1 Introduction

Nonlinearity is a fascinating flavor of nature and many scientists believe that nonlinear science is the best way to understand the essence of physical processes. The mathematical knowledge of complicated systems that vary over time necessitates advancement in the study of nonlinear ordinary and partial differential equations from a variety of angles (Lan 2024; Baskonus et al. 2022). These styles are created in a variety of scientific and technical sectors, including neural networks, population ecology, fluid dynamics, economics, fiber optics, plasma physics, solid-state physics etc. In recent years, the discovery of soliton solutions to the aforementioned miracles has been intriguing and unfathomable (Chen et al. 2022; Malik et al. 2023; Jannat et al. 2023; Bilal et al. 2023).

Due to its unusual properties, graphene was initially investigated in isolation in 2004 and has since piqued the curiosity of experts all over the world. The strongest and thinnest nano-scale material was found to be a monolayer carbon atoms structured in hexagonal lattice. Because graphene has all of its atoms on the surface, it is particularly sensitive to electrical, optical, thermal and mechanical effects, making it an attractive candidate for industrial applications (Malik et al. 2022; Radsar et al. 2021). At mild temperatures, graphene also exhibits Quantum Hall phenomena (Meng et al. 2013; Chang et al. 2016; Aawani et al. 2018).

In Yildiz et al. (2021), scanning tunnelling microscopy of graphene topography on Si\(O^{2}\) revealed creases in graphene sheets. These wrinkle channels also influence electric conductivity (Deng and Berry 2016). Wrinkles have been identified as a possible possibility for the development of flexible electric sensors and the assembly of graphene nanoribbons (Kang and Lee 2013). Besides the aforementioned contributions, there are a number of unsolved concerns, such as the lack of a theoretical model illustrating the wrinkle creation process and knowledge concerning wrinkle dispersion aspects.

The well-known KdV equation read as

$$\begin{aligned} v_{t}+vv_{x}+v_{xxx}=0. \end{aligned}$$
(1)

The solution to Eq. (1) is \(v=3c{{\,\textrm{sech}\,}}^{2}(\frac{\sqrt{c}}{2})(x-ct)\), where c represents the soliton’s velocity profile. The structural shape of wrinkles at various locations created by MD simulation, according to Guo and Guo (2013), has the following form: \(v_{m}=3c_{m}{{\,\textrm{sech}\,}}^{2}[(\frac{\sqrt{c_{m}}}{2})(x-\frac{at^{2}}{2}-bt)c_{m}]\), where a depends on t, b is a constant, and \(c_{m}\) can be adjusted to make it consistent to the wrinkle’s shape. The motion equation that \(v_{m}\) satisfies is as follows:

$$\begin{aligned} \frac{\partial v_{m}}{\partial t}+v_{m}\frac{\partial v_{m}}{\partial x}+\frac{\partial ^{3} v_{m}}{\partial x^{3}}+(a(t)+b-c_{m})\frac{\partial v_{m}}{\partial x}=0. \end{aligned}$$
(2)

When Eq. (2) is compared to the Eq. (1) equation, the extra component indicates the driving force collected by heat gradients and is attributed to wrinkle acceleration. The first term represents the temporal evolution of the wrinkle amplitude. The second term accounts for the thermophoretic motion induced by temperature gradients. The third term captures the diffusion of wrinkles within the graphene sheet.

Equation (1) is widely known to be solved using the inverse scattering approach and therefore entirely integrable (Calogero 1975). The Painlevé analysis (Raza and Yasmeen 2021; Raza et al. 2021), extended algebraic method (Raza and Zubair 2019; Hussain et al. 2024), Bäcklund transformation (Wahlquist and Estabrook 1973), F-expansion method (Bashir and Alhakim 2013), sine-Gordon expansion method (Yan et al. 2021; Ciancio et al. 2022), lump wave solutions (Sivasundaram et al. 2024), symmetry analysis (Abdel-Gawad et al. 2021), unified and generalized unified method (Kumar and Kumar 2023; Kour and Kumar 2018; Raza et al. 2021), Lax pairs and similarity reductions (Clarkson and Kruskal 1989) have all been discovered. The focus of this study is to check the integrability of the TM equation, for that purpose, we applied Painlevé integrable test to Eq. (2). Furthermore, via truncated Painlevé expansion, we derive an auto-Bäcklund transformation (ABT) and certain soliton solutions of the TM equation. We identify multiple regular kink (MRK) and multiple singular kink (MSK) solutions using the Hirota bilinear approach and the Cole-Hopf transformation.

