1 Introduction

With the developing technology, magneto-optical and ferromagnetic materials are gaining importance. Iron, cobalt, nickel and various forms of their alloys are known as ferromagnetic materials or ferromagnetism, which are substances with a relative magnetic permeability greater than one (Gao et al. 2021a; Gawronski et al. 2020; Gorkov et al. 2020). Magneto-optics concerns itself with the impact of magnetic fields on matter throughout the processing stage, in computer data storage and waveguides (Gao et al. 2021a, b; Polatkan et al. 2020; Nishimori et al. 1998). Additionally, fluid mechanics deals with the mechanics of liquids, gases and plasmas and the forces acting on them (Gao et al. 2021a; Pinar Izgi 2022). With view of mechanics for these materials, (3+1)-dimensional variable coefficient modified Kadomtsev–Petviashvili (vcmKP) system is proposed

$$ u_{t} + \theta_{1} (t)u_{xxx} - \theta_{2} (t)u^{2} u_{x} + \theta_{3} (t)u_{x} \upsilon + \theta_{4} (t)\upsilon_{y} + \theta_{5} (t)u_{x} \omega + \theta_{6} (t)\omega_{z} = 0, $$
(1)
$$ \upsilon_{x} = u_{y} ,\;\omega_{x} = u_{z} , $$

that simulates electromagnetic, water, and powder-acoustic/ion-acoustic/dust-ion-acoustic waves (Gao et al. 2021a, 2022a).\(\theta_{i} (t)\)\(\left( {i = 1,2} \right)\) are non-zero functions whereas \(\theta_{j} (t)\)\(\left( {j = 3,4,5,6} \right)\) are all the real differentiable functions of \(t\).These processes can be classified respect to the coefficients of Eq. (1):

  1. I)

    \(\theta_{1} (t) = 1\), \(\theta_{2} (t) = \frac{3}{2}\), \(\theta_{3} (t) = 0\), \(\theta_{4} (t) = 0\),\(\theta_{5} (t) = - 3\), \(\theta_{6} (t) = 3\) and with \(u(x,y,t)\) being first derivative of the angle formed as a result of uniform magnetization of the medium, it’s possible application in magneto-optical recording and the expression for electromagnetic waves in a thin, uncharged, isotropic ferromagnetic film is expressed by Eq. (1) (Gao et al. 2021a, 2022a; Veerakumar and Daniel 2003; Luo and Chen 2016; Jiang et al. 2013; Sun et al. 2009).

  2. II)

    For \(\theta_{1} (t) = - \frac{1}{4}\), \(\theta_{2} (t) = - \frac{3}{2}\), \(\theta_{3} (t) = \pm \frac{3}{2}\), \(\theta_{4} (t) = - \frac{3}{4}\), \(\theta_{5} (t) = 0\), \(\theta_{6} (t) = 0\), two process can be seen; ion-acoustic waves (Luo and Chen 2016; Jiang et al. 2013; Xu et al. 2008; Hao and Zhang 2010) or water waves (Gao et al. 2021a, 2022a; Luo and Chen 2016; Jiang et al. 2013; Ren and Lin 2020; Wazwaz 2011) such that \(u(x,y,t)\) denotes the amplitude or height of the wave expressed in Eq. (1).

  3. III)

    For \(\theta_{1} (t) = \alpha\), \(\theta_{2} (t) = \beta\), \(\theta_{3} (t) = 0\), \(\theta_{4} (t) = \sigma\), \(\theta_{5} (t) = 0\), \(\theta_{6} (t) = \sigma\), where \(\alpha\), \(\beta\), \(\sigma\) are real constants expressing the phase velocity of the plasma-acoustic mode and the normalized electron densities in the equilibrium state, maintaining the charge neutrality condition, Eq. (1) denotes a non-interacting multi-component plasma process that models the presence of cold ions and two-temperature electrons with distinct Maxwell distributions by employing two Boltzmann relationships (Gao et al. 2021a; Das and Sarma 1998; Tariq and Seadawy 2017); where \(u(x,y,z,t)\) is a first-order perturbation of the normalized electrostatic current potential.

