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Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39251730
71821
10.1038/s41598-024-71821-5
Article
Soliton and lump and travelling wave solutions of the (3 + 1) dimensional KPB like equation with analysis of chaotic behaviors
Gu Yongyi 1
Zhang Xiaoting zhangxiaoting1985@163.com

2
Huang Zhishang 1
Peng Liudi 1
Lai Yongkang 1
Aminakbari Najva 3
1 grid.443372.5 0000 0001 1922 9516 School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou, 510320 China
2 Department of Basic Courses Teaching, Software Engineering Institute of Guangzhou, Guangzhou, 510990 China
3 https://ror.org/05ar8rn06 grid.411863.9 0000 0001 0067 3588 School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006 China
6 9 2024
6 9 2024
2024
14 2096620 5 2024
30 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
Nonlinear evolution equations (NLEEs) have a wide range of applications in various fields, including physics, engineering, biology, economics, and more. These equations are used to describe complex systems that change over time, where the state or behavior of the system not only depends on the current situation, but may also be related to the past. A deep understanding of the solutions to NLEEs is of great significance for predicting and controlling the behavior of complex systems. The (3 + 1)-dimensional Kadomtsev-Petviashvili-Boussinesq-like (KPB-like) equation represents a partial differential equation that describes nonlinear wave phenomena in multi-dimensional spaces. This model is commonly employed in fields such as fluid dynamics, plasma physics, and nonlinear optics to study wave behaviors. This paper investigates the KPB-like equations, which have meaningful physical implications in describing nonlinear wave phenomena. Hirota bilinear method is employed to derive the bilinear form for the KPB-like equation and 1-soliton, 2-soliton, lump, and travelling wave solutions are studied to this kind of nonlinear model. In addition, the chaotic behaviors of soliton and lump solutions are explored via applying the Duffing chaotic system. To have a better understanding of dynamic structures, 3D, line, density and contour map plots of begotten results are displayed. Our results indicate the directness and effectiveness of the applied method for analyzing high-dimensional differential equations that arise in nonlinear science.

Keywords

Nonlinear evolution equations
Hirota bilinear approach
Exact solutions
Chaotic analysis
Subject terms

Mathematics and computing
Physics
http://dx.doi.org/10.13039/501100001809 National Natural Science Foundation of China 11901111 Gu Yongyi issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

In recent years, research on nonlinear partial differential equations (NLPDEs) has seen significant growth, notably in fields like oceanography and optics1–6. These equations are pivotal for describing complex nonlinear phenomena and can often be derived from Hirota bilinear7 and generalized bilinear approaches8,9. For instance, well-known equations such as the Korteweg-de Vries (KdV) and Boussinesq equations can be generated and solved using these methods. Hirota bilinear method is particularly effective in constructing various solutions, including rational solutions10,11 to these equations. This approach is also applied to nonintegrable equations12, exploring the possibilities of rational solutions and enhancing our understanding of their behavior in nonlinear science, impacting multiple disciplines including quantum mechanics, fluid dynamics, etc. This integrative approach not only streamlines the discovery of solutions but also enriches the theoretical foundation across different scientific fields.

NLPDEs have attracted the interest of numerous scientists due to their importance in accurately depicting the dynamics of different real-world systems. In recent times, many scholars in mathematics and physics have established various methods to construct and analyze exact solutions of NLPDEs. Successful methods include the exp-functions method13, Lie symmetry method14–16, generalized auxiliary equation approach17, unified method18–20, bell polynomial method21, trial function method22, ansatz method23, homotopy perturbation scheme24, semi-inverse variational principle25, F-expansion method26, Jacobi elliptic function method27, (G′/G)-expansion method28,29, tanh method30, tan-expansion method31, q-homotopy analysis method32, generalized method33, etc.

