1 Introduction

Most of nonlinear phenomena can be easily studied by nonlinear partial differential equations (NPDEs) [1,2,3,4,5,6,7,8]. Researchers have studied various kinds of waves through this powerful tool, such as mixed lump wave [9], multi-waves [10], three-wave [11], breather [12], rogue waves [13,14,15], multiple complex soliton [16], bright and dark soliton [17], complex wave [18], soliton solution [19,20,21,22], traveling wave solutions [23], lump solution [24,25,26,27,28], dark waves [29], double-wave solutions [30], interaction solution [31,32,33,34,35,36,37,38]. At the same time, various methods have been developed to study these NPDEs, such as, Hirota bilinear method [39], the general bilinear techniques [40], bilinear neural network method [41,42,43,44], the tanh method [45], extended tanh method [46], improved \((G'/G)\)-expansion [47], sine-cosine method [48], tanh-coth method [49], Lie group method [50], modified transformed rational function method [51]. Most of these methods listed above can be regarded as trial function method. Considering following general form of NPDEs,

$$\begin{aligned} P\left( \psi , \psi _{t}, \psi _{s}, \psi _{s s}, \ldots \right) =0, \end{aligned}$$
(1)

where \(\psi \) is a complex function. To find the analytical solutions of Eq. (1), the trial function is constructed as follows:

$$\begin{aligned} \psi =\left[ \varPsi _0+\varPsi \left( \xi \right) \right] \mathrm {e}^{i\left( c_{2} t+k_{2} s\right) } , \end{aligned}$$
(2)

where \(\xi =c_{1} t+k_{1} s\), \(\varPsi \left( \xi \right) \) can be any function with the independent variable \(\xi \), such as \(\tanh (\xi )\), \(\cos (\xi )\), \(\tanh (\xi )+\cos (\xi )\) and so on. \(\varPsi \left( \xi \right) \) can even be an arbitrary function \(F(\xi )\) or \(F(\xi )+F^2(\xi )\) and the like. Next, substituting Eq. (2) into Eq. (1), extracting the coefficients of \(\mathrm {e}^{i\left( c_{2} t+k_{2} s\right) }\) and then collecting the coefficients of \(\varPsi \left( \xi \right) \) in both real part and imaginary part, the system of equations can be obtained. Solving these equations, the constraint solutions of the coefficients in the original equation Eq. (2) and the trial function Eq. (1) will be obtained. By introducing these coefficient solutions into the trial function Eq. (1), the explicit solution \(\psi \) of Eq. (1) will be obtained. However, it is rare to study nonlinear option equations by using these powerful tools.

In this paper, we investigate the following Ivancevic option pricing model:

$$\begin{aligned} \mathrm {i} \partial _{t} \psi =-\frac{1}{2} \sigma \partial _{s s} \psi -\beta |\psi |^{2} \psi , \quad (\mathrm {i}=\sqrt{-1}). \end{aligned}$$
(3)

This is a wave-form, nonlinear, stochastic and adaptive option pricing model. This model was first proposed by Ivancevic in Ref. [52] to satisfy both behavioral and efficient markets, where \(\sigma \) means the volatility, which represents either stochastic process itself or just a constant. Landau coefficient \(\beta \) = \(\beta (r, w)\) means the adaptive market potential. In simplest nonadaptive scenario, \(\beta \) is equal to the r, which represents interest rate, while in the adaptive case, \(\beta (r, w)\) can be related to the market temperature and it depends on the set of adjustable parameters \(\{W_i\}\). The independent variable t represents time, and s represents asset price. Response variable \(\psi (s,t)\) represents the option price wave function, and it is the probability density function \(|\psi (s,t)|^2\) that represents the potential field. A novel analytical technique for the solution of time-fractional Ivancevic option pricing model has been studied by Jena et. al. [53].

The organization of this paper is as follows. In Sect. 2, dark wave solutions of Eq. (3) will be obtained through the tanh expansion method. In Sect. 3, rogue wave solutions of Ivancevic option pricing model will be obtained via trial function method. The dynamical characteristics of corresponding rogue waves will be exhibited through curve plots, 3D plots, density plots and contour plots. In Sect. 4, perturbation solutions are obtained through direct perturbation method. Section 5 will conclude this paper.

