Stability Analysis of the Rational Solutions, Periodic Cross-Rational Solutions, Rational Kink Cross-Solutions, and Homoclinic Breather Solutions to the KdV Dynamical Equation with Constant Coefficients and Their Applications

Abstract

We explore various analytical rational solutions with symbolic computation using the ansatz transformation functions. We gain a variety of rational solutions such as M-shaped rational solutions (MSRs), periodic cross-rationals (PCRs), multi-wave solutions, rational kink cross-solutions (RKCs), and homoclinic breather solutions (HBs), and by using the appropriate values for the relevant parameters, their dynamics are visualized in figures. Additionally, two different types of interactions between MSRs and kink waves are analyzed. Furthermore, we examine the stability of the obtained solutions and create a corresponding table. We analyze the stability of these solutions and the movement role of the wave by making graphs as two-dimensional, three-dimensional and density graphs as well as contour visual and stream plots.

1. Introduction

A partial differential equation (PDE) depicts the relationships between several partial derivatives of a variety of multivariate functions. Physics and engineering are two mathematics-based sciences that frequently use PDEs [1,2,3,4]. The fundamentals of contemporary scientific logic are formed by them for several ideas, including heat, sound, diffusion, electrodynamics [5,6,7], electrostatics, elastic, hydrodynamics, and quantum mechanics [8,9,10]. A wide range of scientific fields are interested in studying nonlinear wave phenomena [11,12,13,14]. This has to do with comprehending actual water waves, the way light interacts with matter, how optical fibers transmit light, how traffic moves, how earthquakes happen, and how galaxies grow. Nonlinear wave theory is a recent mathematical field that commonly investigates asymptotic regimes (including fluctuating over several scales, high frequencies, or large amplitudes) that are difficult to approach through numerical simulations [15,16,17,18,19].

The (2+1)-dimensional KdV equation was taken into consideration in this work. For nonlinear partial differential equation (NLPDE) solutions, several techniques have been used in the literature. The pursuit of accurate NLPDE solutions is crucial for comprehending nonlinear physical phenomena [20,21,22,23,24,25,26]. For instance, kink-shaped tanh solutions and bell-shaped sech solutions are frequently used to simulate the nonlinear wave phenomena that are observed in optical fibers, fluid dynamics, and plasma [27,28,29,30,31]. Numerous authors who have an interest in nonlinear physical phenomena have recently looked at the exact solutions of NLPDEs. These authors have provided a large number of effective techniques for the instance inverse scattering technique [32], Darboux transform [33], F-expansion scheme [34], generalized Riccati equation [35], Painleve expansion technique [36], Backlund transform [37], exp-function expansion mechanism [38], extended tanh function approach [39,40], and $(G^{\prime }/G)$-expansion method [41].

The KdV equation with constant coefficients in the (2+1)-dimensional space is given by [42]

$$\begin{array}{cc}& {\mathsf{\Gamma }}_{ty}+{\mathsf{\Gamma }}_{xxxy}+\delta {\mathsf{\Gamma }}_{yx}{\mathsf{\Gamma }}_x+\delta {\mathsf{\Gamma }}_y{\mathsf{\Gamma }}_{xx}+\eta {\mathsf{\Gamma }}_{xx}+\lambda {\mathsf{\Gamma }}_{yy}=0,\end{array}$$

(1)

$$\begin{array}{cc}& \mathsf{\Gamma }(x,y,t)=\Lambda \left(\epsilon \right),\epsilon =x-\rho y-ct.\end{array}$$

(2)

By utilizing the above transformation in Equation (1), we obtain

$$\begin{array}{cc}& \rho c{\Lambda }^{″}-\rho {\Lambda }^{⁗}-2\delta \rho {\Lambda }^{\prime }{\Lambda }^{″}+\eta {\Lambda }^{″}+\lambda {\rho }^2{\Lambda }^{″}=0,\end{array}$$

(3)

If the equation is integrated once into the final scenario, then the following equation is produced:

$$\begin{array}{cc}& (\rho c+\eta +\lambda {\rho }^2){\Lambda }^{\prime }-\delta \rho {\left({\Lambda }^{\prime }\right)}^2-\rho {\Lambda }^{‴}=0.\end{array}$$

(4)

Numerous researchers have worked on the governing model. For instance, Wazwaz et al. examined the Painleve test to look at the conditions for compatibility in order to make sure that the suggested model could be integrated [42]. Ma obtained N-soliton solutions for the (2+1)-dimensional KdV equation, the Kadomtsev–Petviashvili equation, and the (2+1)-dimensional Hirota–Satsuma–Ito equation by analyzing the Hirota N-soliton conditions [43]. Ma et al. presented the diversity of interaction solutions to the (2+1)-dimensional Ito equation, such as the combined multi-wave solutions which were analyzed in [44]. Zayed et al. investigated a two-variable $(\frac{G^{\prime }}{G},\frac{1}{G})$ expansion technique for the nonlinear KdV equation [45]. By using the symbolic computation with various ansatz transformations for Equation (1), this manuscript aims to evaluate MSR and their interactions with one and two kink waves, PCRs, RKCs, multi-wave solutions, and HBs.

This article is set up as follows. In Section 2, we assess the MSRs and examine their associated 3D visualizations. Section 3 provides a concise analysis of the interaction of MSRs with a single exponential function along several 3D, 2D, and contour profiles. In Section 4, we calculate the interaction solution of M-shaped double exponential functions for the provided model using the necessary 3D, 2D, and contour plots. Section 5 evaluates RKCs and their findings for various parameters. In Section 6, we will discuss PCRs, and we will draw some associated graphs. In Section 7, we will explore multi-wave solutions and make graphical visuals with suitable parameters. In Section 8, we will study the interactional solution of MSRs with a rogue wave solution. HBs and their pictorial representations are dispatched in Section 9. The stability property of the solutions is covered in Section 10, along with how it applies to all solutions that were found. In Section 11, we discuss our findings and debate them. Finally, in Section 12, we make our final observations.