Soliton solutions34, travelling wave solutions35, and lump solutions36 are crucial in the study of NLEEs, with extensive applications across fields like optics, plasma physics, and materials science. The Kadomtsev–Petviashvili (KP) equation, introduced in 1970 by Kadomtsev and Petviashvili37, serves as a two-dimensional generalization of famous KdV equation. Soliton, lump and travelling wave solutions have always been the focus of research on the KP equation. For example, Wazwaz studied the soliton solutions of the KP equation38, Schürmann et al. investigated the travelling wave solutions of the KP equation39, and Hamid and colleagues explored the lump solutions of the KP equation40. For more details, please see studies research41–48 and so on. The Boussinesq equation, on the other hand, describes the propagation of long waves on the surface of a fluid with varying depth49,50. It takes into account both nonlinearity and dispersion, providing a more comprehensive model for wave dynamics in shallow water. Combining the KP equation and the Boussinesq equation for research has broader significance.

The KPB-like equations combine elements from both KP and Boussinesq frameworks, enabling the study of multidimensional nonlinear wave phenomena with more complex interactions. These equations are particularly relevant in fields like oceanography, where understanding the behavior of waves in multi-dimensional spaces is momentous. Recently, Sun et al.51 have combined the KP equation with the Boussinesq equation and studied a generalized (3 + 1)-dimensional ((3 + 1)-d) KPB-like equation. The generalized (3 + 1)-d KPB-like equation51 can be read1 c1uxt+c2uyt+c3utt+c432u3ux+32u2uxx+3uxuxx+3uux2+c5uyy+c6uzz=0,

where ci(i=1,…,6) are arbitrary constants. They acquired lump solutions of the above equation in the case of z=t and z=y, and discussed the analyticity and localization of the obtained solutions.

To date, the solutions of the (3 + 1)-d KPB-like equation and its generalized form remain unexplored. Considering the rich physical significance of this equation, studying its solutions is very meaningful. In this work, using Hirota’s bilinear method, the (3 + 1)-d KPB-like equation can be converted into a bilinear formalism, simplifying the exploration of soliton, lump, travelling wave solutions and chaotic behavior. We do not explore interaction solutions and conduct sensitivity analysis, which can be investigated in future study.

The paper is organized as follows. In “Soliton solutions” section, starting from a bilinear expansion, we investigated the 1-soliton and 2-soliton solutions of the (3 + 1)-d KPB-like equation, and plotted corresponding graphs. In “Lump solutions” section, we explored the variations in the lump solutions of the (3+1)-d KPB-like equation with respect to changes in t and parameters. In “Travelling wave solutions” section, we examined the variations in travelling wave solutions of the mentioned equation. In “Chaotic behaviors” section, we studied the chaotic behaviors of the 1-soliton and lump solutions. The last section summarizes and discusses the findings.

Soliton solutions

By using a dependent variable transformation2 u=2[lnf(x,y,z,t)]x=2fx(x,y,z,t)f(x,y,z,t).

Equation (1) can be mapped into3 (c1D3,xD3,t+c2D3,yD3,t+c3D3,t2+c4D3,x4+c5D3,y2+c6D3,z2)f·f=0,

where Dx,Dy, Dz, and Dt are Hirota’s billnear derivatives defined by4 Dp,x1Dp,y2Dp,z3Dp,t4f·g=∂∂x+αp∂∂x′n1∂∂y+αp∂∂y′n2∂∂z+αp∂∂z′n3∂∂t+αp∂∂t′n4f(x,y,z,t)g(x′,y′,z′,t′)|x′=x,y′=y,z′=z,t′=t,

with p is a prime number, arbitrary parameters n1, n2, n3 and n4 are all nonnegative integers, and αps=(-1)rp(s), where rp(s)≡smodp with 0≤rp(s)<p,s≥0.

This is equivalent to5 2[c1(fxtf-fxft)+c2(fytf-fyft)+c3(fttf-ft2)+3c4fxx2+c5(fyyf-fy2)+c6(fzzf-fz2)]=0.

We generalize Eq. (5) and rewrite it in the following form6 2[c1(fxtf-fxft)+c2(fytf-fyft)+c3(fttf-ft2)+3c4fxx2+c5(fyyf-2fy2)+c6(fzzf-2fz2)]=0.

According to the transformation (2), we obtain7 2[c1(fxtf-fxft)+c2(fytf-fyft)+c3(fttf-ft2)+3c4fxx2+c5(fyyf-2fy2)+c6(fzzf-2fz2)]f·ff2x=c1uxt+c2uyt+c3utt+c4(32u3ux+32u2uxx+3uxuxx+3uux2)+c5(uyy-uuy)+c6(uzz-uuz)=0.