This paper is sorted as follows: Sects. 1 and 2, briefly explains the Painlevé integrability test for the TM equation, while in Sect. 3, exact soliton solution have been extracted via Painlevé–Bäcklund transformations. In Sect. 4, MRK and MSK solutions have been obtained, and Sect. 5, displays the graphs of the solutions. Section 5, explains the results and discussion of the applied techniques and finally this paper has been concluded in Sect. 7.

2 Description of Painlevé integrability test

The Painlevé test is conducted to identify the integrability of a nonlinear partial differential equation (NPDE), whereas the Laurent series is truncated at constant level to study the exact soliton solution for NPDEs. Weiss, Tabor, and Carnevale (WTC) have demonstrated how the integrability of partial differential equations (PDEs) is tied to the equation’s Painlevé property (Musette 1999; Pickering 1993). When the solutions of a PDE are single-valued around the moveable, singular manifold \(\psi (l_{1},l_{2},...l_{k})=0\), the PDE has this property. Taking into account an NPDE

$$\begin{aligned} P(v,v_{l_{1}},v_{l_{2}},v_{l_{1}l_{2}},v_{l_{1}l_{1}},v_{l_{2}l_{2}},...)=0. \end{aligned}$$
(3)

The solution to Eq. (3) is assumed by Laurent series as

$$\begin{aligned} v(l_{1},l_{2},...,l_{k})=\psi ^{\delta }(l_{1},l_{2},...l_{k})\sum _{m=0}^{\infty }v_{m}(l_{1},l_{2},...,l_{k})\psi ^{m}(l_{1},l_{2},...l_{k}), \end{aligned}$$
(4)

where \(\psi\) and \(v_{m}\) are analytic functions of \((l_{1},l_{2},...,l_{k})\) and \(\delta\) is negative. The following are the steps in this method

1.:

Plug in \(v=v_{0}\psi ^{\delta }\) into Eq. (3). Evaluate the feasible leading orders \(\delta\) by comparing two or more PDE terms so that they dominate the others for that \(\delta\).

2.:

Ignore the non-leading components from the equation and solve for nonzero \(v_{0}\). This can lead to a number of different paths.

3.:

Evaluate the resonance values by substituting into \(v=v_{0}\psi ^{\delta }+\sum _{m=1}^{\infty }v_{m}\psi ^{m+\delta }\) into the PDE.

4.:

Insert Eq. (3) into the PDE and equate like powers of \(\psi\) to zero to produce an over determined system of equations for \(\psi\), \(v_{m}(m=0,1,...)\) and their derivatives.

5.:

Check whether the resonance values are compatible or not. The compatibility criterion is that the coefficients \(v_{m}\) match resonance values that would arise as arbitrary. Thus, the Eq. (3) is Painlevé integrable if the compatibility conditions are satisfied at the resonance values.

The Painlevé property is distinguished by:

  • m is a negative integer,

  • At positive integer values of m, all resonance values exist and are compatible.