2. Rational $\mathit{M}$-Shaped Solution (MSRs)

Let us use the transformation [46]

$$\begin{array}{c}\hfill \Lambda \left(\epsilon \right)=2{\left(ln\omega \right)}_{\epsilon },\end{array}$$

(5)

If this transformation is substituted into Equation (5), hten the bilinear form shown below is obtained:

$$\begin{array}{cc}& -2\eta {\omega }^2{\omega }^{\prime 2}-2c\rho {\omega }^2{\omega }^{\prime 2}-2\lambda {\rho }^2{\omega }^2{\omega }^{\prime 2}+12{\rho }^4-4\delta \rho {\omega }^{\prime 4}+2\eta {\omega }^3{\omega }^{″}+2c\rho {\omega }^3{\omega }^{″}+2\lambda {\rho }^2{\omega }^3{\omega }^{″}-\\ & 24\rho \omega {\omega }^{\prime 2}{\omega }^{″}+8\delta \rho \omega {\omega }^{\prime 2}+6\rho {\omega }^2{\omega }^{″2}-4\delta \rho {\omega }^2{\omega }^{″2}+8\rho {\omega }^2{\omega }^{\prime }{\omega }^{″}-2\rho {\omega }^3{\omega }^{⁗}=0.\end{array}$$

(6)

Now, we use this bilinear form to evaluate various rational and interactional solutions. For MSRs, we consider ‘ω’ as [47]

$$\begin{array}{cc}& \omega ={\epsilon }_1^2+{\epsilon }_2^2+m_5,\end{array}$$

(7)

where

$$\begin{array}{c}\hfill {\epsilon }_1=m_1\epsilon +m_2,{\epsilon }_2=m_3\epsilon +m_4.\end{array}$$

However, $m_j(1\le j\le 5)$ represents any constants. By using Equation (7) in Equation (6) and collecting the coefficients of $\epsilon$, we then solve the system of equations to find the values of the parameters:

$$\begin{array}{cc}& \delta =\frac{-13}{2},m_1=-1,m_3=m_2=0,m_5=\frac{-(\lambda {\rho }^2{m}_4^2+\rho m_4^{2}c+\eta m_4^2-36\rho )}{\lambda {\rho }^2+\rho c\eta }.\end{array}$$

(8)

Via the above parametric values, we have

$$\begin{array}{c}\hfill \omega =m_4^2+{\epsilon }^2+\frac{(-m_4^2\eta +36\rho -m_4^{2}c\rho -m_4^2\lambda )}{\eta +c\rho +\lambda {\rho }^2}.\end{array}$$

(9)

By using Equation (9) in Equation (5), we obtain

$$\begin{array}{c}\hfill \Lambda \left(\epsilon \right)=\frac{4\epsilon }{m_4^2+{\epsilon }^2+\frac{(-m_4^2\eta +36\rho -m_4^{2}c\rho -\lambda {\rho }^2)}{\eta +c\rho +\lambda {\rho }^2}}.\end{array}$$

(10)

By replacing Equation (2) with Equation (10), we obtain the M-shape rational solution of Equation (1):

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_1(x,y,t)=\frac{4(-ct+x-y\rho )}{m_4^2+{(-ct+x-y\rho )}^2+\frac{(-m_4^2\eta +36\rho -m_4^{2}c\rho -\lambda {\rho }^2)}{\eta +c\rho +\lambda {\rho }^2}}.\end{array}$$

(11)

Equation (11) specifies the MSRs, and $\mathsf{\Gamma }(x,y,t)$ is graphed with $\eta =-1,c=2,$ $\rho =2,$ $\lambda =2$, and $m_4=9.$

3. MSR with a One-Kink Solution

In this section, we explore MSRs with kink rational solutions using the following transformation [48]

$$\begin{array}{cc}& \omega ={\kappa }_1^2+{\kappa }_2^2+z_1{\kappa }_3,\end{array}$$

(12)

where

$$\begin{array}{cc}& {\kappa }_1=g_1\epsilon +g_2,{\kappa }_2=g_3\epsilon +g_4,{\kappa }_3=g_5\epsilon +g_6.\end{array}$$

However, $g_j(1\le j\le 6)$, where all are assumed to be parameters. By inserting Equation (12) into Equation (6), and by using mathematica, we can evaluate all the coefficients $\epsilon$. We obtain equations which provide the following values:

$$\begin{array}{cc}& g_2=\frac{-1}{2}{g}_1\frac{(4\delta +11)}{\delta \sqrt{\frac{-(-\lambda {\rho }^2-\rho c-\eta )}{\delta }}},g_5=\sqrt{\frac{-(-\lambda {\rho }^2-\rho c-\eta )}{\delta }},g_3=0.\end{array}$$

(13)

By utilizing the above values, we obtain

$$\begin{array}{c}\hfill \omega =g_4^2+e^{g_6+\epsilon \sqrt{c+\frac{\eta }{\rho }+\lambda \rho }}{z}_1+\left(g_1\epsilon -\frac{g_1(11+4\delta )}{2\delta \sqrt{c+\frac{\eta }{\rho }+\lambda \rho }}\right).\end{array}$$

(14)

Inserting Equation (14) into Equation (5) yields

$$\begin{array}{c}\hfill \Lambda \left(\epsilon \right)=\frac{2\left(e^{g_6+\epsilon \sqrt{c+\frac{\eta }{\rho }+\lambda \rho }}{z}_1\sqrt{c+\frac{\eta }{\rho }+\rho }+2g_1\left(g_1\epsilon -\frac{g_1(11+4\delta )}{2\delta \sqrt{c+\frac{\eta }{\rho }+\lambda \rho }}\right)\right)}{g_4^2+e^{g_6+\epsilon \sqrt{c+\frac{\eta }{\rho }+\lambda \rho }}{z}_1+\left(g_1\epsilon -\frac{g_1(11+4\delta )}{2\delta \sqrt{c+\frac{\eta }{\rho }+\lambda \rho }}\right)}.\end{array}$$