In addition to the cases specified by the parameter definition, Eq. (1) is used for cold, collisionless, non-magnetized and bi-ion temperature, electrons, low/high temperature ions, and dust-acoustic waves in extremely massive highly negatively charged dust plasma and charged dust grains of different sizes (Lin and Duan 2005); for powder-ion-acoustic waves in a magnetized dusty plasma, whose components consist of electrons with certain uncommon velocity distributions, large dust grains and negatively charged ions (Saha and Chatterjee 2014), and generally in processes in fiber optic communication (Yin et al. 2019, 2020).

As seen from the first two cases, with the chosen coefficients, Eq. (1) reduces into (2+1)- dimensional vcmKP system, whereas third one is the (3+1)- dimensional vcmKP system. A special model is represented by the(2+1)- dimensional vcmKP system (Gao et al. 2022b) which can be considered as the extension of Case I;

$$ u_{y} = \omega_{x} , $$
$$ \begin{gathered} u_{t} - \theta_{1} (t)\left( {u_{xxx} - 6u^{2} u_{x} } \right) - \theta_{2} (t)(xu_{x} + u) - 6\theta_{1} (t)\left( {\lambda f_{1} (t)^{2} + f_{0} (t) + xf(t) - g(t)\omega } \right)u_{x} \hfill \\ - \theta_{3} (t)u_{x} - 3\theta_{1} (t)g(t)^{2} \omega_{y} - \left( {\theta_{4} (t)y + \theta_{5} (t)} \right)u_{y} - 6\theta_{1} (t)f(t)u = 0. \hfill \\ \end{gathered} $$
(2)

Such that \(u\left( {x,y,t} \right)\) corresponds to the first derivative of the angle between the electromagnetic wave propagation direction, \(\lambda\) is a real constant, and \(\omega \left( {x,y,t} \right)\), \(\theta_{i} (t)\)\(\left( {i = 1,2,3,4,5} \right)\), \(f_{1} (t)\), \(f_{0} (t)\), \(f(t)\), \(g(t)\) are real differentiable functions. Regarding magnetic induction and magnetization together with the dielectric constant and magnetic permeability of the medium, Eq. (2) can be reduced from the (2+1) dimensional Maxwell–Landau–Lifshitz system for an isotropic ferromagnetic medium in the absence of static and moving charges, but in the presence of an external magnetic field in terms of spin–spin exchange interaction and Zeeman energy together with the Bohr magneton and gyromagnetic ratio (Gao et al. 2022b).

In the literature, auto-Backlund and and Darboux transformations (Wang and Wang 2018; Gao et al. 2022c; Zhu 1993), soliton-like solutions (Wang and Wang 2018; Gao et al. 2022c; Zhu 1993; Gao and Tian 2001a, b), Lax pairs (Wang and Wang 2018; Zhu 1993), bilinear forms (Wang and Wang 2018; Zhu 1993), infinite conservation laws (Gao et al. 2022b; Zhu 1993) have been studied for Eq. (1) (Case I and Case II) and Eq. (2).In this work, the case III of Eq. (1), generalized form of Eq. (1) and general form of Eq. (2) have been considered. As to the novelty of this paper, the travelling wave, soliton solutions of the considered systems [Eqs. (1), (2)] are hold by using Bernoulli method which is the well-known ansatz-based method and the analytical method. The obtained solutions are seen for the first time in this study as far as we know and are important for the development of the use of magneto-optical and ferromagnetic materials in industry and applied sciences, especially fiber optic communication. The second section provides a summary of the methodology, while the third section presents the obtained solutions and 3D plots.