2 Dark wave of Ivancevic option pricing model

To get the financial dark wave solutions of Eq. (3), a transformation is given as follows:

$$\begin{aligned} \psi =e^{i\left( c_{2} t+k_{2} s\right) } \varPsi \left( \xi \right) , \end{aligned}$$
(4)

where \(\xi =c_{1} t+k_{1} s\). Substituting transformation (4) into Eq. (3), we get a complex equation,

$$\begin{aligned}&e^{i (t c_{2}+s k_{2})}\left( \frac{\left( \frac{d^{2}}{d \xi ^{2}} \varPsi (\xi )\right) \sigma k_{1}^{2}}{2}+\left( i \sigma k_{1} k_{2}+i c_{1}\right) \right. \nonumber \\&\qquad \left. \left( \frac{d}{d \xi } \varPsi (\xi )\right) +\varPsi (\xi )\left( \beta |\varPsi (\xi )|^{2}-\frac{\sigma k_{2}^{2}}{2}-c_{2}\right) \right) \nonumber \\&\quad =0 . \end{aligned}$$
(5)

The real and imaginary parts of Eq. (5) are extracted as follows:

$$\begin{aligned}&\left( \sigma k_{1} k_{2}+c_{1}\right) \left( \frac{d}{d \xi } \varPsi (\xi )\right) , \end{aligned}$$
(6)
$$\begin{aligned}&\frac{\left( \frac{d^{2}}{d \xi ^{2}} \varPsi (\xi )\right) \sigma k_{1}^{2}}{2}+\beta \varPsi (\xi )|\varPsi (\xi )|^{2}\nonumber \\&\quad -\frac{\varPsi (\xi ) \sigma k_{2}^{2}}{2}-\varPsi (\xi ) c_{2}. \end{aligned}$$
(7)

Making the following transformation to Eq. (7),

$$\begin{aligned} \varPsi (\xi )=a_{0}+a_{1} \tanh (\xi ), \end{aligned}$$
(8)

The real part of Eq. (5) is transformed as

$$\begin{aligned}&\tanh ^{3}(\xi ) \sigma a_{1} k_{1}^{2}-a_{1} \tanh (\xi ) \sigma k_{1}^{2}+\tanh (\xi )\left| a_{0}\nonumber \right. \\&\left. \quad +a_{1} \tanh (\xi )\right| ^{2} \beta a_{1}\nonumber \\&\quad +\left| a_{0}+a_{1} \tanh (\xi )\right| ^{2} \beta a_{0} -\frac{\tanh (\xi ) \sigma a_{1} k_{2}^{2}}{2}\nonumber \\&\quad -\frac{\sigma a_{0} k_{2}^{2}}{2}-\tanh (\xi ) a_{1} c_{2}-a_{0} c_{2}. \end{aligned}$$
(9)

The term of \(\frac{d}{d \xi } \varPsi (\xi )\) in Eq. (8) and the terms in Eq. (9) having same order of \(\tanh (\xi )\) are collected. Then, equating these equations to 0, a system of equations for concerned parameters is obtained as follows:

$$\begin{aligned} \begin{aligned}&\sigma k_{1} k_{2}+c_{1}=0,\\&\left( \beta a_{0}^{2}-\frac{\sigma k_{2}^{2}}{2}-c_{2}\right) a_{0}=0,\\&\beta a_{1}^{3}+\sigma a_{1} k_{1}^{2}=0,\\&3 \beta a_{0} a_{1}^{2}=0, \\&3 a_{1}\left( \beta a_{0}^{2}+\left( -\frac{k_{1}^{2}}{3}-\frac{k_{2}^{2}}{6}\right) \sigma -\frac{c_{2}}{3}\right) =0. \end{aligned} \end{aligned}$$
(10)

The following three sets of solutions of Eq. (10) are obtained

$$\begin{aligned}&\text{ Case1: } \left\{ a_{0}=a_{0}, a_{1}=0, c_{1}=-\sigma k_{1} k_{2}, c_{2}\right. \nonumber \\&\left. \qquad =\beta a_{0}^{2}-\frac{\sigma k_{2}^{2}}{2}, k_{1}=k_{1}, k_{2}=k_{2}\right\} ,\nonumber \\&\text{ Case2: } \left\{ a_{0}=0, a_{1}=\sqrt{-\frac{\sigma }{\beta }} k_{1}, c_{1}\right. =-\sigma k_{1} k_{2}, c_{2}\nonumber \\&\qquad \left. =-\sigma k_{1}^{2}-\frac{1}{2} \sigma k_{2}^{2}, k_{1} =k_{1}, k_{2}=k_{2} \right\} ,\nonumber \\&\text{ Case3: } \left\{ a_{0}=0, a_{1}=-\sqrt{-\frac{\sigma }{\beta }} k_{1}, c_{1}=-\sigma k_{1} k_{2}, c_{2}\right. \nonumber \\&\left. \qquad =-\sigma k_{1}^{2}-\frac{1}{2} \sigma k_{2}^{2}, k_{1}=k_{1}, k_{2}=k_{2}\right\} . \end{aligned}$$
(11)