(15)

By substituting Equation (15) into Equation (2), we achieve the MSR with a one-kink solution for Equation (1):

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_2(x,y,t)=\frac{2\left(e^{g_6+(-ct+x-y\rho )\sqrt{c+\frac{\eta }{\rho }+\lambda \rho }}{z}_1\sqrt{c+\frac{\eta }{\rho }+\rho }+2g_1\left(g_1(-ct+x-y\rho )-\frac{g_1(11+4\delta )}{2\delta \sqrt{c+\frac{\eta }{\rho }+\lambda \rho }}\right)\right)}{g_4^2+e^{g_6+(-ct+x-y\rho )\sqrt{c+\frac{\eta }{\rho }+\lambda \rho }}{z}_1+\left(g_1(-ct+x-y\rho )-\frac{g_1(11+4\delta )}{2\delta \sqrt{c+\frac{\eta }{\rho }+\lambda \rho }}\right)}.\end{array}$$

(16)

4. MSR with a Two-Kink Solution

In this section, we work on MSRs with double kinks which consist of two exponential functions. We consider [2]

$$\begin{array}{cc}& \omega ={\zeta }_1^2+{\zeta }_2^2+z_1{e}^{{\zeta }_3}+z_2{e}^{-{\zeta }_4},\end{array}$$

(17)

where

$$\begin{array}{cc}& {\zeta }_1=u_1\epsilon +u_2,{\zeta }_2=u_3\epsilon +u_4,{\zeta }_3=u_5\epsilon +u_6,{\zeta }_4=u_7\epsilon +u_8.\end{array}$$

However, $u_j(1\le j\le 8)$ are all constants. We put Equation (17) into Equation (6) using mathematica and calculated the coefficients of $\epsilon$ and the exponential functions. We obtain a system of equations that provides the values of the constants:

$$\begin{array}{cc}& u_1=\frac{1}{5}\sqrt{2}\sqrt{5},u_7=\sqrt{\frac{-1}{2}\frac{(\lambda {\rho }^2+\rho c+\eta )}{\rho }},u_2=\frac{-8}{5}\frac{(\delta +2)\sqrt{2}\sqrt{5}}{(4\delta +3)\sqrt{\frac{-1}{2}\frac{({\rho }^2+\rho c+\eta )}{\rho }}},\\ & u_4=\frac{\sqrt{\frac{-(-624{\delta }^2\rho -3336\delta \rho -3651\rho )}{20\lambda {\rho }^2+20\rho c+20\eta }}}{(4\delta +3)},u_8=u_3=0,u_5=\sqrt{\frac{(-\rho {\rho }^2-\rho c-\eta )}{2\rho }}.\end{array}$$

(18)

Using the obtained values in Equation (17), we obtain

$$\begin{array}{cc}& \omega =e^{u_6+\frac{\epsilon \sqrt{\frac{-\eta +c\rho +{\rho }^3}{\rho }}}{\sqrt{2}}}{z}_1+e^{-u_8-\frac{\epsilon \sqrt{\frac{-\eta +\rho (c+\lambda \rho )}{\rho }}}{\sqrt{2}}}{z}_2+{\left(u_3\epsilon +\frac{2\sqrt{\frac{3}{5}}\sqrt{\frac{(1217+1112\delta +208{\delta }^2)\rho }{\eta +\rho (c+\lambda \rho )}}}{(3+4\delta )}\right)}^2\\ & +{\left(\sqrt{\frac{2}{5}}\epsilon -\frac{16(2+\delta )}{\sqrt{5}(3+4\delta )\sqrt{\frac{-\eta +\rho (c+\lambda \rho )}{\rho }}}\right)}^2.\end{array}$$

(19)

Inserting Equation (19) into Equation (5) gives

$$\begin{array}{c}\hfill \Lambda \left(\epsilon \right)=\frac{2\left(\frac{4}{5}\epsilon +\frac{e^{u_6+\frac{\epsilon \vartheta }{\sqrt{2}}}{z}_1\vartheta }{\sqrt{2}}-\frac{6\sqrt{2}(2+\delta )}{5(3+4\delta )\vartheta }-\frac{e^{-u_8-\frac{\epsilon \vartheta }{\sqrt{2}}}{z}_2\vartheta }{\sqrt{2}}+2u_3\mu \right)}{e^{u_6+\frac{\epsilon \vartheta }{\sqrt{2}}}{z}_1+e^{-u_8-\frac{\epsilon \vartheta }{\sqrt{2}}}{z}_2+{\left(u_3\epsilon +\frac{2\sqrt{\frac{3}{5}}\sqrt{\frac{(1217+1112\delta +208{\delta }^2)\rho }{\eta +\rho (c+\lambda \rho )}}}{(3+4\delta )}\right)}^2+{\left(\sqrt{\frac{2}{5}}\epsilon -\frac{16(2+\delta )}{\sqrt{5}(3+4\delta )\vartheta }\right)}^2},\end{array}$$

(20)

where

$$\begin{array}{c}\hfill \mu =\left(u_3\epsilon +\frac{2\sqrt{\frac{3}{5}}\sqrt{\frac{(1217+1112\delta +208{\delta }^2)\rho }{\eta +\rho (c+\lambda \rho )}}}{(3+4\delta )}\right),\vartheta =\sqrt{\frac{-\eta +c\rho +{\rho }^3}{\rho }}.\end{array}$$