2.1 Painlevé analysis of thermopherotic motion equation

By Ref. Estévez and Prada (2005); Zhou et al. (2022) we suppose that the solution to the TM equation around the singular manifold (\(\psi (x,t)=0\)) as the Laurent series expansion is as follows (Kumar and Malik 2022)

$$\begin{aligned} v=\psi ^{\delta }\sum _{m=0}^{\infty }v_{m}\psi ^{m}, \end{aligned}$$
(5)

where \(\delta\) is a negative integer and v, \(v_{m}\) such that \(v_{0}\ne 0\) are the analytic functions of x and t. Inserting \(v=v_{0}\psi ^{\delta }\) into Eq. (2) to determine the values of \(\delta\) and \(v_{0}\), yields

$$\begin{aligned} \delta =-2,\qquad v_{0}=-12\psi _{xx}^{2}. \end{aligned}$$
(6)

Inserting the Laurent series

$$\begin{aligned} v=v_{0}\psi ^{-2}+\sum _{m=0}^{\infty }v_{m}\psi ^{m-2}=0, \end{aligned}$$
(7)

into Eq. (2), one gets the recursion relation. By setting the coefficients of \(\psi ^{m-8}\) to zero, the resonance values can be computed.

$$\begin{aligned} m^{3}-9m^{2}+14m+24=0. \end{aligned}$$

Thus the values are \(m=-1,4,6.\) The highest value of resonance is at \(m=6\). So, we truncate the Laurent series in Eq. (5) as below

$$\begin{aligned} v=v_{0}\psi ^{-2}+v_{1}\psi ^{-1}+v_{2}+v_{3}\psi +v_{4}\psi ^{2}+v_{5}\psi ^{3}+v_{6}\psi ^{4}. \end{aligned}$$
(8)

By substituting Eq. (8) for Eq. (2) and comparing equivalent powers of \(\psi\) to zero, the coefficients are as follows:

$$\begin{aligned} v_{1}= & {} 12\psi _{xx}, \end{aligned}$$
(9)
$$\begin{aligned} v_{2}= & {} -\frac{1}{\psi _{x}^{2}}\bigg (4\psi _{x}\psi _{xxx}-3\psi _{xx}^{2} +\psi _{x}\psi _{t}+(a(t)+b-c)\psi _{x}^{2}\bigg ), \end{aligned}$$
(10)
$$\begin{aligned} v_{3}= & {} -\frac{1}{\psi _{x}^{3}}\bigg (4\psi _{xx}\psi _{xxx}+\psi _{t}\psi _{xx} -\psi _{x}\psi _{xxxx}-\psi _{x}\psi _{xt}-3\frac{\psi _{xx}^{3}}{\psi _{x}}\bigg ), \end{aligned}$$
(11)
$$\begin{aligned} v_{5}= & {} -\frac{1}{6\psi _{x}^{8}}\bigg (-6v_{4x}\psi _{x}^{7}-6v_{4}\psi _{x}^{6}\psi _{xx} +a\psi _{x}^{5}+\psi _{x}^{4}\psi _{xxxxxx}+2\psi _{x}^{4}\psi _{xxxt} +\psi _{x}^{4}\psi _{tt}\nonumber \\{} & {} -9\psi _{xx}\psi _{xxxxx}\psi _{x}^{3}-9\psi _{xx}\psi _{xxt}\psi _{x}^{3} -17\psi _{xxx}\psi _{xxxx}\psi _{x}^{3}-8\psi _{xt}\psi _{xxx}\psi _{x}^{3} -2\psi _{xxxx}\psi _{t}\psi _{x}^{3}\nonumber \\{} & {} -2\psi _{x}^{3}\psi _{t}\psi _{xt}+48\psi _{x}^{2}\psi _{xx}^{2}\psi _{xxxx}+21\psi _{x}^{2}\psi _{xx}^{2}\psi _{xt} +70\psi _{x}^{2}\psi _{xxx}^{2}\psi _{xx}+17\psi _{x}^{2}\psi _{xx}\psi _{xxx}\psi _{t}\nonumber \\{} & {} +\psi _{x}^{2}\psi _{t}^{2}\psi _{xx}-174\psi _{xx}^{3}\psi _{xxx}\psi _{x} -21\psi _{x}\psi _{t}\psi _{xx}^{3}+81\psi _{xx}^{5}\bigg ). \end{aligned}$$
(12)

The resonance value \(m=-1\), happens as arbitrary around the singular manifold. When the coefficients of \(\psi ^{-1}\) and \(\psi ^{1}\) are collected, the result is zero. Absence of \(v_{4}\) and \(v_{6}\) corresponds that \(v_{4}\) and \(v_{6}\) are arbitrary. Thus, the resonance values are at \(m=4\) and \(m=6\). As a result, Painlevé test’s compatibility conditions are met and Eq. (2) is Painlevé integrable.