Substituting case 1 of Eq. (11) into (8), the explicit solution \(\psi _1\) of Eq. (3) via transformation (4) is obtained,

$$\begin{aligned} \psi _1=\mathrm {a}_{0} e^{\frac{i\left( 2 t \beta a_{0}^{2}-t \sigma k_{2}^{2}+2 s k_{2}\right) }{2}}. \end{aligned}$$
(12)

Substituting case 2 of Eq. (11) into (8), the explicit solution \(\psi _2\) of Eq. (3) via transformation (4) is obtained,

$$\begin{aligned}&\psi _2=e^{-\frac{i}{2}\left( 2 t \sigma k_{1}^{2}+t \sigma k_{2}^{2}-2 k_{2} s\right) } \nonumber \\&\quad \sqrt{-\frac{\sigma }{\beta }} k_{1} \tanh \left( -\sigma t k_{1} k_{2}+k_{1} s\right) . \end{aligned}$$
(13)
Fig. 1
figure 1

(Color online) The three-dimensional plots, density plot, curve plots and contour plot for the strength \(|\psi _3|\) of dark wave solutions

Full size image

Substituting case3 of Eq. (11) into (8), the explicit solution \(\psi _3\) of Eq. (3) via transformation (4) is obtained,

$$\begin{aligned} \psi _3=&-e^{-\frac{i}{2}\left( 2 t \sigma k_{1}^{2}+t \sigma k_{2}^{2}-2 k_{2} s\right) } \nonumber \\&\sqrt{-\frac{\sigma }{\beta }} k_{1} \tanh \left( c_{1} t+k_{1} s\right) . \end{aligned}$$
(14)

Some appropriate values in Eq. (14) are given as: \( \beta =5,\sigma =3,k_2=2,k_1=4, \) to analyze the dynamics properties briefly. The wave function \(\psi _3\) with only two independent variables of time t and asset price s is obtained as follows:

$$\begin{aligned} \psi _3=\frac{4 \sqrt{15}}{5} \mathrm {i} \tanh (4 s-24 t) \mathrm {e}^{\mathrm {i}(2 (s-27 t))}. \end{aligned}$$
(15)

Figure 1 shows the three-dimensional plots, density plot, curve plots and contour plot for the strength \(|\psi _3|\) of dark wave solutions for Eq. (15).

3 Rogue wave of Ivancevic option pricing model

To obtain the analytical solutions of Eq. (3), a transformation is given as follows:

$$\begin{aligned} \psi =\mathrm {e}^{\mathrm {i}(\mathrm {ps}+\mathrm {qt})}\left( \psi _{0}+v_{y}\right) , \end{aligned}$$
(16)

where

$$\begin{aligned} v=a \ln \left( b+y^{2}\right) , \quad y=\mathrm {e}^{\mathrm {ks}-\omega \mathrm {t}}, \end{aligned}$$

from Eq. (16), the terms in Eq. (3) are obtained as follows:

