Introduction

Nonlinear ultrasonic waves are becoming more important in the realm of diagnosing and managing medical problems. Modern advances in ultrasound imaging have made it possible to more accurately assess the blood flow to organs and tumors, as well as to perform high-intensity targeted ultrasound ablation of diseased tissues1, as well as the use of micro/nano agents to facilitate the gene and medication flow using ultrasound. We are particularly interested in ultrasound-based devices because of their adaptability, which makes them ideal for monitoring patients during surgeries. For example, newborns who undergo open-heart surgery to treat fetal heart problems have significantly higher rates of brain and kidney dysfunction. Depending on the kind of cardiac surgery, several studies have shown that organ damage occurs at a frequency of 35–75 percent. To successfully monitor perfusion during surgery, there is an ongoing need for more advanced and portable imaging methods. The complex physiological structures surrounding arteries limit the use of ultrasonography in assessing perfusion status. The goal of developing specialized microbubble contrast agents is to enhance blood vessel visualization while reducing the effects of media heterogeneity. When these enhancing drugs are injected into the circulatory system, they cause nonlinear responses when they interact with ultrasound waves. Nonlinearity results in vibration frequencies that differ from the resonant frequency. The phenomena described above can be measured and examined to provide a visual description of nonlinear sources while also reducing interference caused by various qualities of the surrounding medium2.

The presence of nonlinearity in natural processes has prompted many scholars to acknowledge nonlinear study as a significant and cutting-edge domain, serving as an ongoing endeavor to comprehend the intricacy of the natural environment3,4,5. In order to thoroughly evaluate complex systems with time variability, it is necessary to explore a wide range of nonlinear partial differential equations (NLPDEs)6,7,8,9 and nonlinear fractional partial differential equations (FNLPDEs)10. These nonlinear equations potentially help us in comprehending the riddle by providing insight into the underlying processes that are elucidated and emphasizing the complexities that are not readily grasped by usual differential equations and conventional approaches. The most widely used class of FNLPDEs is the integrable soliton equations, which will be the focus of our study. These equations have extraordinary applications in several fields of applied mathematics and physical systems, including fluid mechanics, nonlinear optics, plasma physics, hydrodynamics and other related fields. These integrable systems are considered as an important class of FNLPDEs and soliton theory. Solving the soliton theory problems is one of the most significant physical challenge for these models. The nonlinear sciences face a significant challenge in the form of developing mathematical methods to formulate exact solutions to FNLPDEs. As a result, many researchers are in search to develop new techniques to face these challenges and get accurate exact solutions.

More importantly, the study of soliton theory is a very fascinating field of research in nonlinear sciences. Soliton theory have various applications in optoelectronics, such as magneto-optics, metamaterials, birefringent fibers, and so on. The theory of solitons has attracted the scientific community’s attention due to its significance in current research in telecommunications engineering and mathematical physics. There are several types of solitons, such as dark, bright, peakons, anti-kink, kink, temporal, spatiotemporal, spatial, singular and so on. Consequently, they have an important role in the field of nonlinear sciences. Various studies have shown different features of soliton solutions and their applications in scientific and technical fields11. Soliton solutions have important applications in many fields, including nonlinear optics, plasma physics, fluid dynamics, coastal engineering, and communications engineering. Mathematicians have devised sophisticated analytical and numerical techniques to find soliton solutions of various NLPDEs.

Exact solutions help with the investigation of the physical behaviors of the underlying system, these solutions are of paramount importance as they serve as a foundation for further investigations and research12,13. To obtain the exact solution of any physical system, it is needed to model the behaviors of the considered system as PDE or ODE. Where most of the natural or industrial complex phenomena can be modeled with NLPDEs or FNLPDEs. To extract the exact solutions in these situations, many researchers developed various approaches and techniques to deal with different situations. Here, for instance we list some of the well-known methods recently used in the literature, including Hirota bilinear method14, Darboux transformation15, the Backlund transformation16, the inverse scattering method17, tanh-coth method18, new sub equation method and modified Khater’s method19, iterative transform method20, modified simple equation technique21, bifurcation analysis22,23,24,25, Lie symmetry technique26, improved F-expansion function method27, Riccati equation mapping method28, truncated Painlevé approach29, Lie classical approach26, Adomian decomposition technique30, simplest equation technique31, \(tan(\frac{\phi }{2})\) technique32, multiple exp-function approach33, Bernoulli \(\frac{G^{\prime }}{G}\)-expansion method34, \(\frac{G^{\prime }}{G}\)-expansion method35 and36,37,38,39,40.