The 1-soliton solution

With the purpose of searching one-soliton solution for the (3+1)-d KPB-like equation, supposing that f is of the form8 f=1+eη1,

where9 η1=a1(x+a2y+a3t+a4z)+a5,

with a1, a2, a3, a4 and a5 are arbitrary constants.

Substitute Eq. (8) with Eq. (9) into Eq. (6), and make all coefficients of the exponential functions to be zero, one can get10 a1=-a3c1-a32c3-a2a3c23c4,a4=-a3c1-a32c3-a2a3c2-a22c5c6.

Then substituting Eq. (8) to Eq. (10) into transformation u=2[lnf]x, yields the one-soliton solution of Eq. (7) as below11 u=2-a3c1-a32c3-a2a3c23c4e-a3c1-a32c3-a2a3c23c4ta3+ya2+z-a3c1-a32c3-a2a3c2-a22c5c6+x+a51+e-a3c1-a32c3-a2a3c23c4ta3+ya2+z-a3c1-a32c3-a2a3c2-a22c5c6+x+a5.

If setting a2=4,a3=2,a5=-5,c1=1,c2=0,c3=0,c4=-1,c5=0,c6=1, we can acquire one-kink soliton solution and give the wave shape in different planes, which is shown in Fig. 1.Figure 1 Plots of the 1-soliton solution for a2=4,a3=2,a5=-5,c1=1,c2=0,c3=0,c4=-1,c5=0,c6=1, where (a) y=0,z=0, (b) x=0,z=0, (c) t=0,z=0. (All figures are generated by using MATLAB, Version 9.14.0.2306882 (R2023a), https://www.mathworks.com/products/new_products/release2023a.html.

The 2-soliton solution

With the purpose of searching two-soliton solution for the (3 + 1)-d KPB-like equation, supposing that f is of the form12 f=1+eη1+eη2+A12eη1+η2,

where13 η1=a1(x+a2y+a3t+a4z)+a5,η2=a6(x+a7y+a8t+a9z)+a10,

with ai(i=1,…,10) and A12 are arbitrary constants.

Inserting Eqs. (12) and (13) into Eq. (6), letting all coefficients of the exponential functions to be zero, and then putting Eq. (12) with Eq. (13) into transform u=2[lnf]x, yields the following 2-soliton solution14 u=2-a6eM+a6eN)1+eM+eN+a32-6a3a8+a82eM+Na32-2a3a8+a82,

with15 M=-a6ta3+y3c4c5a6+x+a5,N=a6ta8+y3c4c5a6+x+a10,

and16 A12=a32-6a3a8+a82a32-2a3a8+a82,a1=-a6,a2=3c4c5a6,a4=0,a7=a2,a9=0,c1=-a63c4c5a3a8c2+3a3a6c4+3a6a8c4a8a3,c3=3a62c4a8a3.

If setting a3=2,a5=2.5,a6=5,a8=1,a10=1,c2=1,c4=1,c5=1,c6=1, then we have two-kink soliton solution, and the wave shapes at z=0,t=0; z=0,t=2; z=0,t=5; z=0,t=10; z=0,t=15; z=0,t=20; z=0,t=25 and z=0,t=35 are given in Fig. 2.

In Fig. 2, at t=0, the image appears strange wave, as t increases, the strange wave disappears, and from t=2 onwards, the image appears kink solution, and the kink shape becomes more and more obvious.Figure 2 Plots of the 2-soliton solution for z=0,a3=2,a5=2.5,a6=5,a8=1,a10=1,c2=1,c4=1,c5=1,c6=1, where (a1–a8) correspond to t=0, t=2, t=5, t=10, t=15, t=20, t=25 and t=35, respectively.

Lump solutions

Supposing that Eq. (6) has lump solutions as follows17 g=α1x+α2y+α3t+α4z+α5,h=α6x+α7y+α8t+α9z+α10,f=g2+h2+α11,

where αi (1≤i≤11) are constants to be determined later. Restricting c1=1, c4=1, and then inserting Eq. (17) into Eq. (6), we achieve18 α2=0,α3=0,α8=-4α9c6α4α1,α11=-3(α14+2α12α62+α64)2c6α42.