$$\begin{aligned} \begin{aligned} \psi&=\frac{\mathrm {exp}\left( {\mathrm {i}(\mathrm {ps}+\mathrm {qt})}\right) \left( \psi _0 y^{2}+2 a y+b \psi _0\right) }{y^{2}+b},\\ |\psi |&=\frac{\left( y^{2}+b\right) \psi _0+2 a y}{y^{2}+b}, \\ \frac{\partial }{\partial t} \psi&=\left( \frac{i q\left( \psi _0 y^{2}+2 a y+b \psi _0\right) }{y^{2}+b} \right. \\&\quad +\left( \frac{-2 \psi _0 \omega y^{2}-2 a \omega y}{y^{2}+b}\right. \\&\quad \left. \left. +\frac{2 \omega y^{2}\left( \psi _0 y^{2}+2 a y+b \psi _0\right) }{\left( y^{2}+b\right) ^{2}}\right) \right) \mathrm {exp}\left( {\mathrm {i}(\mathrm {ps}+\mathrm {qt})}\right) ,\\ \frac{\partial }{\partial s} \psi&=\left( \frac{{\text {i p}} \left( \psi _0 y^{2}+2 a y+b \psi _0\right) }{y^{2}+b}\right. \\&\quad \left. +\frac{2 k y a\left( -y^{2}+b\right) }{\left( y^{2}+b\right) ^{2}}\right) \mathrm {exp}\left( {i(p s+q t)}\right) , \\ \frac{\partial ^{2}}{\partial s^{2}} \psi&=\left( -\frac{\left( \psi _0 y^{2}+2 a y+b \psi _0\right) p^{2}}{y^{2}+b}\right. \\&\quad +\frac{4 I k y a\left( -y^{2}+b\right) p}{\left( y^{2}+b\right) ^{2}} \\&\quad \left. +\frac{2 a k^{2} y\left( y^{4}-6 b y^{2}+b^{2}\right) }{\left( y^{2}+b\right) ^{3}}\right) \mathrm {exp}\left( {\mathrm {i}(\mathrm {ps}+\mathrm {qt})}\right) . \end{aligned} \end{aligned}$$
(17)

Substituting Eq. (17) into Eq. (3), we get the following algebraic equation,

$$\begin{aligned} \begin{aligned}&\frac{-q \left( \psi _0 \,y^{2}+2 a y +b \psi _0 \right) }{y^{2}+b}+\left( \frac{-2 \omega \psi _0 \,y^{2}-2 a \omega y}{y^{2}+b} +\frac{2 \omega \,y^{2} \left( \psi _0 \,y^{2}+2 a y +b \psi _0 \right) }{\left( y^{2}+b \right) ^{2}}\right) \mathrm {i}\\&\qquad +\frac{\sigma \left( -\frac{ \left( \psi _0 \,y^{2}+2 a y +b \psi _0 \right) p^{2}}{y^{2}+b}+\frac{4 \,\mathrm {i} k y a \left( -y^{2}+b \right) p}{\left( y^{2}+b \right) ^{2}}+\frac{2 a \,k^{2} y \left( y^{4}-6 b \,y^{2}+b^{2}\right) }{\left( y^{2}+b \right) ^{3}}\right) }{2} +\frac{\beta \left( \left( y^{2}+b \right) \psi _0 +2 a y \right) ^{2} \left( \psi _0 \,y^{2}+2 a y +b \psi _0 \right) }{\left( y^{2}+b \right) ^{3}} =0. \end{aligned} \end{aligned}$$
(18)

We collect the terms in Eq. (18) having same order of y and make them zero; the system of equations are obtained as follows:

$$\begin{aligned} \begin{aligned}&2 \beta \, \psi _0 ^{3}-p^{2} \sigma \psi _0 -2 q \psi _0 =0,\\&2 b^{3} \beta \, \psi _0 ^{3}-b^{3} p^{2} \sigma \psi _0 -2 b^{3} q \psi _0 =0, \\&6 b \beta \, \psi _0 ^{3}-3 b \,p^{2} \sigma \psi _0 +24 a^{2} \beta \psi _0 -6 b q \psi _0 =0,\\&6 b^{2} \beta \, \psi _0 ^{3}-3 b^{2} p^{2} \sigma \psi _0 +24 a^{2} b \beta \psi _0 -6 b^{2} q \psi _0 =0, \\&24 a b \beta \, \psi _0 ^{2}-12 a b \,k^{2} \sigma -4 a b \,p^{2} \sigma \\&\quad +16 a^{3} \beta -8 a b q=0,\\&\quad -4 \,\mathrm {i} a k p \sigma +12 a \beta \, \psi _0 ^{2}+2 a \,k^{2} \sigma -2 a \,p^{2} \sigma \\&\quad +4 \,\mathrm {i} a \omega -4 a q=0, \\&4 \,\mathrm {i} a \,b^{2} k p \sigma +12 a \,b^{2} \beta \, \psi _0 ^{2}+2 a \,b^{2} k^{2} \sigma -2 a \,b^{2} p^{2} \sigma \\&\quad -4 \,\mathrm {i} a \,b^{2} \omega -4 a \,b^{2} q=0. \end{aligned} \end{aligned}$$
(19)
Fig. 2
figure 2