Inserting Equation (20) into Equation (2) gives the MSR with a double-kink solution to Equation (1):

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_3(x,y,t)=\frac{2\left(-\frac{4}{5}(ct-x+y\rho )+\frac{e^{u_6+\frac{(ct-x+y\rho )\vartheta }{\sqrt{2}}}{z}_1\vartheta }{\sqrt{2}}-\frac{6\sqrt{2}(2+\delta )}{5(3+4\delta )\vartheta }-\frac{e^{-u_8-\frac{(ct-x+y\rho )\vartheta }{\sqrt{2}}}{z}_2\vartheta }{\sqrt{2}}+2u_3\mu \right)}{e^{u_6+\frac{(ct-x+y\rho )\vartheta }{\sqrt{2}}}{z}_1+e^{-u_8-\frac{(ct-x+y\rho )\vartheta }{\sqrt{2}}}{z}_2+{\nu }^2+{\left(\sqrt{\frac{2}{5}}(ct-x+y\rho )-\frac{16(2+\delta )}{\sqrt{5}(3+4\delta )\vartheta }\right)}^2}\end{array}$$

(21)

where

$$\begin{array}{cc}& \nu =\left(u_3(ct-x+y\rho )+\frac{2\sqrt{\frac{3}{5}}\sqrt{\frac{(1217+1112\delta +208{\delta }^2)\rho }{\eta +\rho (c+\lambda \rho )}}}{(3+4\delta )}\right),\vartheta =\sqrt{\frac{-\eta +c\rho +{\rho }^3}{\rho }},\\ & \mu =\left(u_3\epsilon +\frac{2\sqrt{\frac{3}{5}}\sqrt{\frac{(1217+1112\delta +208{\delta }^2)\rho }{\eta +\rho (c+\lambda \rho )}}}{(3+4\delta )}\right).\end{array}$$

5. Rational Kink Cross Solution (RKCs)

We use the transformation [1]

$$\begin{array}{cc}& \omega =e^{-{\sigma }_1}+m_0{e}^{{\sigma }_1}+{\sigma }_2^2+{\sigma }_3^2+q_7,\end{array}$$

(22)

where

$$\begin{array}{cc}& {\sigma }_1=q_1\epsilon +q_2,{\sigma }_2=q_3\epsilon +q_4,{\sigma }_3=q_5\epsilon +q_6.\end{array}$$

However, $g_j(1\le j\le 5)$ are all real-valued constants. We insert Equation (22) into Equation (6) and compute all the coefficients of the exponential functions and powers of $\epsilon$. We gain equations which provide the following values:

$$\begin{array}{cc}& m_4=0,\delta =\frac{-9}{4},q_1=\sqrt{\frac{-1}{9}\frac{(2\lambda {\rho }^2+2\rho c+2\eta )}{\delta }},q_6=\frac{2(q_3^2+q_5^2)}{\sqrt{\frac{-1}{9}\frac{(2\lambda {\rho }^2+2\rho c+2\eta )q_5}{\delta }}}.\end{array}$$

(23)

We then use Equation (23) in Equation (22):

$$\begin{array}{c}\hfill \omega =q_7+e^{-q_2-\frac{1}{3}\sqrt{2}\epsilon \sqrt{\frac{-\eta +\rho (c+\lambda \rho )}{\rho }}}+e^{q_2+\frac{1}{3}\sqrt{2}\epsilon \sqrt{\frac{-\eta +\rho (c+\lambda \rho )}{\rho }}}{m}_0+q_3^2{\epsilon }^2+{(q_5\epsilon +\frac{3\sqrt{2}(q_3^2+q_5^2)}{q_5\sqrt{\frac{-\eta +\rho (c+\lambda \rho )}{\rho }}})}^2.\end{array}$$

(24)

By substituting Equation (24) into Equation (5), we obtain

$$\begin{array}{c}\hfill \Lambda \left(\epsilon \right)=\frac{\left(2\left(2q_3^2\epsilon +2q_5^2\epsilon +\frac{6\sqrt{2}(q_3^2+q_5^2)}{\phi }-\frac{1}{3}\sqrt{2}\tau \phi +\frac{1}{3}\sqrt{2}{e}^{q_2+\frac{1}{3}\sqrt{2}\epsilon \phi }{m}_0\phi \right)\right)}{q_7+\tau +e^{q_2+\frac{1}{3}\sqrt{2}\epsilon \phi }{m}_0+q_3^2{\epsilon }^2+{(q_5\epsilon +\frac{3\sqrt{2}(q_3^2+q_5^2)}{q_5\phi })}^2},\end{array}$$

(25)

where

$$\begin{array}{c}\hfill \tau =e^{-q_2-\frac{1}{3}\sqrt{2}\epsilon \sqrt{\frac{-\eta +\rho (c+\lambda \rho )}{\rho }}},\phi =\sqrt{\frac{-\eta +\rho (c+\lambda \rho )}{\rho }}.\end{array}$$

We then substitute Equation (25) into Equation (2) to find the RKCs of Equation (1):

$$\begin{array}{cc}& {\mathsf{\Gamma }}_4(x,y,t)=\left(2\right(2q_3^2(ct-x+\rho y)+2q_5^2(ct-x+\rho y)+\frac{6\sqrt{2}(q_3^2+q_5^2)}{\phi }-\frac{1}{3}\sqrt{2}\tau \phi +\frac{1}{3}\sqrt{2}\\ & e^{q_2+\frac{1}{3}\sqrt{2}(ct-x+\rho y)\phi }{m}_0\phi \left)\right)/q_7+\tau +e^{q_2+\frac{1}{3}\sqrt{2}(ct-x+\rho y)\phi }{m}_0+q_3^2{(ct-x+\rho y)}^2+\\ & {(q_5(ct-x+\rho y)+\frac{3\sqrt{2}(q_3^2+q_5^2)}{q_5\phi })}^2.\end{array}$$