3 Exact travelling wave solutions by Painlevé-Bäcklund transformation

The truncated Painlevé expansion technique is the most commonly utilized way of constructing the auto-Bäcklund transformation (ABT) (Yan 2003; Singh and Ray 2022). When a PDE has the Painlevé property, truncating the series (5) about the singular manifold at the constant term can provide the ABT, which is given as

$$\begin{aligned} v=v_{0}\psi ^{-2}+v_{1}\psi ^{-1}+v_{2}. \end{aligned}$$
(13)

By substituting Eq. (13) into Eq. (2) yields

$$\begin{aligned}&\psi ^{-5}:-2v_{0}^{2}\psi _{x}-24v_{0}\psi _{x}^{3}=0.\nonumber \\&~~~~~~~~~\text {Hence},\qquad v_{0}=-12\psi _{x}^{2}, \end{aligned}$$
(14)
$$\begin{aligned}&\psi ^{-4}:18v_{0x}\psi _{x}^{2}+v_{0}v_{0x}-6v_{1}\psi _{x}^{3} -3v_{0}v_{1}\psi _{x}+18v_{0}v_{1}\psi _{x}\psi _{xxx}=0. \nonumber \\&~~~~~~~~~\text {Hence},\qquad v_{1}=12\psi _{xx}, \end{aligned}$$
(15)
$$\begin{aligned}&\psi ^{-3}:-6v_{0x}\psi _{xx}-6v_{0xx}\psi _{x}-2v_{0}\psi _{xxx}+v_{0}v_{1x}+v_{1}v_{0x} -v_{1}^{2}\psi _{x}+6v_{1x}\psi _{x}^{2}-2v_{0}\phi _{t} \nonumber \\&~~~~~~~-2v_{2}v_{0}\psi _{x}+6v_{1}\psi _{x}\psi _{xx}-2v_{0}\psi _{x}(a(t)+b-c)=0, \end{aligned}$$
(16)
$$\begin{aligned}&\psi ^{-2}:-3v_{1xx}\psi _{x}-3v_{1x}\psi _{xx}-v_{1}\psi _{xxx}+v_{0}v_{2x}+v_{1}v_{1x}+v_{2}v_{0x}+(a(t)+b-c)v_{0x} \nonumber \\&~~~~~~~-v_{1}\psi _{t}+v_{0xxx}+v_{0t}-v_{2}v_{1}\psi _{x}-(a(t)+b-c)v_{1}\psi _{x}=0, \end{aligned}$$
(17)
$$\begin{aligned}&\psi ^{-1}:v_{1}v_{2x}+v_{2}v_{1x}+v_{1t}+v_{1xxx}+(a(t)+b-c)v_{1x}=0, \end{aligned}$$
(18)
$$\begin{aligned}&\psi ^{0}:v_{2t}+v_{2}v_{2x}+v_{2xxx}+(a(t)+b-c)v_{2x}=0. \end{aligned}$$
(19)

By using Eqs. (14) and (15), Eqs. (16–18) become

$$\begin{aligned}&24\psi _{x}(-3\psi _{x}\psi _{xx}^{2}+4\psi _{x}^{2}\psi _{xxx}+\psi _{x}^{2}\psi _{t} +\psi _{x}^{3}v_{2}+(a(t)+b-c)\psi _{x}^{3})=0, \end{aligned}$$
(20)
$$\begin{aligned}&\frac{\partial }{\partial x}(-3\psi _{x}\psi _{xx}^{2}+4\psi _{x}^{2}\psi _{xxx}+\psi _{x}^{2}\psi _{t} +\psi _{x}^{3}v_{2}+(a(t)+b-c)\psi _{x}^{3}) \nonumber \\&\quad +\psi _{x}\int (v_{2x}\psi _{xx}+v_{2}\psi _{xxx}+\psi _{xxxxx} +\psi _{xxt}+(a(t)+b-c)\psi _{xxx})dx=0, \end{aligned}$$
(21)
$$\begin{aligned}&12(v_{2x}\psi _{xx}+v_{2}\psi _{xxx}+\psi _{xxxxx}+\psi _{xxt}+(a(t)+b-c)\psi _{xxx})=0, \end{aligned}$$
(22)