To ensure the rational analysis and localization of the function f in the (x, y)-plane, some conditions must be met: α1≠0, 2c6α42≠0, 4c6α4α92≠0, and α7≠0. These conditions lead to the quadratic function solution of Eq. (6).19 f=(α1x+α4z+α5)2+(α6x+α7y-4α9c6α4α1t+α9z+α10)2-3(α14+2α12α62+α64)2c6α42.

The solution corresponding to the (3+1)-d integrable KPB-like Eq. (7) is given by:20 u=22(xα1+zα4+α5)α1+2-4α9c6α4α1t+xα6+yα7+zα9+α10α6xα1+zα4+α52+-4α9c6α4α1t+xα6+yα7+zα9+α102-3α14+2α12α62+α642α42c6.

If setting the parameters α1=1,α4=1,α5=1,α6=0.6,α7=1,α9=1,α10=1,c6=-1, we can derive lump solutions that produce unusual wave patterns. These patterns can be observed at coordinates z=0,t=0; z=0,t=1; z=0,t=1.5 and z=0,t=2.5. These specific instances are depicted in Fig. 3. Additionally, if setting α1=0.1,α4=1,α5=1,α6=6,α7=1,α9=1,α10=1,c6=1, we can obtain corresponding lump solutions and give the breather type waves that are propagating through the position axis at z=0,t=0; z=0,t=0.5; z=0,t=1 and z=0,t=1.5, which are shown in Fig. 4. Finally, By setting α1=1,α4=1,α5=1,α6=0.6,α7=1,α9=1,α10=1,c6=1, we can obtain corresponding lump solutions and give the weird waves that have multiple peaks at z=0,t=0; z=0,t=0.5; z=0,t=1 and z=0,t=1.5, which are represented in Fig. 5.

In Fig. 3, comparing Figures (a1), (a2), (a3) and (a4) reveals that as time t increases, both lumps exhibit a tendency to move outward and gradually dissipate. Figures (b1), (b2), (b3) and (b4) show the corresponding density maps of these 3D plots in the xoy plane.

In Fig. 4, a comparison of Figure (a1) through Figure (a4) shows the dynamics of singular waves over time. As t increases, the singular waves on the left side gradually converge inward, while those on the right diverge outward and eventually fade, causing the desirable range of u to narrow. Correspondingly, Figure (b1) through Figure (b4) display density maps that illustrate these changes in the 3D plots. These density maps track the inward movement of the singular wave on the left and the outward dispersion and fading of the wave on the right.

In Fig. 5, a comparison of Figures (a1), (a2), (a3) and (a4) shows that as t increases, the strange wave clusters gradually move outwards and the maximum peak of the strange waves varies in the xoy plane; the density maps shown in Figures (b1) to (b4) illustrate these variations in the 3D plots. From Figs. 4 and 5, it can be seen that when c6>0, singular solutions may occur.Figure 3 Plots of the lump solutions (20) for α1=1,α4=1,α5=1,α6=0.6,α7=1,α9=1,α10=1,c6=-1, where (a1,b1) z=0,t=0, (a2,b2) z=0,t=1, (a3,b3) z=0,t=1.5 and (a4,b4) z=0,t=2.5.

Figure 4 Plots of the lump solutions (20) for α1=0.1,α4=1,α5=1,α6=6,α7=1,α9=1,α10=1,c6=1, where (a1,b1) z=0,t=0, (a2,b2) z=0,t=0.5, (a3,b3) z=0,t=1 and (a4,b4) z=0,t=1.5.

Figure 5 Plots of the lump solutions (20) for α1=1,α4=1,α5=1,α6=0.6,α7=1,α9=1,α10=1,c6=1, where (a1,b1) z=0,t=0, (a2,b2) z=0,t=0.5, (a3,b3) z=0,t=1 and (a4,b4) z=0,t=1.5.

Travelling wave solutions

In this part, we use the (G′/G)-expansion method52 to acquire exact travelling wave solutions for Eq. (1). The dynamics of some solutions will be demonstrated by computer simulation.