The main goal of this work is to study the \(\beta\)-fractional Westervelt equation and identify its various soliton solutions in the field of medical imgaging and therapy. The equation under study is of great relevance in the field of high-frequency ultrasound imaging, especially related to the propagation of nonlinear waves. To obtain the exact solutions of the proposed model, we use efficient integration methods, namely the modified generalized Riccati equation mapping method (MGREMM)41, modified generalized exponential rational function method (MGERFM)42 and logarithmic transformation43. The proposed methods enhance the understanding of various physical processes in related fields by computing new solutions of mathematical forms.

The outline of this article is given as: Sect. 2, describes the fundamental definitions and some of the key characteristics of the fractional derivative. Sect. 3, presents the governing equation of the Westervelt equation, whereas Sect. 4, presents the calculation of soliton solutions by applying the proposed methods with numerical simulation. Sect. 5, provides the discussion and concluding remarks.

Fractional order derivative

Various natural systems and scenarios can be mathematically represented by PDEs. These equations are employed for modelling a wide range of problems. Although new technology progresses, systems inherently grow more intricate, whereas differential equations offer a fundamental and straightforward model of the real process. To arrive at practical solutions for these intricate systems, it is necessary to introduce novel and enhanced mathematical tools. In order to address these issues, fractional PDEs are presented. Along with the improvement of the fractional calculus, a novel notion of fractional order derivatives called \(\beta\)-derivative is introduced. These derivatives of fractional order have garnered significant interest from scholars. Studies have recently demonstrated that these fractional derivatives offer a more precise comprehension of the solitary wave characteristics of nonlinear systems compared to other fractional derivatives. A particular category of derivatives offers scholars and experts a valuable instrument for comprehending and examining a range of physical, biological, and engineering systems.

The \(\beta\)-derivative

Definition 2.1

Let \(\kappa (t): [c,\infty )\rightarrow {\mathbb {R}}\), then the \(\beta\)-derivative44 of \(\kappa\) is given as:

$$\begin{aligned} {\mathscr {D}}_{t}^{\beta }\{\kappa (t)\}=\lim _{\epsilon \rightarrow 0}\frac{\kappa \left( t+\epsilon \left( t+\frac{1}{\Gamma (\beta )}\right) ^{1-\beta }\right) -\kappa (t)}{\epsilon },~~~\beta \in (0,1]. \end{aligned}$$
(1)

Theorem 2.1

44 Let \(h\ne 0\) and \(\kappa\) are two \(\beta\)-differentiable functions with \(\beta \in (0,1]\). Then:

  1. 1.

    \({\mathscr {D}}_{t}^{\beta }\left( c\{\kappa (t)\}+d\{ h(t)\}\right) =c{\mathscr {D}}_{t}^{\beta }\{\kappa (t)\}+d{\mathscr {D}}_{t}^{\beta }\{h(t)\},~\forall c,~d\in {\mathbb {R}}\).

  2. 2.

    \({\mathscr {D}}_{t}^{\beta }\left( \{\kappa (t)\}\times \{h(t)\}\right) = \{h(t)\}{\mathscr {D}}_{t}^{\beta }\left( \{\kappa (t)\}\right) +\{\kappa (t)\}{\mathscr {D}}_{t}^{\beta }\left( \{h(t)\}\right)\).

  3. 3.