respectively. It is clear that Eqs. (16–19) are obeyed, provided that \(\psi _{x}\ne 0\)

$$\begin{aligned}&-3\psi _{x}\psi _{xx}^{2}+4\psi _{x}^{2}\psi _{xxx}+\psi _{x}^{2}\psi _{t}+\psi _{x}^{3}v_{2}+(a(t)+b-c)\psi _{x}^{3}=0, \end{aligned}$$
(23)
$$\begin{aligned}&v_{2x}\psi _{xx}+v_{2}\psi _{xxx}+\psi _{xxxxx}+\psi _{xxt}+(a(t)+b-c)\psi _{xxx}=0, \end{aligned}$$
(24)
$$\begin{aligned}&v_{2t}+v_{2}v_{2x}+v_{2xxx}+(a(t)+b-c)v_{2x}=0. \end{aligned}$$
(25)

Thus, we acquire an ABT of Eq. (2) as below,

$$\begin{aligned} v=12\frac{\partial ^{2}}{\partial x^{2}}(\ln \psi )+v_{2}. \end{aligned}$$
(26)

  • Setting \(v_{2}(x,y,t)=0\) in Eq. (26), yields Cole-Hopf transformation

    $$\begin{aligned} v=12\frac{\partial ^{2}}{\partial x^{2}}(\ln \psi ), \end{aligned}$$
    (27)

where Eqs. (23) and (24) are satisfied by \(\psi (x,t)\), and \(v_{2}\) is the solution of Eq. (2). Various solutions can be obtained using the aforementioned ABT and varied values of \(v_{2}\) and \(\psi\).

3.1 Extraction of exact solutions via ABT

Considering a general case to obtain solitary wave solutions of the dynamical thermophoretic equation

$$\begin{aligned}&\psi (x,t)=1+\exp (\iota (l(t)x+m(t))), \end{aligned}$$
(28)
$$\begin{aligned}&v_{2}(x,t)=p(t), \end{aligned}$$
(29)

where l, m and p are functions of t. Inserting Eqs. (28), (29) into Eqs. (23–25) yields the system of equations as below

$$\begin{aligned}&l^{4}(t)-l(t)l'(t)x-l(t)m'(t)-p(t)l^{2}(t)-l^{2}(t)(a(t)+b-c)=0, \end{aligned}$$
(30)
$$\begin{aligned}&l^{4}(t)-l(t)l'(t)x+\iota l'(t)-l(t)m'(t)-p(t)l^{2}(t)-l^{2}(t)(a(t)+b-c)=0,\end{aligned}$$
(31)
$$\begin{aligned}&p'(t)=0. \end{aligned}$$
(32)

Solving the above equations, we obtain

$$\begin{aligned} g(t)=\gamma , \qquad l(t)=\beta ,\qquad m(t)=\beta t(\beta ^{2}-\frac{a(t)}{2}-b+c-\gamma )+\alpha , \end{aligned}$$
(33)

where \(\alpha\), \(\beta\) and \(\gamma\) are integration constants. Thus, analytic solution is evaluated via Eq. (26) by plugging Eq. (33) into (28) and (29) as shown in Fig. 2