We utilize the following transform21 ξ=px+qy+rz+st,u(x,y,z,t)=w(ξ),

where p, q, r, s are constants. Applying (21) to (1) and integrating once, we can obtain an ordinary differential equation22 c1ps+c2qs+c3s2+c5q2+c6r2wξ+c432p3wξ2+32p2wξw2+38pw4+δ=0,

where δ is an integrate constant. By balancing wξ2,wξw2 and w4, in (22), we have N=1. Thus, we assume the solution is of the form23 w(ξ)=b0+b1G′(ξ)G(ξ),

where b1≠0, and24 G′′(ξ)+λG′(ξ)+μG(ξ)=0.

Then, inserting (21) together with (23) and (24) into (22), we gain the following algebraic equations by collecting the coefficients of G′(ξ)G(ξ)i,3pc42p-b12b12=0,12pλp-b0c42p-b1b12=0,2b1(6λ2p3b1c4-12λp2b0b1c4+12μp3b1c4-6p2b12c4μ-6p2c4b02+9pc4b02b1-4psc1-4c5q2-4c2qs-4c6r2-4c3s2)=0,4b1(6λμp3b1c4-3p2c4λb02-6μp2b0b1c4+3pb03c4-2λpsc1-2λq2c5-2λqsc2-2λr2c6-2λs2c3)=0,12μ2p3b12c4-12μp2b02b1c4+3c4pb04-8μpsb1c1-8μq2b1c5-8μqsb1c2-8μr2b1c6-8μs2b1c3+8δ=0.

We get the following results by solving the above equations with the aid of Maple.

Family 1: b1=2p,r=±3c4-λ2+4μp4-psc1-q2c5-qsc2-c3s2c6,b0=λp,δ=-6-λ24+μ2p5c4.

With above parameters, the solutions to Eq. (1) are given as follows:

Case 1. If λ2-4μ>0, we gain25 u1=pλ2-4μC1sinhλ2-4μ2ξ+C2coshλ2-4μ2ξC1coshλ2-4μ2ξ+C2sinhλ2-4μ2ξ,

where ξ=px+qy±3c4-λ2+4μp4-psc1-q2c5-qsc2-c3s2c6z+st, and C1,C2 are constants. By choosing specific values of C1 and C2, we can extract diverse known solutions, such as If C1=0,C2≠0, then u11=pλ2-4μcothλ2-4μ2ξ.

If C1≠0,C2=0, then u12=pλ2-4μtanhλ2-4μ2ξ.

If C2≠0,C12<C22, then u13=pλ2-4μtanhξ1+λ2-4μ2ξ,

where ξ1=tanh-1C1C2.

If C1≠0,C22<C12, then u14=pλ2-4μcothξ1+λ2-4μ2ξ,

where ξ1=coth-1C2C1.

Case 2. If λ2-4μ<0, we get26 u2=p4μ-λ2-C1sin4μ-λ22ξ+C2cos4μ-λ22ξC1cos4μ-λ22ξ+C2sin4μ-λ22ξ,

where ξ=px+qy±3c4-λ2+4μp4-psc1-q2c5-qsc2-c3s2c6z+st, and C1,C2 are constants.

Case 3. If λ2-4μ=0, we see27 u3=2pC2C1+C2ξ,

where ξ=px+qy±-psc1-q2c5-qsc2-c3s2c6z+st, and C1,C2 are constants.

Family 2: b1=2p,r2=3c4-λ2+4μp4-psc1-q2c5-qsc2-c3s2c6,b0=λ±-λ2+4μp,δ=-6-λ24+μ2p5c4.

With above parameters, the solutions to Eq. (1) are represented as follows:

Case 4. If λ2-4μ<0, we obtain28 u4=p4μ-λ2±1+-C1sin4μ-λ22ξ+C2cos4μ-λ22ξC1cos4μ-λ22ξ+C2sin4μ-λ22ξ,

where ξ=px+qy±3c4-λ2+4μp4-psc1-q2c5-qsc2-c3s2c6z+st, and C1,C2 are constants. By considering particular values of C1 and C2, we can deduce various known solutions, for example, If C1=C1=1, then u41=p4μ-λ2±1+-sin4μ-λ22ξ+cos4μ-λ22ξcos4μ-λ22ξ+sin4μ-λ22ξ.

If C1=0,C2≠0, then u42=p4μ-λ2±1+cot4μ-λ22ξ.

If C1≠0,C2=0, then u43=p4μ-λ2±1+tan4μ-λ22ξ.