(26)

where

$$\begin{array}{c}\hfill \tau =e^{-q_2-\frac{1}{3}\sqrt{2}\epsilon \sqrt{\frac{-\eta +\rho (c+\lambda \rho )}{\rho }}},\phi =\sqrt{\frac{-\eta +\rho (c+\lambda \rho )}{\rho }}.\end{array}$$

6. Periodic Cross-Rational Solutions (PCRs)

We use the transformation [49]

$$\begin{array}{cc}& \omega ={\varpi }_1^2+{\varpi }_2^2+m_{1}cos\left({\varpi }_3\right)+m_{2}cosh\left({\varpi }_4\right)+e_9,\end{array}$$

(27)

where

$$\begin{array}{cc}& {\varpi }_1=e_1\epsilon +e_2,{\varpi }_2=e_3\epsilon +e_4,{\varpi }_3=e_5\epsilon +e_6,{\varpi }_4=e_7\epsilon +e_8.\end{array}$$

However, $e_j(1\le j\le 9)$ are all real-valued parameters. We put Equation (27) into Equation (6) and collect equations by comparing the coefficients of the trigonometric function to obtain

$$\begin{array}{cc}& \delta =\frac{15}{8},e_1=I{e}_3,e_5=\sqrt{\frac{-1}{3}\frac{(2\lambda {\rho }^2+2\rho c+2\eta )}{\rho }},e_7=\sqrt{\frac{-1}{3}\frac{(-2\lambda {\rho }^2-2\rho c-2\eta )}{\rho }}.\end{array}$$

(28)

By putting Equation (28) into Equation (27), we obtain

$$\begin{array}{c}\hfill \omega =e_9+{(e_2+i{e}_3)}^2+{(e_4+e_3\epsilon )}^2+m_{1}cos\left(e_6+\frac{\epsilon \sqrt{\frac{-2\eta +2c\rho +2\lambda {\rho }^2}{\rho }}}{\sqrt{3}}\right)+m_{2}cos\left(e_8+\frac{\epsilon \sqrt{\frac{2\eta +2c\rho +2\lambda {\rho }^2}{\rho }}}{\sqrt{3}}\right).\end{array}$$

(29)

By replacing Equation (29) with Equation (5), we obtain

$$\begin{array}{c}\hfill \Lambda \left(\epsilon \right)=\frac{12e_3(i{e}_2+e_4)-2\sqrt{6}{m}_1\sqrt{\frac{-\eta +\rho (c+\lambda \rho )}{\rho }}sin\left(\beta \right)+2\sqrt{6}\sqrt{c+\frac{\eta }{\rho }+\lambda \rho }sinh\left(\varrho \right)}{3\left(e_2^2+e_4^2+e_9+2i{e}_2{e}_3\epsilon +2e_3{e}_4\epsilon +m_{1}cos\left(\beta \right)+m_{2}cosh\left(\varrho \right)\right)}.\end{array}$$

(30)

where

$$\begin{array}{c}\hfill \beta =e_6+\sqrt{\frac{2}{3}}\epsilon \sqrt{\frac{-\eta +\rho (c+\lambda \rho )}{\rho }},\varrho =e_8+\sqrt{\frac{2}{3}}\epsilon \sqrt{c+\frac{\eta }{\rho }+\lambda \rho }.\end{array}$$

Replacing Equation (30) with Equation (2) to find the PCRs of Equation (1) yields

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_5(x,y,t)=\frac{12e_3(i{e}_2+e_4)-2\sqrt{6}{m}_1\sqrt{\frac{-\eta +\rho (c+\lambda \rho )}{\rho }}sin\left(\varphi \right)+2\sqrt{6}\sqrt{c+\frac{\eta }{\rho }+\lambda \rho }sinh\left(\psi \right)}{3\left(e_2^2+e_4^2+e_9-2i{e}_2{e}_3(ct-x+y\rho )-2e_3{e}_4(ct-x+y\rho )+m_{1}cos\left(\varphi \right)+m_{2}cosh\left(\psi \right)\right)}.\end{array}$$

(31)

where

$$\begin{array}{c}\hfill \varphi =e_6+\sqrt{\frac{2}{3}}(ct-x+y\rho )\sqrt{\frac{-\eta +\rho (c+\lambda \rho )}{\rho }},\psi =e_8+\sqrt{\frac{2}{3}}(ct-x+y\rho )\sqrt{c+\frac{\eta }{\rho }+\lambda \rho }.\end{array}$$

7. Multi-Wave Solution

We use the transformation [50]

$$\begin{array}{cc}& \omega =l_{0}cosh\left(\varrho \right)+l_{1}cos\left({\varrho }_1\right)+l_{2}cos\left({\varrho }_3\right),\end{array}$$

(32)

where

$$\begin{array}{cc}& \varrho =m_1\epsilon +m_2,\varrho =m_3\epsilon +m_4,{\varrho }_3=m_5\epsilon +m_6.\end{array}$$

However, $m_i(1\le i\le 6)$ are constants. Inserting Equation (32) into Equation (6) yields the following values:

$$\begin{array}{cc}& m_1=\sqrt{\frac{-(-3\lambda {\delta }^2-3\rho c-3\eta )}{4\lambda \delta -3\rho }},m_5=0,m_3=\sqrt{\frac{-(\lambda {\rho }^2+\rho c+\eta )}{2\delta \rho -2\rho }}.\end{array}$$

(33)

By inserting Equation (33) into Equation (32),we obtain

$$\begin{array}{c}\hfill \omega =l_{1}cos(m_4+\epsilon \sqrt{\frac{-\eta -c\rho -\lambda {\rho }^2}{-2\rho +2\delta \rho }})+l_{2}cosh\left(m_6\right)+l_{0}cosh\left(m_2+\epsilon \sqrt{\frac{3\eta +3c\rho +3\lambda {\rho }^2}{-3\rho +4\delta \rho }}\right).\end{array}$$