    \({\mathscr {D}}_{t}^{\beta }\left( \frac{\{\kappa (t)\}}{\{h(t)\}}\right) =\frac{\{h(t)\}{\mathscr {D}}_{t}^{\beta }\{\kappa (t)\}-\{\kappa (t)\}{\mathscr {D}}_{t}^{\beta }\{h(t)\}}{\left( \{h(t)\}\right) ^2}\).

  4. 4.

    \({\mathscr {D}}_{t}^{\beta }\{\kappa (t)\}=\frac{d\{\kappa (t)\}}{dt}\left( t+\frac{1}{\Gamma (\beta )}\right) ^{1-\beta }\).

  5. 5.

    \({\mathscr {D}}_{t}^{\beta }\{c(t)\}=0, ~~{\text {where } c \text { is a constant}}.\)

The governing equation

The propagation of ultrasonic waves and their interactions with different mediums have been simulated using the Westervelt equation, often referred to as the nonlinear full-wave equation. This equation is widely recognized for its various applications in many fields, including the treatment of uterine fibroids, tumors, and tremor, incorporating diverse therapeutic modalities. The classical Westervelt equation45,46,47 read as:

$$\begin{aligned} 2k\left( uu_{tt}+(u_t)^2\right) +\nu ^2u_{xx}+\varrho u_{xxt}-u_{tt}=0, \end{aligned}$$
(2)

where \(u=u(x,t)\) is a real wave function reflecting the system behaviors over the space variable x and time t. Moreover, the parameters k is the nonlinearity parameter with \(k=\frac{1+\frac{B}{2A}}{\delta }\) and \(\delta\) is bulk bulk modulus with \(\delta =\nu \eta ^2\) where \(\eta\) is the mass density. Whereas, \(\nu\) is the speed of sound and \(\varrho\) is the diffusivity of sound.

Moreover, the exact solutions of (2), are obtained by applying different approaches, such as in45 the authors studied the exact solutions by applying generalized Kudryashov and modified Kudryashov methods, where in46 the fractional form of Westervelt equation is studied by two different analytical methods, namely the \(\text {exp}_a\) function and modified simplest equation technique. In47 some new exact solutions extracted by using modified exponential rational functional method and modified \(G'/G^2\)-model expansion method. In addition, Eq. (2) with \(\beta\)-derivative can be written as:

$$\begin{aligned} 2k\left( u{\mathscr {D}}_{tt}^{2\beta }u+\left( {\mathscr {D}}_{t}^{\beta }u\right) ^2\right) +\nu {\mathscr {D}}_{xx}^{2\beta }u+\varrho {\mathscr {D}}_{xxt}^{3\beta }u-{\mathscr {D}}_{tt}^{2\beta }u=0. \end{aligned}$$
(3)

Now, in the next section, we extract different forms of solutions of the proposed model.

Extraction of solutions

To solve Eq. (2), we use the transformation defined by:

$$\begin{aligned} u=u(x,t)=\Phi (\xi ); \end{aligned}$$
(4)
$$\begin{aligned} \xi =\frac{\varphi }{\beta }\left( x+\frac{1}{\Gamma (\beta )}\right) ^{\beta }+\frac{\zeta }{\beta }\left( t+\frac{1}{\Gamma (\beta )}\right) ^{\beta },~~~\text {( for } \beta -\text {derivative)} \end{aligned}$$
(5)

where the symbols \(\varphi ,~\zeta ,\) are the spatio and temporal coefficients. By applying the aforementioned transformations in Eq. (2), we have:

$$\begin{aligned} \zeta \varphi ^2 \varrho \Phi ^{'''}+\nu ^2 \varphi ^2 \Phi ^{''}+\zeta ^2\left( 2k\Phi \Phi ^{''}+2k\left( \Phi ^{'}\right) ^2-\Phi {''}\right) =0. \end{aligned}$$
(6)

Furthermore, we have to solve Eq. (6), by applying the suggested methods. Now, by applying the balance principle between the terms in Eq. (6) gives, \(n=1\).