$$\begin{aligned} v(x,t)&=\frac{12\beta ^{2}\bigg (\exp (\iota (\beta x+\beta t(\beta ^{2}-\frac{a(t)}{2}-b+c-\gamma )+\alpha ))\bigg )^{2}}{\bigg (1+\exp (\iota (\beta x+\beta t(\beta ^{2}-\frac{a(t)}{2}-b+c-\gamma )+\alpha ))\bigg )^{2}}\nonumber \\&-\frac{12\beta ^{2}\exp (\iota (\beta x+\beta t(\beta ^{2}-\frac{a(t)}{2}-b+c-\gamma )+\alpha ))}{1+\exp (\iota (\beta x+\beta t(\beta ^{2}-\frac{a(t)}{2}-b+c-\gamma )+\alpha ))}+\gamma . \end{aligned}$$
(34)
Fig. 1
figure 1

a Bright soliton for v(x, t) in Case B for \(\alpha =3.2\), \(\gamma =1\), \(\beta =-0.4\), \(a(t)=\sin (t)\), \(c=2\), \(b=0.9\). b 2D plot at \(t=1,-1\)

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4 Multiple regular and singular kink solutions

We largely employ Hirota’s direct technique, where it is demonstrated that soliton solutions are merely polynomials of exponentials, to construct N-kink solutions of any totally integrable equation. We extract MRK and MSK solutions for the TM equation in this section (Dey 1986; Zhao 2024). Inserting

$$\begin{aligned} v(x,t)=e^{g_{k}x-h_{k}t}, \end{aligned}$$
(35)

into Eq. (2) linear terms. The dispersion relation between \(g_{k}\) and \(h_{k}\) is given by

$$\begin{aligned} h_{k}=(a(t)+b-c)g_{k}+g_{k}^{3}, \end{aligned}$$
(36)

as a consequence, we establish

$$\begin{aligned} \Theta _{k}=g_{k}x-h_{k}t=g_{k}x-((a(t)+b-c)g_{k}+g_{k}^{3})t. \end{aligned}$$
(37)

By employing the Cole-Hopf transformation, the multi-soliton solutions of Eq. (2) are supposed to be

$$\begin{aligned} v(x,t)=12\frac{\partial ^{2}}{\partial x^{2}}(\ln \psi ). \end{aligned}$$
(38)

For the regular and singular one kink solution the auxiliary function \(\psi (x,t)\) is as below, respectively

$$\begin{aligned} \psi (x,t)=1\pm e^{g_{1}x-((a(t)+b-c)g_{1}+g_{1}^{3})t}, \end{aligned}$$
(39)

The regular and singular one kink solutions are obtained by entering Eq. (39) into Eq. (2) respectively, as follows

$$\begin{aligned} v(x,t)=\pm \frac{12g_{1}^{2}e^{-g_{1}(a(t)t+tg_{1}^{2}+bt-ct-x)}}{(1\pm e^{-g_{1}(a(t)t+tg_{1}^{2}+bt-ct-x)})^{2}}. \end{aligned}$$
(40)

For the regular and singular two kink solution we assume the functions respectively, as follows

$$\begin{aligned} \psi (x,t)&=1\pm e^{\Theta _{1}}\pm e^{\Theta _{2}}+A_{12}e^{\Theta _{1}+\Theta _{2}}\nonumber \\&=1\pm e^{g_{1}x-((a(t)+b-c)g_{1}+g_{1}^{3})t}\pm e^{g_{2}x-((a(t)+b-c)g_{2}+g_{2}^{3})t}\nonumber \\&+A_{12}e^{(g_{1}+g_{2})x-((a(t)+b-c)(g_{1}+g_{2})+(g_{1}^{3}+g_{2}^{3}))t}. \end{aligned}$$
(41)

Using Eq. (41) in Eq. (38) and entering the obtained result in Eq. (2), yields

$$\begin{aligned} A_{12}=0, \end{aligned}$$
(42)

and thus we devise

$$\begin{aligned} A_{jk}=0,\qquad 1\le j<k\le 3. \end{aligned}$$
(43)