(34)

Substituting Equation (34) into Equation (5) yields

$$\begin{array}{c}\hfill \Lambda \left(\epsilon \right)=\frac{2(-l_1\varsigma sin(m_4+\epsilon \varsigma +l_0\tau sinh(m_2+\epsilon \tau )))}{l_{1}cos(m_4+\epsilon \varsigma )+l_{2}cosh\left(m_6\right)+l_{0}cosh(m_2+\epsilon \tau )}.\end{array}$$

(35)

where

$$\begin{array}{c}\hfill \varsigma =\sqrt{\frac{-\eta -c\rho -\lambda {\rho }^2}{-2\rho +2\delta \rho }},\tau =\sqrt{\frac{3\eta +3c\rho +3\lambda {\rho }^2}{-3\rho +4\delta \rho }}.\end{array}$$

We then insert Equation (35) into Equation (2) to obtain the multi-wave solution:

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_6(x,y,t)=\frac{2(-l_1\varsigma sin(m_4+(-ct+x-y\rho )\varsigma +l_0\tau sinh(m_2+(-ct+x-y\rho )\tau )))}{l_{1}cos(m_4+(-ct+x-y\rho )\varsigma )+l_{2}cosh\left(m_6\right)+l_{0}cosh(m_2+(-ct+x-y\rho )\tau )}.\end{array}$$

(36)

where

$$\begin{array}{c}\hfill \varsigma =\sqrt{\frac{-\eta -c\rho -\lambda {\rho }^2}{-2\rho +2\delta \rho }},\tau =\sqrt{\frac{3\eta +3c\rho +3\lambda {\rho }^2}{-3\rho +4\delta \rho }}.\end{array}$$

8. MSR Interaction with Rogue Waves

We explore the interaction between MSRs and rogue waves using the transformation

$$\begin{array}{cc}& \omega =o_1^2+o_2^2+e^{o_3}+l_{0}cosh\left(o_4\right)+p_9,\end{array}$$

(37)

where

$$\begin{array}{cc}& o_1=p_1\epsilon +p_2,o_2=p_3\epsilon +p_4,o_3=p_5\epsilon +p_6.\end{array}$$

However, $p_i(1\le i\le 9)$ are the parameters. Substituting Equation (37) into Equation (6) yields the values of the assumed parameters:

$$\begin{array}{cc}& p_1=\frac{-1}{6}\sqrt{2}\sqrt{3}\sqrt{\delta },p_2=\frac{1}{3}\frac{\sqrt{2}{\sqrt{3}}^{\frac{3}{2}}{p}_5}{(p_5^2-p_7^2)},p_3=p_8=0,p_9=\frac{-1}{3}\frac{(3p_4^2{p}_5^4-6p_4^2{p}_5^2{p}_7^2+3p_4^2{p}_7^4+2{\delta }^3{p}_5^2)}{p_5^4-2p_5^2{p}_7^2+p_7^4}.\end{array}$$

(38)

Putting Equation (38) in Equation (37) yields

$$\begin{array}{c}\hfill \omega =p_4^2+e^{p_6+p_5\epsilon }-(\frac{(3p_4^2{p}_5^4-6p_4^2{p}_5^2{p}_7^2+3p_4^2{p}_7^4+2{\delta }^3{p}_5^2)}{p_5^4-2p_5^2{p}_7^2+p_7^4})+{\left(\frac{\sqrt{\frac{3}{2}{p}_5{\delta }^{\frac{3}{2}}}}{p_5^2-p_7^2}-\frac{\sqrt{\lambda }}{\sqrt{6}}\right)}^2+l_{0}cosh\left(p_6\epsilon \right).\end{array}$$

(39)

By inserting Equation (39) into Equation (5), we obtain

$$\begin{array}{c}\hfill \Lambda \left(\epsilon \right)=\frac{2\left(p_5{e}^{p_6+p_5\epsilon }-\sqrt{\frac{2}{3}}\sqrt{\delta }\left(\frac{\sqrt{\frac{3}{2}{p}_5{\delta }^{\frac{3}{2}}}}{p_5^2-p_7^2}-\frac{\sqrt{\lambda }}{\sqrt{6}}\right)+p_7{l}_{0}sinh\left(p_7\epsilon \right)\right)}{p_4^2+e^{p_6+p_5\epsilon }-(\frac{(3p_4^2{p}_5^4-6p_4^2{p}_5^2{p}_7^2+3p_4^2{p}_7^4+2{\delta }^3{p}_5^2)}{p_5^4-2p_5^2{p}_7^2+p_7^4})+{\left(\frac{\sqrt{\frac{3}{2}{p}_5{\delta }^{\frac{3}{2}}}}{p_5^2-p_7^2}-\frac{\sqrt{\lambda }}{\sqrt{6}}\right)}^2+l_{0}cosh\left(p_6\epsilon \right)}.\end{array}$$

(40)

We replace Equation (40) with Equation (2) to find the interaction between the rogue wave and MSR:

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_7(x,y,t)=\frac{2\left(p_5\left(\upsilon -\frac{2{\delta }^2}{p_5^2-p_7^2}\right)+\delta \left(\varphi \right)+3p_{7}l+0sinh\left(p_7\left(\varphi \right)\right)\right)}{3(p_4^2+\upsilon -\frac{3p_4^2{(p_5^2-p_7^2)}^2+2p_5^2{\delta }^3}{3{(p_5^2-p_7^2)}^2}+\frac{1}{6}\delta {\varphi }^2+l_{0}cosh\left(\varphi \right))}.\end{array}$$