Modified generalized Riccati equation mapping method

The general form of the solution for MGREMM41 is expressed as

$$\begin{aligned} \begin{aligned} \Phi (\xi )=\phi _{0}+\sum _{r=1}^{n}\phi _{r}\Omega ^{r}(\xi )+\sum _{r=1}^{n}\psi _r\left( \frac{\Omega ^{'}(\xi )}{\Omega (\xi )}\right) ^{r}. \end{aligned} \end{aligned}$$
(7)

For \(n=1\), we get

$$\begin{aligned} \Phi (\xi )=\phi _0+\phi _1 \Omega (\xi )+\psi _1\frac{\Omega ' (\xi )}{\Omega (\xi )}, \end{aligned}$$
(8)

and \(\Omega ^{'}(\xi )=\delta _0+\delta _1 \Omega (\xi )+\delta _2 \Omega (\xi )^2\). On putting Eq. (8) in Eq. (6), we get:

(I): On taking \(\delta _1^2-4\delta _0\delta _2>0\), \(\delta _1\delta _2\ne 0\), \(\delta _0\delta _2\ne 0\) and considering the values of the parameters as:

\(\bullet\) If \(\phi _0=\frac{-\delta _1 \zeta \varphi ^2 \varrho +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k},~\phi _1=-\frac{\delta _2 \varphi ^2 \varrho }{\zeta k},~\psi _1=\frac{\varphi ^2 \varrho }{\zeta k},\) we have the following solutions:

The solitary wave solutions

$$\begin{aligned} u_1(x,t)=\left( \frac{-\delta _1 \zeta \varphi ^2 \varrho +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k}+\frac{S^2 \varphi ^2 \varrho \text {sech}^2\left( \frac{\xi S}{2}\right) }{2 \zeta k \left( \delta _1+S \tanh \left( \frac{\xi S}{2}\right) \right) }+\frac{\varphi ^2 \varrho \left( \delta _1+S \tanh \left( \frac{\xi S}{2}\right) \right) }{2 \zeta k}\right) ,~~~~~~~ ~~~~~~~ \end{aligned}$$
(9)
$$\begin{aligned} u_2(x,t)=\left( \frac{-\delta _1 \zeta \varphi ^2 \varrho +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k}-\frac{S^2 \varphi ^2 \varrho \text {csch}^2\left( \frac{\xi S}{2}\right) }{2 \zeta k \left( \delta _1+S \coth \left( \frac{\xi S}{2}\right) \right) }+\frac{\varphi ^2 \varrho \left( \delta _1+S \coth \left( \frac{\xi S}{2}\right) \right) }{2 \zeta k}\right) ,~~~~~~~~~~~~~~ \end{aligned}$$
(10)

The bright-dark soliton solution

$$\begin{aligned} u_3(x,t)=\left( \frac{\zeta ^2-\nu ^2 \varphi ^2-\frac{2 i \zeta S^2 \varphi ^2 \varrho }{S \cosh (\xi S)+\delta _1 (\sinh (\xi S)-i)}+\zeta S \varphi ^2 \varrho (\tanh (\xi S)+i \text {sech}(\xi S))}{2 \zeta ^2 k}\right) .~~~~~~~~~ ~~~~~~~~~~~~~ \end{aligned}$$
(11)

\(\bullet\) When \(\phi _0=\frac{-\delta _1 \zeta \varphi ^2 \varrho +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k},~\phi _1=-\frac{\delta _2 \varphi ^2 \varrho }{\zeta k},~\psi _1=0,\) then we have the following solutions: The explicit soliton solution

$$\begin{aligned} u_4(x,t)=\left( \frac{-\delta _1 \zeta \varphi ^2 \varrho +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k}+\frac{\varphi ^2 \varrho \left( \delta _1+S (\coth (\xi S)+\text {csch}(\xi S))\right) }{2 \zeta k}\right) ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(12)