2 Methods

Finding solutions to complex and nonlinear problems that exemplify real-world problems is very important and difficult, and each method used for this has its own limitations and disadvantages. Ansatz-based methods are among the most frequently used methods to obtain exact solutions of nonlinear PDEs. The method used does not require initial or boundary conditions and can therefore be used effectively and methodically to find various classes of nonlinear models that exemplify science and technology-related phenomena. The general process structure is same for all of them; the only difference is the choice of auxiliary equation. The first step for the ansatz-based methods (Pinar 2021, 2020; Pinar and Özis 2015; Pinar et al. 2020) is to obtain some exact solutions of the considered models is the reduction: the classical wave transformation is applied. For the Bernoulli’s type differential equation, which is the auxiliary equation,\(z^{\prime}\left( \xi \right) = Pz\left( \xi \right) + Qz^{2} \left( \xi \right)\), \(P\), \(Q\) represent parameters, and its solution is denoted as \(z\left( \xi \right) = - \frac{P}{{\exp \left( { - P\xi } \right)PC_{1} - Q}}\), where \(C_{1}\) is an integration constant. Also the solution is supposed as \(u\left( \xi \right) = \sum\limits_{i = 0}^{N} {g_{i} z^{i} \left( \xi \right)}\),\(g_{i}\)\((i = 0,...,N)\) are parameters that will be determined with the aim of the method and the definition of N is established through the application of the classical balancing principle that is balancing the linear term of highest order in the equation with the highest order nonlinear term) work only for positive integer values (Pinar and Özis 2015). After all, the solution assumption, and the Bernoulli-type differential equations, if necessary, are substituted into the reduced equation which is transformed into a nonlinear algebraic system by setting the coefficient of each power of \(z\left( \xi \right)\) to zero. The mentioned parameters are determined by solving the system of algebraic equations. Ultimately, by substituting the solution sets of the system of algebraic equations and the solution of the Bernoulli-type equation into the assumed solution, the exact solution can be obtained in the explicit form.

3 Results

3.1 The normalized electro-static potential model (case III)

With the assumptions Case III for Eqs. (1) turns into Eqs. (3)

$$ u_{t} + \alpha u_{xxx} - \beta u^{2} u_{x} + \sigma \upsilon_{y} + \sigma \omega_{z} = 0,\;\upsilon_{x} = u_{y} ,\;\omega_{x} = u_{z} . $$
(3)

Equation (3) is reduced by using \(\xi = \mu x + \eta y + \rho z - ct\), \(\mu \ne 0\), \(\eta \ne 0\), \(\rho \ne 0\),\(c \ne 0\) and as a result

$$ \mu \upsilon^{\prime } = \eta u^{\prime } ,\;\mu \omega^{\prime } = \rho u^{\prime } , $$
$$ - cu^{\prime } + \alpha \mu^{3} u^{\prime \prime \prime } - \beta \mu u^{2} u^{\prime } + \frac{{\sigma \left( {\eta + \rho } \right)}}{\mu }u^{\prime } = 0, $$
(4)

is obtained where \(u^{\prime} = \frac{\partial u}{{\partial \xi }}\), \(\omega^{\prime} = \frac{\partial \omega }{{\partial \xi }}\) and \(\upsilon^{\prime} = \frac{\partial \upsilon }{{\partial \xi }}\).The assumption is \(u\left( \xi \right) = g_{0} + g_{1} z\left( \xi \right)\),\(g_{i}\)\((i = 0,1)\) are parameters. Applying the procedure, the meaningful solution set is hold as \(c = \frac{{ - P\mu^{4} \alpha + 2\eta^{2} \sigma + 2\rho \sigma^{2} }}{2\mu }\), \(g_{0} = \pm \frac{{P\mu \sqrt {6\beta \alpha } }}{2\beta }\), \(g_{1} = \pm \frac{{Q\mu \sqrt {6\beta \alpha } }}{\beta }\). Other solution sets give trivial solutions.

The 3D plots of the blow-up and soliton solutions \(u(x,y,z,t)\) are shown in Fig. 1a and b for each solution, respectively for the parameters \(\beta = 1,\alpha = 1,\sigma = - 0.2,\mu = 1,\eta = - 1,\rho = 0.1,Q = 1.2\), \(C_{1} = 1\) and \(y = z = 5\).