(41)

where

$$\begin{array}{c}\hfill \upsilon =e^{p_6+p_5(2p_7^{2}t+x+\frac{t}{\alpha \delta }\beta -y\rho +t\lambda \rho )},\varphi =2p_7^{2}t+x+\frac{t}{\delta }\beta -y\rho +t\lambda \rho .\end{array}$$

9. Homoclinic Breather Solutions (HBs)

For HBs, we use [51]

$$\begin{array}{cc}& \omega =e^{-{\mu }_1}+m_1{e}^{{\mu }_2}+m_{2}cos\left({\mu }_3\right),\end{array}$$

(42)

where

$$\begin{array}{cc}& {\mu }_1=g(n_1\epsilon +n_2),{\mu }_2=g(n_3\epsilon +n_4),{\mu }_3=g_1(n_5\epsilon +n_6).\end{array}$$

However, $n_i(1\le i\le 6)$ are all real-valued constants. We use Equation (42) with Equation (6) and evaluate the values of the assumed constants:

$$\begin{array}{cc}& \delta =\frac{-15}{4},n_1=\frac{\frac{-1}{6}\frac{(\lambda {\rho }^2+c\rho +\eta )}{\rho }}{g},n_3=0,n_5=\frac{\frac{-1}{2}\frac{(-\lambda {\rho }^2-c\rho -\eta )}{\rho }}{g_1}.\end{array}$$

(43)

Putting Equation (43) into Equation (42) yields

$$\begin{array}{c}\hfill \omega =e^{-g(n_2+\frac{\epsilon \sqrt{\frac{-\eta +c\rho +\lambda {\rho }^2}{\rho }}}{\sqrt{6}g})}+e^{n_{4}g}{m}_1+m_{2}cos\left(g_1\left(n_6+\frac{\epsilon \sqrt{\frac{-(-\eta -\rho x-\lambda {\rho }^2)}{\rho }}}{\sqrt{2}{g}_1}\right)\right).\end{array}$$

(44)

By inserting Equation (44) into Equation (5), we obtain

$$\begin{array}{c}\hfill \Lambda \left(\epsilon \right)=\frac{-\sqrt{6}\sqrt{\frac{-\eta +c\rho +\lambda {\rho }^2}{\rho }}+3\sqrt{2}{e}^{n_{2}g+\frac{\epsilon \sqrt{\frac{-\eta +c\rho +\lambda {\rho }^2}{\rho }}}{\sqrt{6}}}{m}_2\sqrt{\frac{-\eta +c\rho +\lambda {\rho }^2}{\rho }}sin\left(n_6{g}_1+\frac{\epsilon \sqrt{\frac{-\eta +c\rho +\lambda {\rho }^2}{\rho }}}{\sqrt{2}}\right)}{3\left(1+e^{n_{2}g+n_{4}g+\frac{\epsilon \sqrt{\frac{-\eta +\rho (c+\lambda \rho )}{\rho }}}{\sqrt{6}}}{m}_1+e^{n_{2}g+\frac{\epsilon \sqrt{\frac{-\eta +\rho (c+\lambda \rho )}{\rho }}}{\sqrt{6}}}{m}_{2}cos\left(n_6{g}_1+\frac{\epsilon \sqrt{\frac{-\eta +\rho (c+\lambda \rho )}{\rho }}}{\sqrt{2}}\right)\right)}.\end{array}$$

(45)

We then substitute Equation (45) into Equation (2) to find the HBs:

$$\begin{array}{c}\hfill {\mathsf{\Gamma }}_8(x,y,t)=\frac{-\sqrt{6}\sqrt{\frac{-\eta +c\rho +\lambda {\rho }^2}{\rho }}+3\sqrt{2}\phi m_2\sqrt{\frac{-\eta +c\rho +\lambda {\rho }^2}{\rho }}sin\left(\varphi \right)}{3(1+e^{n_{2}g+n_{4}g+\frac{(-ct+x-\rho y)\sqrt{\frac{-\eta +\rho (c+\lambda \rho )}{\rho }}}{\sqrt{6}}}{m}_1+\phi m_{2}cos\left(\varphi \right))},\end{array}$$

(46)

where

$$\begin{array}{c}\hfill \varphi =n_6{g}_1+\frac{(-ct+x-\rho y)\sqrt{\frac{-\eta +c\rho +\lambda {\rho }^2}{\rho }}}{\sqrt{2}},\phi =e^{n_{2}g+\frac{(-ct+x-\rho y)\sqrt{\frac{-\eta +c\rho +\lambda {\rho }^2}{\rho }}}{\sqrt{6}}}.\end{array}$$

10. Stability Property of Solutions

Now, using a Hamiltonian approach, we will examine the stability property for a $(2+1)$-dimensional KdV equation of constant coefficients. The Hamiltonian methodology “K” is given by

$$\begin{array}{c}\hfill K=\frac{1}{2}{\int }_{-h}^h{{\rm Y}}^2\left(z\right)dz.\end{array}$$

(47)

The solutions’ stability condition can be assessed as follows:

$$\begin{array}{c}\hfill \frac{\partial K}{\partial \rho }>0.\end{array}$$

(48)

The wave velocity is $\rho$, and K stands for the momentum in the Hamiltonian system. Using the Hamiltonian system, the stability’s condition is stated, and all feasible solutions are then determined (

Table 1. Stability properties of the solutions ${\mathsf{\Gamma }}_i(x,y,t)$, where ($i=1,2,3,...,8$).