The dark-singular soliton solutions

$$\begin{aligned} u_5(x,t)=\left( \frac{-\delta _1 \zeta \varphi ^2 \varrho +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k}+\frac{\varphi ^2 \varrho \left( 2 \delta _1+S \left( \tanh \left( \frac{\xi S}{4}\right) +\coth \left( \frac{\xi S}{4}\right) \right) \right) }{4 \zeta k}\right) ,~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(13)

The explicit solitary wave solutions

$$\begin{aligned} u_6(x,t)=\left( \frac{-\delta _1 \zeta \varphi ^2 \varrho +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k}-\frac{\varphi ^2 \varrho \left( \frac{S\sqrt{ \left( c^2+d^2\right) }-c S \cosh (\xi S)}{c \sinh (\xi S)+d}-\delta _1\right) }{2 \zeta k}\right) ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(14)
$$\begin{aligned} u_7(x,t)=\left( \frac{-\delta _1 \zeta \varphi ^2 \varrho +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k}-\frac{\varphi ^2 \varrho \left( -\frac{S\sqrt{ \left( c^2+d^2\right) }+c S \cosh (\xi S)}{c \sinh (\xi S)+d}-\delta _1\right) }{2 \zeta k}\right) , ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(15)

\(\bullet\) When \(\phi _0=\frac{-\delta _1 \zeta \varphi ^2 \varrho +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k},~\phi _1=-\frac{\delta _2 \varphi ^2 \varrho }{\zeta k},~\psi _1=\frac{\varphi ^2 \varrho }{\zeta k},\) we have the following solutions: The kink-type solution

$$\begin{aligned} u_8(x,t)=\Bigg (\frac{\zeta ^2-\nu ^2 \varphi ^2+\zeta S \varphi ^2 \varrho \tanh \left( \frac{\xi S}{2}\right) }{2 \zeta ^2 k}\Bigg ),~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(16)

The singular soliton solution

$$\begin{aligned} u_9(x,t)=\left( \frac{\zeta ^2-\nu ^2 \varphi ^2+\zeta S \varphi ^2 \varrho \coth \left( \frac{\xi S}{2}\right) }{2 \zeta ^2 k}\right) ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(17)

The exponential function solution

$$\begin{aligned} u_{10}(x,t)=\left( \frac{1+\frac{\varphi ^2 \left( -\nu ^2+\zeta S \varrho \left( 1-\frac{2}{1+i e^{\xi S}}\right) \right) }{\zeta ^2}}{2 k}\right) ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(18)

where \(S=\sqrt{\delta _1^2-4 \delta _0 \delta _2},\) and \(\xi\) is defined for \(\beta\)-derivative in Eq. (5)

(II):On taking \(\delta _1^2-4\delta _0\delta _2<0\), \(\delta _1\delta _2\ne 0\), \(\delta _0\delta _2\ne 0\) and considering the values as:

\(\bullet\) If \(\phi _0=\frac{-\delta _1 \zeta \varphi ^2 \varrho +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k},~\phi _1=-\frac{\delta _2 \varphi ^2 \varrho }{\zeta k},~\psi _1=0,\) then the following periodic solutions

$$\begin{aligned} u_{11}(x,t)=\left( \frac{-\delta _1 \zeta \varphi ^2 \varrho +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k}-\frac{\varphi ^2 \varrho \left( A \tan \left( \frac{A \xi }{2}\right) -\delta _1\right) }{2 \zeta k}\right) ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(19)
$$\begin{aligned} u_{12}(x,t)=\left( \frac{\varphi ^2 \varrho \left( A \cot \left( \frac{A \xi }{2}\right) +\delta _1\right) }{2 \zeta k}+\frac{-\delta _1 \zeta \varphi ^2 \varrho +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k}\right) , ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(20)
$$\begin{aligned} u_{13}(x,t)=\left( \frac{-\delta _1 \zeta \varphi ^2 \varrho +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k}-\frac{\varphi ^2 \varrho \left( A (\tan (A \xi )+\sec (\xi \text { A}))-\delta _1\right) }{2 \zeta k}\right) ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(21)
$$\begin{aligned} u_{14}(x,t)=\left( \frac{\varphi ^2 \varrho \left( A (\cot (A \xi )+\csc (A \xi ))+\delta _1\right) }{2 \zeta k}+\frac{-\delta _1 \zeta \varphi ^2 \varrho +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k}\right) ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~ \end{aligned}$$
(22)