(Color online) The three-dimensional plots, density plot, curve plots and contour plot of the strength \(|\psi |\) of rogue wave solutions for Eq. (21) by choosing \(\omega =2,k=3,b=3,\beta =2,\sigma =4\)

Full size image

Solving Eq. (19), we get the constraint relationship between the coefficients as follows:

$$\begin{aligned} \begin{array}{l} \text{ case1: } \left\{ a=\sqrt{\frac{b \sigma }{\beta }} k, p=\frac{\omega }{k \sigma }, q=\frac{k^{4} \sigma ^{2}-\omega ^{2}}{2 k^{2} \sigma }, \psi _0=0\right\} , \\ \text{ case2: } \left\{ a=-\sqrt{\frac{b \sigma }{\beta }} k, p=\frac{\omega }{k \sigma }, q=\frac{k^{4} \sigma ^{2}-\omega ^{2}}{2 k^{2} \sigma }, \psi _0=0\right\} . \end{array}\nonumber \\ \end{aligned}$$
(20)

Substituting the case 1 in Eq. (20) into Eq. (17), the explicit solution \(\psi \) of Eq. (3) via transformation (16) is obtained,

$$\begin{aligned} \psi =\frac{2 e^{i\left( \frac{\omega s}{k \sigma }+\frac{\left( k^{4} \sigma ^{2}-\omega ^{2}\right) t}{2 k^{2} \sigma }\right) } \sqrt{\frac{b \sigma }{\beta }} k \mathrm {e}^{\mathrm {ks}-\omega \mathrm {t}}}{\left( e^{k s-\omega t}\right) ^{2}+b}. \end{aligned}$$
(21)

In order to analyze the dynamics of the solution, some parameters in Eq. (21) are given as follows:

$$\begin{aligned} \omega =2,k=3,b=3,\beta =2,\sigma =4. \end{aligned}$$
(22)

Figure 2 shows the three-dimensional plots, density plot, curve plots and contour plot of the strength \(|\psi |\) of rogue wave solutions for Eq. (21).

4 Perturbation solutions of Ivancevic option pricing model

As we all know, there is white noise in option model. In order to restore the real situation of the option model, we add a perturbation term to the Ivancevic option pricing model Eq. (3) and the Ivancevic option pricing model with loss is obtained as follows:

$$\begin{aligned} \mathrm {i} \partial _{t} \psi +\frac{1}{2} \sigma \partial _{s s} \psi +\beta |\psi |^{2} \psi =-i \epsilon \psi , \quad (\mathrm {i}=\sqrt{-1}).\nonumber \\ \end{aligned}$$
(23)

To obtain the perturbation solutions, \(\psi \) is expanded as follows:

$$\begin{aligned} \begin{aligned} \psi&=\mathrm {e}^{\epsilon (a+\mathrm {i} b)} \psi ^{\prime }(\xi , \tau , \epsilon ) \\&=\mathrm {e}^{\epsilon (a+\mathrm {i} b)}\left[ \psi _{0}(\xi , \tau )+\epsilon \psi _{1}(\xi , \tau )+O\left( \epsilon ^{2}\right) \right] , \end{aligned}\nonumber \\ \end{aligned}$$
(24)

where \(a=a(t,s), b=b(t,s), \xi =\xi (t,s,\varGamma ), \tau =\tau (t,s,\varGamma )\) and \(\{\xi ,\tau \}\) satisfy the following relationship,

$$\begin{aligned} \{\xi , \tau \} {\mathop {\longrightarrow }\limits ^{\epsilon \rightarrow 0}}\{t, s\}. \end{aligned}$$
(25)