If C2≠0,C12<C22, then u44=p4μ-λ2±1+cotξ1+4μ-λ22ξ,

where ξ1=tan-1C1C2.

If C1≠0,C22<C12, then u45=p4μ-λ2±1+tanξ1-4μ-λ22ξ,

where ξ1=tan-1C2C1.

Case 5. If λ2-4μ=0, we acquire29 u5=2pC2C1+C2ξ,

where ξ=px+qy±-psc1-q2c5-qsc2-c3s2c6z+st, and C1,C2 are constants.

Since u2 and u4 differ by a constant term and u3, u5, have the same form, we only list some figures about u2. Figure 6 illustrates the singular-periodic soliton behavior of solution u2, with using y=2,p=2,q=2,s=1,λ=1,μ=4,c1=2,c2=1,c3=2,c4=2,c5=3,c6=1,C1=2,C2=3.Figure 6 Plot of solution u2 where ξ satisfies “+” of “±”, for y=2,p=2,q=2,s=1,λ=1,μ=4,c1=2,c2=1,c3=2,c4=2,c5=3,c6=1,C1=2,C2=3, and (a) z=-1, (b) z=0, (c) z=1.

Chaotic behaviors

In a damped elastic system, the potential energy function adheres to the specified governing equation.30 d2Xdt2+αdXdt+βX3=0.

When subjected to a periodic external force, the system in question exhibits chaotic dynamics, resulting in a modification of its governing equation to:31 d2Xdt2+αdXdt+βX3=ρcost.

This particular formulation represents the renowned Duffing chaotic system. To instantiate this system, we assign specific values to its parameters as follows:α=0.05,β=5.5,ρ=7.5.

Additionally, we establish the initial conditions for the system:X′(0)=1,X(0)=1.

With these parameter settings and initial conditions, the Duffing chaotic system is fully defined and poised to display its characteristic chaotic behavior when subjected to the periodic external force.

A chaotic solution for the Duffing system is demonstrated, as depicted in Fig. 7. Figure 7a shows the relationship between X and dXdt. The trajectory of the system’s motion is clearly non-periodic and stochastic, consistently following varied paths while orbiting around one or two attractors, thereby displaying characteristics of both determinism and randomness, blending these two aspects. Figure 7b,c illustrate, respectively, the relationship between t and X, and dXdt, presenting non-periodic waveforms with varying amplitudes and wave widths.Figure 7 Plots of X-dXdt, t-X and t-dXdt.

Chaotic behavior of 1-soliton solution

If the variable x in Eq. (11) is assigned the chaotic derivative dXdt from the Duffing system (31), an alternative unidirectional chaotic Dromion soliton structure in the x direction is formed, as seen in Fig. 8.

Replacing the variable y in Eq. (11) with the chaotic solution X from the Duffing system (31) produces a unidirectional chaotic Dromion soliton structure across both the y direction, as depicted in Fig. 9.

By setting the variable y in Eq. (11) as the chaotic derivative dXdt from the Duffing system (31), another unidirectional chaotic Dromion soliton structure emerges in the y direction, showcased in Fig. 10.Figure 8 Plots of 1-soliton solution where x assumes the chaotic solution dXdt from the Duffing system (20) , with parameters t=0,z=0,a2=6,a3=1,a5=1,c1=1,c2=1,c3=1,c4=-10,c5=1,c6=1.

Figure 9 Plots of 1-soliton solution where y assumes the chaotic solution X from the Duffing system (20) , with parameters t=0,z=0,a2=6,a3=1,a5=1,c1=1,c2=1,c3=1,c4=-10,c5=1,c6=1.

Figure 10 Plots of 1-soliton solution where y assumes the chaotic solution dXdt from the Duffing system (20) , with parameters t=0,z=0,a2=6,a3=1,a5=1,c1=1,c2=1,c3=1,c4=-10,c5=1,c6=1.

Chaotic behavior of lump solution

By substituting the variable x in Eq. (20) with the chaotic solution X from the Duffing system (31), one can derive a unidirectional chaotic Dromion soliton structure in the x direction, as shown in Fig. 11. This unique chaotic structure consists of a sequence of vibrating Dromion solitons, each distinguished by its size, shape, orientation, and other distinct features. The key characteristic of this structure is that no two solitons are identical, which exemplifies the fundamental nature of chaos.