\(\bullet\) When \(\phi _0=\frac{-\delta _1 \zeta \varphi ^2 \varrho +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k};\phi _1=-\frac{\delta _2 \varphi ^2 \varrho }{\zeta k};\psi _1=\frac{\varphi ^2 \varrho }{\zeta k},\) we have the following solutions:

$$\begin{aligned} u_{15}(x,t)=\Bigg (-\frac{\varphi ^2 \varrho \left( A \left( \tan \left( \frac{A \xi }{4}\right) -\cot \left( \frac{A \xi }{4}\right) \right) -2 \delta _1\right) }{4 \zeta k}+\frac{A \varphi ^2 \varrho \left( \frac{1}{4} A \csc ^2\left( \frac{A \xi }{4}\right) +\frac{1}{4} A \sec ^2\left( \frac{A \xi }{4}\right) \right) }{\zeta k \left( A \left( \tan \left( \frac{A \xi }{4}\right) -\cot \left( \frac{A \xi }{4}\right) \right) -2 \delta _1\right) }~~\nonumber \\+\frac{-\delta _1 \zeta \varphi ^2 \varrho +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k}\Bigg ), \end{aligned}$$
(23)
$$\begin{aligned} u_{16}(x,t)=\left( \frac{\phi _1 \left( -A\sqrt{ \left( d^2-c^2\right) }-A c \cos (A \xi )\right) }{2 \delta _2 (c \sin (A \xi )+d)}-\frac{\nu ^2 \varphi ^2}{\zeta ^2}+1\right) ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~ \end{aligned}$$
(24)
$$\begin{aligned} u_{17}(x,t)=\left( \frac{-A \zeta \varphi ^2 \varrho \tan \left( \frac{A \xi }{2}\right) +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k}\right) ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(25)
$$\begin{aligned} u_{18}(x,t)=\left( \frac{A \zeta \varphi ^2 \varrho \cot \left( \frac{A \xi }{2}\right) +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k}\right) ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(26)

\(\bullet\) When \(\phi _0=\frac{-\delta _1 \zeta \varphi ^2 \varrho +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k},~\phi _1=-\frac{\delta _2 \varphi ^2 \varrho }{\zeta k},~\psi _1=0,\) we have the following solution:

$$\begin{aligned} u_{19}(x,t)=\left( \frac{2 \delta _0 \delta _2 \varphi ^2 \varrho \cos (A \xi )}{\zeta k \left( \delta _1 \cos (A \xi )+A \sin (A \xi )+A\right) }+\frac{-\delta _1 \zeta \varphi ^2 \varrho +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k}\right) ,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(27)

where \(A=\sqrt{4 \delta _0 \delta _2-\delta _1^2}\) and c,  d are real numbers that satisfy the condition \(d^2-c^2>0\). In addition, \(\xi\) is defined for \(\beta\)-derivative in Eq. (5).

\(\bullet\)Numerical simulation:The Fig. 1 which represents the different aspects physically for the solution (11) has been sketched with the assistance of the values of the parameters \(\delta _0=0.3, \delta _1=2.7, \delta _2=0.7, \zeta =0.03, k=1.5, \nu =2.5, \varphi =2.27, \varrho =2\) and Fig. 2 show the solitary wave behavior to the solution (14) for the parameters \(\delta _0=0.8, \delta _1=2.7, \delta _2=0.07, \zeta =0.99, \nu =3.9, \varphi =3, \varrho =0.2, k=3.5, c=0.5, d=1.5\). The dynamics of the wave for the solution (18) has been shown in Fig. 3 for the values \(\delta _0=0.08, \delta _1=4.7, \delta _2=0.7, \zeta =0.09, k=1.5, \nu =1.9, \varphi =2.53, \varrho =1.2\), while the parameters \(\delta _0=2.08, \delta _1=0.7, \delta _2=1.07, \zeta =2.9, k=1.5, \nu =1.5, \varphi =1.53, \varrho =0.2\) have been used for the solution (22) to explain the periodic movement of the waves in Fig. 4.