Substituting Eq. (24) into Eq. (23),

$$\begin{aligned}&\mathrm {i}\left( \frac{\sigma }{2} \psi _{0 \tau \tau } \tau _{s}^{2}+\mathrm {e}^{2 \epsilon a} \beta \left| \psi _{0}\right| ^{2} \psi _{0}\right) -\psi _{0 \xi } \xi _{t}\nonumber \\&\quad +\epsilon \left\{ \left[ \mathrm {i}\left( \frac{\sigma }{2} \psi _{1 \tau \tau } \tau _{s}^{2}+e^{2 \epsilon a} \beta \left( 2\left| \psi _{0}\right| ^{2} \psi _{1}+\psi _{0}^{2} \psi _{1}^{*}\right) \right. \right. \right. \nonumber \\&\quad \left. -\psi _{1 \xi } \xi _{t}\right] \nonumber \\&\quad +\left[ \frac{\sigma }{2}\left( \mathrm {i} a_{t t}-b_{t t}\right) -a_{t}-\mathrm {i} b_{t}-1\right] \psi _{0}\nonumber \\&\qquad \left[ \frac{1}{\epsilon }\left( \mathrm {i} \frac{\sigma }{2} \tau _{s s}-\tau _{t}\right) +\sigma \tau _{s}\left( \mathrm {i} a_{s}-b_{s}\right) \right] \psi _{0 \tau } \nonumber \\&\quad \left. + \mathrm {i} \sigma \frac{\xi _{s}}{\epsilon } \tau _{s} \psi _{0 \xi \tau }\right\} =O\left( \epsilon ^{2}\right) . \end{aligned}$$
(26)

Let the coefficient of the same power of \(\epsilon \) be zero, and the following approximate equations are obtained,

$$\begin{aligned}&-\psi _{0 \xi \xi } \xi +\mathrm {i}\left( \frac{\sigma }{2} \psi _{0 \tau \tau } \tau _{s}^{2}+\mathrm {e}^{2 \epsilon a} \beta \left| \psi _{0}\right| ^{2} \psi _{0}\right) =0, \end{aligned}$$
(27)
$$\begin{aligned}&-\psi _{1 \xi } \xi _{t}+\mathrm {i}\left[ \frac{\sigma }{2} \psi _{1 \tau \tau } \tau _{s}^{2}+\mathrm {e}^{2 \epsilon a} \beta \left( 2\left| \psi _{0}\right| ^{2} \psi _{1}+\psi _{0}^{2} \psi _{1}^{*}\right) \right] \nonumber \\&\quad +\left[ \frac{\sigma }{2}\left( \mathrm {i} a_{s s}-b_{s s}\right) -a_{t}-\mathrm {i} b_{t}-1\right] \psi _{0} \nonumber \\&\quad +\left[ \frac{1}{\epsilon }\left( \mathrm {i} \frac{\sigma }{2} \tau _{s s}-\tau _{t}\right) +\sigma \tau _{s}\left( \mathrm {i} a_{s}-b_{s}\right) \right] \psi _{0 \tau } \nonumber \\&\quad + i \sigma \frac{\xi _{s}}{\epsilon } \tau _{s} \psi _{0 \xi \tau }=0. \end{aligned}$$
(28)

Because \(\psi _0\) in Eq. (27) is not explicitly related to \(\epsilon \), the following relationship can be obtained,

$$\begin{aligned} \xi _{t}=\mathrm {e}^{2 \epsilon a}, \tau _{s}=\mathrm {e}^{\epsilon a}, \end{aligned}$$
(29)

so \(\psi _0\) is the exact solution of Eq. (3). From Eq. (24), \(\psi _1\) is the solution of following equation,

$$\begin{aligned}&-\psi _{1 \xi }+\mathrm {i}\left[ \frac{sigma}{2} \psi _{1 \tau \tau }+\beta _{2}\left( 2\left| \psi _{0}\right| ^{2} \psi _{1}+\psi _{0}^{2} \psi _{1}^{*}\right) \right] \nonumber \\&\quad =0. \end{aligned}$$
(30)
Fig. 3
figure 3

(Color online) The three-dimensional plot, density plot and contour plot of the intensity \(|\psi |^2\) of perturbation solutions for Eq. (33) by choosing \(\beta = -4,\) \(\sigma = 2\), \(\epsilon = 0.01\), \(k_1 = 0.3\), \(k_2 = 0.5\)

Full size image

From Ref [54], the solution of Eq. (30) can be \(\psi _1=\psi _{0\xi }\) or \(\psi _1=\psi _{0\tau }\). For a given nontrivial solution \(\psi _0\), Eqs. (28-30) are consistent in any t and s, so the last three terms of Eq. (28) are all equal to zero and we can get,

$$\begin{aligned} \begin{aligned}&\xi _{s}=0,\\&\frac{sigma}{2}\left( \mathrm {i} a_{s s}-b_{s s}\right) -a_{t}-\mathrm {i} b_{t}-1=0, \\&\frac{1}{\epsilon }\left( \mathrm {i} \frac{sigma}{2} \tau _{s s}-\tau _{t}\right) +\sigma _{1} \tau _{s}\left( \mathrm {i} a_{s}-b_{s}\right) =0. \end{aligned} \end{aligned}$$
(31)