Fig. 1
figure 1

Plot for the obtained solution set for \(P = - 2\) and \(P = 2\), respectively

Full size image

The sign of \(P\) is effective in the solution of the Eq. (3) that blow-up and travelling wave solution are obtained, respectively. It can be said that the behavior of the solution changes due to value of \(P\).

3.2 (2+1)- dimensional vcmKP system

To reduce equation Eq. (2), \(\xi = \mu x + \eta y - ct\), \(\mu \ne 0\), \(\eta \ne 0\), \(c \ne 0\) is considered

$$ \eta u^{\prime } = \mu \omega^{\prime } , $$
$$ \begin{gathered} - cu^{\prime } - \theta_{1} (t)\left( {\mu^{3} u^{\prime \prime \prime } - 6\mu u^{2} u^{\prime } } \right) - \theta_{2} (t)(x\mu u^{\prime } + u) - 6\theta_{1} (t)\left( {\lambda f_{1} (t)^{2} + f_{0} (t) + xf(t) - g(t)\frac{\eta }{\mu }u} \right)\mu u^{\prime } \hfill \\ \; - \theta_{3} (t)\mu u^{\prime } - 3\theta_{1} (t)g(t)^{2} \frac{\eta }{\mu }u^{\prime } - \left( {\theta_{4} (t)y + \theta_{5} (t)} \right)\eta u^{\prime } - 6\theta_{1} (t)f(t)u = 0, \hfill \\ \end{gathered} $$
(5)

is obtained where \(u^{\prime } = \frac{\partial u}{{\partial \xi }}\), \(\omega^{\prime } = \frac{\partial \omega }{{\partial \xi }}\).The assumption is \(u\left( \xi \right) = g_{0} + g_{1} z\left( \xi \right)\), \(g_{i}\)\((i = 0,1)\) are the parameters. By applying the aforementioned procedure, the solution sets are obtained and retained.

$$ P = \pm \frac{{\eta g(t) + 2\mu g_{0} }}{{\mu^{2} }},\;f\left( t \right) = - \frac{{\theta_{2} \left( t \right)}}{{6\theta_{1} \left( t \right)}},\;g_{1} = \pm Q\mu , $$
$$ c = \frac{{ - 6f_{1} \left( t \right)\theta_{1} \left( t \right)\lambda \mu^{2} - 2\eta^{2} \theta_{1} \left( t \right)g\left( t \right)^{2} + 2\eta \mu \theta_{1} \left( t \right)g\left( t \right)g_{0} + 2\mu^{2} \theta_{1} \left( t \right)g_{0}^{2} - \theta_{4} \left( t \right)\eta \mu y - 6f_{0} \left( t \right)\theta_{1} \left( t \right)\mu^{2} - \theta_{5} \left( t \right)\eta \mu - \theta_{3} \left( t \right)\mu^{2} }}{\mu }. $$

When the obtained results and \(z\left( \xi \right)\) are substituted into the solution assumption, the solution is obtained for Eqs. (2).

The 3D plots of the blow-up and shock wave solutions \(u(x,y,t)\) are shown in Fig. 2a–c for each solution, respectively for theparameters \(g_{0} = 1\), \(C_{1} = 2\),\(f_{1} \left( t \right) = \exp \left( t \right)\), \(\theta_{1} \left( t \right) = 1\), \(g\left( t \right) = 1\),\(\mu = 1\),\(Q = 1\),\(\eta = 2\), \(\lambda = 1\), \(f_{0} \left( t \right) = t\), \(\theta_{4} \left( t \right) = 1\), \(\theta_{3} \left( t \right) = 1\),\(\theta_{5} \left( t \right) = 1\),for \(t = 0\), \(x = 0\), \(y = 0\), respectively.