Solution	Stable	Unstable	Values of Variables
${\mathsf{\Gamma }}_1(x,y,t)$	✓		$\eta =-1,c=2,\rho =2,\lambda =2,m_4=9,x,y,t\in [-7,7]$
${\mathsf{\Gamma }}_2(x,y,t)$	✓		$g_6=8,c=1,\rho =-1,\eta =1,\lambda =1,g_1=3,$
			$\delta =3,g_4=2,z_1=1,t=5,x,y,t\in [-1,1]$
${\mathsf{\Gamma }}_3(x,y,t)$	✓		$c=1.7,\rho =4,z_1=3,u_8=2,\eta =2,u_3=-1,$
			$z_2=1,\delta =2,\lambda =2,t=1,u_6=9,x,y,t\in [-2,2]$
${\mathsf{\Gamma }}_4(x,y,t)$		✓	Singular solution
${\mathsf{\Gamma }}_5(x,y,t)$		✓	Singular solution
${\mathsf{\Gamma }}_6(x,y,t)$	✓		$l_1=4,\eta =1,c=1,\rho =3,l_0=2,m_2=3,m_6=2,$
			$\lambda =1.2,\delta =2,l_2=12,t=9,m_4=3,x,y,t\in [-3,3]$
${\mathsf{\Gamma }}_7(x,y,t)$	✓		$p_6=3.3,p_5=1,p_7=1.2,\delta =1.5,l_0=2,\eta =3,$
			$\rho =2,\lambda =2,p_4=1,t=-1,x,y,t\in [-11,11]$
${\mathsf{\Gamma }}_8(x,y,t)$		✓	Singular solution

).

11. Results and Discussion

With the help of the proper parameter settings, we were able to successfully produce the desired type of solution, which illustrates a wave discrepancy. In Figure 1, the wave appears in the MSRs with $\eta =-1,c=2,\rho =2,\lambda =2,m_4=9,\mathrm{and}t=11$. We can see how the wave moves and changes its position with various values for the time parameter t. Figure 2 and Figure 3 display the movement of the wave and the stability conditions with $t=9$ and $t=-6$, respectively. In Figure 4, we explore MSRs with exponential functions with 3D, contour, 2D, and stream plots via $g_6=8,c=1,\rho =-1,\eta =1,\lambda =1,$ $g_1=3,\delta =3,g_4=2,z_1=1,\mathrm{and}t=5$. Figure 5 and Figure 6 show a wave with a high amplitude. Figure 7 shows the interaction solution between the MSRs and kink II via 3D, 2D, contour, and stream plots with $c=1.7,\rho =4,z_1=3,u_8=2,\eta =2,$ $u_3=-1,z_2=1,\delta =2,\lambda =2,t=1,\mathrm{and}{u}_6=9$. In Figure 8, various bright and dark lumps appear via the assumed parameters $c=1.7,\rho =4,z_1=3,u_8=2,\eta =2,u_3=-1,$ $z_2=1,\delta =2,\lambda =2,t=1,\mathrm{and}{u}_6=3$. Figure 9 depicts PKCs in which a soliton wave appears with a high amplitude via $g_3=6,c=1,\rho =2,g_5=3,\eta =1,\lambda =1,$ $g_2=5,m_0=5,g_7=11,\mathrm{and}t=3$.

The PKCs in Equation (31) are graphically presented in Figure 10 and Figure 11. Figure 12 shows a visual representation of the multi-wave solution in Equation (36) in which multiple bright lumps appear, considering $l_1=4,\eta =1,c=1,\rho =3,l_0=2,$ $m_2=3,m_6=2,$ $\lambda =1.2,\delta =2,l_2=12,t=9,\mathrm{and}{m}_4=3$. The multi-wave shape profiles are explained via $l_1=4,\eta =1,c=1,\rho =3,l_0=2,m_2=3,m_6=2,\lambda =1.2,\delta =2,l_2=12,$ $t=9,\mathrm{and}{m}_4=2.5$ in Figure 13. For various parametric values, we attained the multi-wave sketches in Figure 14 via $l_1=4,\eta =1,c=1,\rho =3,l_0=2,m_2=3,m_6=2,$ $\lambda =1.2,\delta =2,l_2=12,t=9,\mathrm{and}{m}_4=4.5$. In Figure 15, we constructed the interactional solution between the MSRs and rogue wave profiles via $p_6=3.3,p_5=1,p_7=1.2,$ $\delta =1.5,l_0=2,\eta =3,\rho =2,\lambda =2,p_4=1,\mathrm{and}t=-1$. The dynamical behavior of the solution in Equation (41) via $p_6=3.3,p_5=1,p_7=1.2,$ $\delta =1.5,l_0=2,\eta =3,\rho =2,$ $\lambda =2,p_4=1,\mathrm{and}t=3$ is presented in Figure 16. With $p_6=3.3,p_5=1,p_7=1.2,$ $\delta =1.5,l_0=2,\eta =3,$ $\rho =2,\lambda =2,p_4=1,\mathrm{and}t=5$, the solution to the MSRs with rogue waves via three-dimensional, contour, two-dimensional, and stream plots are shown in Figure 17. We attained HBs using Equation (46) with $\eta =2,\rho =1.3,c=3.1,\lambda =-2,$ $n_2=-2,n_6=3,m_2=-1,m_1=2,m_4=3,g_1=-5,\mathrm{and}t=3$. Figure 18 shows 3D visuals of the homoclinic solution with (1) Figure 18a with $g=-0.4$, (2) Figure 18b with $g=-1.4$, (3) Figure 18c with $g=-2.4$, and (4) Figure 18d with $g=-4.4$. Figure 19 shows the stability conditions corresponding to Figure 18.

12. Conclusions

Some analytical solutions were acquired by symbolic computations. Additionally, a thorough investigation of the solutions’ dynamics was conducted. In this work, the MSRs, multi-wave solitons, RKCs, PCRs, HBs, and interactional solutions such as MSRs with one kink, double kinks, and rogue waves were all studied. Using the Hamilton system features, we identified solutions as stable or unstable solutions. Finally, the findings were graphically analyzed using contour, density, three-dimensional, two-dimensional, and stream plots. We obtained entirely unique outcomes with our research.