Fig. 1
figure 1

Plots of solution (11) for the different values of the fractional parameter.

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Fig. 2
figure 2

Plots of solution (14) for the different values of the fractional parameter.

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Fig. 3
figure 3

Plots of solution (18) for the different values of the fractional parameter.

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Fig. 4
figure 4

Plots of solution (22) for the different values of the fractional parameter.

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Solutions via modified generalized exponential rational function method

The solution for mGERFM42 is expressed as

$$\begin{aligned} \begin{aligned} \Phi (\xi )=d_0+\sum _{r=1}^{n}d_{r}\Bigg (\frac{\Omega '(\xi )}{\Omega (\xi )}\Bigg )^{r}+\sum _{r=1}^{n}f_{r}\Bigg (\frac{\Omega '(\xi )}{\Omega (\xi )}\Bigg )^{-r}, \end{aligned} \end{aligned}$$
(28)

where

$$\begin{aligned} \Omega (\xi )=\frac{r_1 e^{s_1 \xi }+r_2 e^{s_2 \xi }}{r_3 e^{s_3 \xi }+r_4 e^{s_4 \xi }}. \end{aligned}$$
(29)

For \(n=1\), the Eq. (28) is mentioned as:

$$\begin{aligned} \Phi (\xi )=d_{0}+d_{1}\Bigg (\frac{\Omega '(\xi )}{\Omega (\xi )}\Bigg )+f_{1}\Bigg (\frac{\Omega '(\xi )}{\Omega (\xi )}\Bigg )^{-1}. \end{aligned}$$
(30)

\(\bullet\) On taking \(r=[1, 1, 1, 0]\) and \(s=[0, -1, 0, 0]\), in Eq. (29), offers \(\Omega (\xi )=1+e^{-\xi }\), and on putting the Eq. (30) in the Eq. (6) gives \(\lbrace d_0=\frac{d_1 \left( \zeta ^2+\zeta \varphi ^2 \varrho -\nu ^2 \varphi ^2\right) }{2 \zeta \varphi ^2 \varrho }, f_1=0, k=\frac{\varphi ^2 \varrho }{d_1 \zeta } \rbrace\) and \(\lbrace d_0=\frac{\zeta ^2+\zeta \varphi ^2 \varrho -\nu ^2 \varphi ^2}{2 \zeta ^2 k}, d_1=\frac{\varphi ^2 \varrho }{\zeta k}, f_1=0 \rbrace\), then we get:

The exponential soliton solution

$$\begin{aligned} u_1(x, t)=\frac{1}{2} d_1 \left( -\frac{\nu ^2}{\zeta \varrho }+\frac{\zeta }{\varphi ^2 \varrho }-\frac{2}{e^{\frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }+\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }}+1}+1\right) .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(31)

The explicit hyperbolic solution

$$\begin{aligned} u_2(x, t)=\frac{\zeta ^2+\zeta \varphi ^2 \varrho -\nu ^2 \varphi ^2}{2 \zeta ^2 k}-\frac{\varphi ^2 \varrho }{\zeta k \left( \cosh \left( \frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{\beta }+\frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) +\sinh \left( \frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{\beta }+\frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) +1\right) }.~~~~ \end{aligned}$$
(32)