Inserting Eq. (42) and Eq. (41) into Eq. (38), following are the regular and the singular two kink solutions, respectively, of Eq. (2)

$$\begin{aligned} v(x,t)&=\pm \frac{12[g_{1}^{2}e^{g_{1}x-((a(t)+b-c)g_{1}+g_{1}^{3})t}+g_{2}^{2} e^{g_{2}x-((a(t)+b-c)g_{2}+g_{2}^{3})t}]}{(1\pm e^{g_{1}x-((a(t)+b-c)g_{1}+g_{1}^{3})t}\pm e^{g _{2}x-((a(t)+b-c)g_{2}+g_{2}^{3})t})}\nonumber \\&-\frac{12[g_{1}e^{g_{1}x-((a(t)+b-c)g_{1}+ g_{1}^{3})t}+g_{2}e^{g_{2}x-((a(t)+b-c)g_{2}+g_{2}^{3})t}]^{2}}{(1\pm e^{g_{1}x-((a(t)+b-c)g_{1}+ g_{1}^{3})t}\pm e^{g _{2}x-((a(t)+b-c)g_{2}+g_{2}^{3})t})^{2}}. \end{aligned}$$
(44)

For the regular and singular three kink solution we assume the test functions respectively, as follows

$$\begin{aligned} \psi (x,t)&=1\pm e^{\Theta _{1}}\pm e^{\Theta _{2}}\pm e^{\Theta _{3}}+A_{12}e^{\Theta _{1}+\Theta _{2}}+A_{23}e^{\Theta _{2}+\Theta _{3}}\nonumber \\&+A_{13}e^{\Theta _{1}+\Theta _{3}}+B_{123}e^{\Theta _{1}+\Theta _{2}+\Theta _{3}} \end{aligned}$$
(45)
$$\begin{aligned} \psi (x,t)&=1\pm e^{g_{1}x-((a(t)+b-c)g_{1}+g_{1}^{3})t}\pm e^{g_{2}x-((a(t)+b-c)g_{2}+g_{2}^{3})t}\nonumber \\&\pm e^{g_{3}x-((a(t)+b-c)g_{3}+g_{3}^{3})t}. \end{aligned}$$
(46)

The regular and singular three kink solutions of the TM equation are respectively, obtained by inserting Eq. (46) into Eq. (38), which is as follows

$$\begin{aligned} v(x,t)&=\pm \frac{12[g_{1}^{2}e^{g_{1}x-((a(t)+b-c)g_{1}+g_{1}^{3})t}+g_{2}^{2}e^{g_{2}x- ((a(t)+b-c)g_{2}+g_{2}^{3})t}+g_{3}^{2}e^{g_{3}x-((a(t)+b-c)g_{3}+g_{3}^{3})t}]}{(1\pm e^{g_{1}x-((a(t)+b-c)g_{1}+g_{1}^{3})t}\pm e^{g _{2}x-((a(t)+b-c)g_{2}+g_{2}^{3})t}\pm e^{g_{3}x-((a(t)+b-c)g_{3}+g_{3}^{3})t})} \nonumber \\&-\frac{12[g_{1}e^{g_{1}x-((a(t)+b-c)g_{1}+g_{1}^{3})t}+g_{2}e^{g_{2}x- ((a(t)+b-c)g_{2}+g_{2}^{3})t}+g_{3}e^{g_{3}x-((a(t)+b-c)g_{3}+g_{3}^{3})t}]^{2}}{(1\pm e^{g_{1}x-((a(t)+b-c)g_{1}+g_{1}^{3})t}\pm e^{g _{2}x-((a(t)+b-c)g_{2}+g_{2}^{3})t}\pm e^{g_{3}x-((a(t)+b-c)g_{3}+g_{3}^{3})t})^{2}}. \end{aligned}$$
(47)

This demonstrates that for finite N (\(N\ge 1\)), N-kink solutions can be found and the TM equation is totally integrable.

We utilize Eqs. (35), (36) to get the dispersion relation, Eq. (41) to find the value of \(A_{12}\), which is then generalized for the remaining components \(A_{jk}\), Eq. (45) to find \(B_{123}\), which is provided by \(B_{123}=A_{12}A_{23}A_{13}\) for totally integrable equations only. The existence of N-soliton solutions for any order is confirmed by the discovery of three-soliton solutions.