The solutions of Eq. (29) and Eq. (31) are obtained,

$$\begin{aligned} a=-2 t, b=\frac{s^{2}}{\sigma _{1}}, \tau =\mathrm {e}^{-2 \epsilon t} s, \xi =\frac{1}{4 \epsilon }\left( 1-\mathrm {e}^{-4 \epsilon t}\right) .\nonumber \\ \end{aligned}$$
(32)
Fig. 4
figure 4

(Color online) The curve plots of the intensity \(|\psi |^2\) of perturbation solutions for Eq. (33) by choosing \(\beta = -4,\) \(\sigma = 2\), \(\epsilon = 0.01\)(left), \(k_1 = 0.3\), \(k_2 = 0.5\)

Full size image

Substituting the exact solution Eq. (32) into Eq. (14), through transformation (24) and \(\psi _1=\psi _{0\xi }\), the perturbation solutions of Ivancevic option pricing model Eq. (14) can be obtained as follows:

$$\begin{aligned} \begin{aligned}&\psi = -\frac{1}{2}{\mathrm e}^{\frac{\epsilon \left( \mathrm {I} s^{2}-2 t \sigma \right) }{\sigma }} {\mathrm e}^{\frac{\frac{\mathrm {I}}{8} \left( 8 \,{\mathrm e}^{-2 \epsilon t} s k_{2} \epsilon +2 \sigma k_{1}^{2} {\mathrm e}^{-4 \epsilon t}+\sigma k_{2}^{2} {\mathrm e}^{-4 \epsilon t}-2 \sigma k_{1}^{2}-\sigma k_{2}^{2}\right) }{\epsilon }}\\&\qquad \sqrt{-\frac{\sigma }{\beta }}\, k_{1}\\&\qquad \left( 2 \,\mathrm {I} \epsilon k_{1}^{2} \tanh \! \left( \frac{k_{1} \left( 4 \,{\mathrm e}^{-2 \epsilon t} s \epsilon +\sigma k_{2} {\mathrm e}^{-4 \epsilon t}-\sigma k_{2}\right) }{4 \epsilon }\right) \sigma \right. \\&\quad +\mathrm {I} \epsilon \tanh \! \left( \frac{k_{1} \left( 4 \,{\mathrm e}^{-2 \epsilon t} s \epsilon +\sigma k_{2} {\mathrm e}^{-4 \epsilon t}-\sigma k_{2}\right) }{4 \epsilon }\right) \sigma k_{2}^{2}\\&\quad \left. -2 \epsilon \left( \tanh ^{2}\left( \frac{k_{1} \left( 4 \,{\mathrm e}^{-2 \epsilon t} s \epsilon +\sigma k_{2} {\mathrm e}^{-4 \epsilon t}-\sigma k_{2}\right) }{4 \epsilon }\right) \right) \sigma k_{1} k_{2}\right. \\&\quad \left. +2 \epsilon k_{1} \sigma k_{2}-2 \tanh \right. \\&\left. \quad \left( \frac{k_{1} \left( 4 \,{\mathrm e}^{-2 \epsilon t} s \epsilon +\sigma k_{2} {\mathrm e}^{-4 \epsilon t}-\sigma k_{2}\right) }{4 \epsilon }\right) \right) . \end{aligned} \end{aligned}$$
(33)

By choosing \(\beta = -4,\) \(\sigma = 2\), \(\epsilon = 0.01\), \(k_1 = 0.3\), \(k_2 = 0.5\) in Eq. (32), the three-dimensional plot, density plot and contour plot of the intensity \(|\psi |^2\) of perturbation solutions for Eq. (33) are shown well in Fig. 3. Figure 4 shows the curve plots of Eq. (33), from which we can find that perturbation solutions decays rapidly with the increase in \(\epsilon \).

5 Conclusions

In this work, we have constructed the rogue wave solutions and the dark wave solutions of Ivancevic option pricing model by choosing some different trial functions. With the help of symbolic computing technology, the rogue wave solutions of Ivancevic option pricing model are obtained via trial function method and the dark wave solutions of Ivancevic option pricing model are obtained via tanh method. Perturbation solutions are obtained through direct perturbation method. Various curve plots, density plot, three-dimensional plots and contour plots, and dynamical characteristics of these waves are shown well using Maple.