Fig. 2
figure 2

Plots for the negative sign set with assumed parameters for \(t = 0\), \(x = 0\), \(y = 0\), respectively

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3.3 A generalization of (3+1)-dimensional vcmKP system

Equation (1) is reduced by using \(\xi = \mu x + \eta y + \rho z - ct\), \(\mu \ne 0\), \(\eta \ne 0\), \(\rho \ne 0\),\(c \ne 0\) and as a result

$$ \mu \upsilon^{\prime } = \eta u^{\prime } ,\;\mu \omega^{\prime } = \rho u^{\prime } , $$
$$ - cu^{\prime } + \theta_{1} (t)\mu^{3} u^{\prime \prime \prime } - \theta_{2} (t)\mu u^{2} u^{\prime } + \theta_{3} (t)\eta u^{2} + \theta_{4} (t)\frac{\eta }{\mu }u^{\prime } + \theta_{5} (t)\rho u^{2} + \theta_{6} (t)\frac{\rho }{\mu }u^{\prime } = 0, $$
(6)

is obtained where \(u^{\prime } = \frac{\partial u}{{\partial \xi }}\), \(\omega^{\prime } = \frac{\partial \omega }{{\partial \xi }}\) and \(\upsilon^{\prime } = \frac{\partial \upsilon }{{\partial \xi }}\).. The assumption is \(u\left( \xi \right) = g_{0} + g_{1} z\left( \xi \right)\) and \(g_{i}\)\((i = 0,1)\) are parameters. By applying the aforementioned procedure, the solution sets are obtained and retained.

$$ P = \frac{{12Q\mu^{3} g_{0} \theta_{1} \left( t \right) - \eta g_{1}^{2} \theta_{3} \left( t \right)}}{{6\mu^{3} g_{1} \theta_{1} \left( t \right)}}, $$
$$ g_{1} = \pm \frac{{\mu \sqrt { \pm 6\eta Q\theta_{1} \left( t \right)\theta_{2} \left( t \right)\theta_{3} \left( t \right)\left( { \mp \mu g_{0} \theta_{2} \left( t \right) + \sqrt {3\mu^{2} g_{0}^{2} \theta_{2} \left( t \right)^{2} - 6\theta_{2} \left( t \right)\left( {\eta^{2} \theta_{4} \left( t \right) + \eta \mu \theta_{5} \left( t \right) + \rho^{2} \theta_{6} \left( t \right) - c\mu } \right)} } \right)} }}{{\eta \theta_{2} \left( t \right)\theta_{3} \left( t \right)}}. $$

The 3D plots of the soliton solutions \(u(x,y,z,t)\) are shown in Fig. 3a and b for each solution, respectively for the parameters \(C_{1} = 1\), \(\theta_{1} \left( t \right) = 1\),\(\theta_{2} \left( t \right) = 1\), \(\theta_{3} \left( t \right) = t^{2}\), \(\theta_{4} \left( t \right) = - 2\), \(\theta_{5} \left( t \right) = - t\), \(\theta_{6} \left( t \right) = t\)\(\eta = 1\), \(\rho = 1\), \(c = 1\), \(g_{0} = 1\), \(Q = 2\),\(z = 0\), \(y = 0\),\(\mu = 1\), and \(\mu = 0.5\), respectively.

Fig. 3
figure 3

Plots for different values of \(\mu\) respectively

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4 Conclusion

With the developing technology, magneto-optical and ferromagnetic materials are gaining importance and are used magneto-optics, ferromagnetism, fluid mechanics, etc. These processes are modeled via Kadomtsev–Petviashvili-type models. In this work, a generalized (3+1)-dimensional vcmKP system for various three cases are considered. As to the novelty of the funding, the analytical solutions as the soliton solutions, travelling solutions and blow-up solutions are hold for the observed problems by using the ansatz-based methods. The problem has been addressed for the first time from this point of view in the literature. Hence, the gap on analytical solutions is filled in the literature. As far as we know, the obtained solutions are seen for the first time in this study and are important for the development of the use of magneto-optical and ferromagnetic materials in industry, applied sciences and fiber optic communication fields. They will create a new perspective for the problems in magneto-optics, plasma physics, ferromagnetism, fluid mechanics, etc. and will be vital to guide theoretical development.