was supported by the key project of Beijing Social Science Foundation “strategic research on improving the service quality of capital logistics based on big data technology (18GLA009).”
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

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Received: 2023-04-21

Revised: 2023-06-21

Accepted: 2023-07-10

Published Online: 2023-07-29

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2023-0272

Keywords for this article

Hirota bilinear; long wave limit; generalized (3+1)-dimensional Hirota–Satsuma–Ito equation; multiple lump solutions

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