5 Graphical representation of regular and singular kink solutions

Fig. 2
figure 2

a 3D wave profile for regular one kink soliton Eq. (40) against \(g_1=-2\), \(a(t)=\cos (t)\), \(c=2\), \(b=-0.6\). b Contour plot of Eq. (40)

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Fig. 3
figure 3

a 3D wave profile for regular two kink soliton Eq. (44) against \(g_1=-2\),\(g_2=3.3\), \(a(t)=\cos (t)\), \(c=2\), \(b=-0.6\). b Contour plot of Eq. (44)

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Fig. 4
figure 4

a 3D wave profile for regular three kink soliton Eq. (47) against \(g_1=-2\), \(g_2=3.3\), \(g_3=4\), \(a(t)=\cos (t)\), \(c=2\), \(b=-0.6\). b Contour plot of Eq. (47)

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Fig. 5
figure 5

a 3D wave profile for singular one kink soliton Eq. (40) against \(g_1=-2\), \(a(t)=\cos (t)\), \(c=2\), \(b=-0.6\). b Contour plot of Eq. (40)

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Fig. 6
figure 6

a 3D wave profile for singular two kink soliton Eq. (44) against \(g_1=-2\),\(g_2=3.3\), \(a(t)=\cos (t)\), \(c=2\), \(b=-0.6\). b Contour plot of Eq. (44)

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Fig. 7
figure 7

a 3D wave profile for singular three kink soliton Eq. (47) against \(g_1=-2\), \(g_2=3.3\), \(g_3=4\), \(a(t)=\cos (t)\), \(c=2\), \(b=-0.6\). b Contour plot of Eq. (47)

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6 Results and discussion

In conclusion, incorporating Painlevé analysis into the study of a dynamical thermopherotic equation governing wrinkle propagation can provide a systematic and powerful tool for assessing integrability, obtaining analytical solutions by auto-Bäcklund transformation and gaining insights into the underlying dynamics and the novelty of multiple regular and multiple singular kink solutions lies in the intricate dynamics, bifurcation phenomena, and the mathematical and physical insights they provide for understanding complex systems. The dynamics of the obtained solutions to the underlying problem have been thoroughly discussed and displayed by 3-D and 2-D wave profiles. Figure 1, is drawn against solution Eq. (34), which is a bright soliton. Bright solitons might represent specific, localized wrinkle structures that have a pronounced impact on the electrical conductivity of the graphene sheet. These structures could create variations in the conductivity profile due to changes in the material’s geometry and thermal properties. Bright solitons in the context of thermopherotic equations may suggest nonlinear thermal effects that influence the electrical conductivity of graphene sheets in a localized and persistent manner. Figures 2, 3, 4 represents the 3D and contour plots of regular one-kink, two-kink and three-kink soliton solutions of Eqs. (40), (44) and (47), respectively. Figures 5, 6, 7 represents the 3D and contour plots of singular one-kink, two-kink and three-kink soliton solutions of Eqs. (40), (44) and (47), respectively.

7 Conclusion

This study investigates multi-soliton solutions to the TM equation, which addresses the thermophoresis of creases in graphene sheets. We computed the Painlevé integrability of (2) using the Painlevé analysis. The Painlevé–Bäcklund transformation was then obtained using the truncated Painlevé expansion. Soliton and periodic wave solutions were found using the Painlevé auto-Bäcklund transformation with constant seed solutions. 3-D, 2-D and contour profiles of these solutions have been illustrated, which show travelling wave-like and periodic solutions. Multiple regular kink and singular kink solutions were explicitly deduced by employing the Hirota bilinear method. The ABT technique is more effective, easy and uncomplicated to discover the exact solutions of NPDEs and the Hirota bilinear technique is a well-known and effective way to find kink solutions. In future, studying the coupling of bright solitons and kinks with other nonlinear phenomena, such as solitary waves or breathers and explore how the interaction between different nonlinear structures affects the overall dynamics and stability of wrinkle patterns will be much more interesting.