If the variable x in Eq. (20) is selected as the chaotic derivative dXdt from the Duffing system (31), a different unidirectional chaotic Dromion soliton structure in the x direction emerges, as illustrated in Fig. 12.

Substituting the variable y in Eq. (20) with the chaotic solution X of the Duffing system (31) results in a unidirectional chaotic Dromion soliton structure in the y direction, as depicted in Fig. 13.

Choosing the variable y in Eq. (20) to be the chaotic derivative dXdt of the Duffing system (31) leads to another unidirectional chaotic Dromion soliton structure in the y direction, as shown in Fig. 14.Figure 11 Plots of lump solutions where x assumes the chaotic solution X from the Duffing system (20) , with parameters α1=1,α4=1,α5=1,α6=0.6,α7=1,α9=1,α10=1,c6=-1. These plots correspond to the conditions: (a1,b1) at z=0,t=0, (a2,b2) at z=0,t=1 and (a3,b3) at z=0,t=2.

Figure 12 Plots of lump solutions where x assumes the chaotic solution dXdt from the Duffing system (20) , with parameters α1=1,α4=1,α5=1,α6=0.6,α7=1,α9=1,α10=1,c6=-1. These plots correspond to the conditions: (a1,b1) at z=0,t=0, (a2,b2) at z=0,t=1 and (a3,b3) at z=0,t=2.

Figure 13 Plots of lump solutions where y assumes the chaotic solution X from the Duffing system (20) , with parameters α1=1,α4=1,α5=1,α6=0.6,α7=1,α9=1,α10=1,c6=-1. These plots correspond to the conditions: (a1,b1) at z=0,t=0, (a2,b2) at z=0,t=1 and (a3,b3) at z=0,t=2.

Figure 14 Plots of lump solutions where y assumes the chaotic solution dXdt from the Duffing system (20) , with parameters α1=1,α4=1,α5=1,α6=0.6,α7=1,α9=1,α10=1,c6=-1. These plots correspond to the conditions: (a1,b1) at z=0,t=0, (a2,b2) at z=0,t=1 and (a3,b3) at z=0,t=2.

Discussion and conclusion

The (3 + 1)-d KPB-like equations are a class of NLPDEs that extend the traditional KP and Boussinesq equations, which are derived from fundamental principles governing fluid dynamics and wave propagation. The solutions to these equations, including lump solutions, provide insights into localized wave structures that can arise in various contexts. The study of KPB-like equations contributes to a deeper understanding of nonlinear wave phenomena and their manifestations in different physical settings. In Ref.51, Sun et al. proposed a generalized (3 + 1)-d KPB-like equation and obtained its lump solutions in the restrictive condition of z=t and z=y. In this work, we studied the solutions of equations without the mentioned constraints. The equations studied have a wide range, enriching the research on the KPB-like equations.

In this paper, we applied the Hirota bilinear method to derive the bilinear form of the (3 + 1)-d KPB-like equation. Soliton solutions, including one- and two-soliton solutions, lump solutions are achieved via the Hirota bilinear form and visually presented. Travelling wave solutions of the (3 + 1)-d KPB-like equation are acquire by the (G′/G)-expansion method. The chaotic behaviors of the obtained solutions are investigated by implementing the Duffing chaotic system. We explored the evolution of these solutions under different parameters. Additionally, using Matlab, we illustrated the physical structure and characteristics of these solutions through three-dimensional, two-dimensional, and contour images. To our knowledge, these solutions have not been observed in other literature. The obtained solutions have potential values in the research of wave behaviors. The strategies employed in this study are readily applicable to other nonlinear evolution equations that arise in mathematical physics and nonlinear wave phenomena.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (11901111, 12171232).

Author contributions

Writing—original draft, Methodology, Supervision, Writing—review & editing: Yongyi Gu, Xiaoting Zhang. Formal analysis, Validation, Writing—original draft: Zhishang Huang. Software, Methodology, Resources: Liudi Peng, Yongkang Lai. Conceptualization, Validation, Writing—review & editing: Najva Aminakbari.

Data availability

Data is provided within the manuscript.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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