\(\bullet\) Now, on considering \(r=[2, 0, 1, 1]\) and \(s=[0, 0, 1, -1]\), in Eq. (29), offers \(\Omega (\xi )=\text {sech}(\xi )\) and on solving the Eqs. (30) and (6) provide \(\lbrace d_0=\frac{(\zeta -\nu \varphi ) (\zeta +\nu \varphi )}{2 \zeta ^2 k}, f_1=d_1, \varrho =-\frac{d_1 \zeta k}{\varphi ^2} \rbrace\), \(\lbrace f_1=0, \varrho =-\frac{d_1 (\zeta -\nu \varphi ) (\zeta +\nu \varphi )}{2 d_0 \zeta \varphi ^2}, k=\frac{(\zeta -\nu \varphi ) (\zeta +\nu \varphi )}{2 d_0 \zeta ^2} \rbrace\) and \(\lbrace d_1=0, d_0=-\frac{f_1 (\zeta -\nu \varphi ) (\zeta +\nu \varphi )}{2 \zeta \varphi ^2 \varrho }, k=-\frac{\varphi ^2 \varrho }{\zeta f_1} \rbrace\) and the following solutions are expressed as:

The combined soliton solution

$$\begin{aligned} u_3(x, t)=\frac{-2 d_1 \zeta ^2 k \left( \coth ^2\left( \frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }+\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) +1\right) \tanh \left( \frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }+\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) +\zeta ^2-\nu ^2 \varphi ^2}{2 \zeta ^2 k}.~~~~~~~~~~~~~~ \end{aligned}$$
(33)

The kink-type soliton solution

$$\begin{aligned} u_4(x, t)=d_0-d_1 \tanh \left( \frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }+\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(34)

The soliton solution

$$\begin{aligned} u_5(x, t)=-\frac{f_1 \left( \zeta ^2-\nu ^2 \varphi ^2+2 \zeta \varphi ^2 \varrho \coth \left( \frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }+\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) \right) }{2 \zeta \varphi ^2 \varrho }.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(35)

\(\bullet\) Next on considering \(r=[1, 1, 2, 0]\) and \(s=[i, -i, -1, 0]\), then the Eq. (29), offers \(\Omega (\xi )=e^{\xi } \cos (\xi )\) and on solving the Eqs. (30) abd (6) provide \(\lbrace d_0=-\frac{f_1 \left( \zeta ^2-\varphi ^2 \left( \nu ^2-2 \zeta \varrho \right) \right) }{4 \zeta \varphi ^2 \varrho }, d_1=0, k=-\frac{2 \varphi ^2 \varrho }{\zeta f_1} \rbrace\) and \(\lbrace d_0=\frac{1}{2} \left( \frac{1-\frac{\nu ^2 \varphi ^2}{\zeta ^2}}{k}-2 d_1\right) , f_1=0, \varrho =\frac{d_1 \zeta k}{\varphi ^2} \rbrace\) then we get the following periodic solutions:

$$\begin{aligned} u_6(x, t)=\frac{1}{4} f_1 \left( \frac{\nu ^2-\frac{\zeta ^2}{\varphi ^2}}{\zeta \varrho }-\frac{4}{\tan \left( \frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }+\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) -1}-2\right) .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(36)
$$\begin{aligned} u_7(x, t)=\frac{1}{2} \left( \frac{1-\frac{\nu ^2 \varphi ^2}{\zeta ^2}}{k}-2 d_1\right) +d_1 \sec \left( \frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{\beta }+\frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) \nonumber ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ \times \left( \cos \left( \frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{\beta }+\frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) -\sin \left( \frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{\beta }+\frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) \right) . \end{aligned}$$
(37)

\(\bullet\) Suppose \(r=[1, -1, 2, 0]\) and \(s=[2, 0, 0, 0]\), then the Eq. (29), offers \(\Omega (\xi )=e^{\xi } \sinh (\xi )\) and on managing the Eqs. (30) abd (6) provide \(\lbrace d_1=\frac{\zeta \varrho \left( 1-2 d_0 k\right) }{k \left( 2 \zeta \varrho +\nu ^2\right) }, f_1=0, \varphi =-\frac{\sqrt{\zeta ^2 \left( 1-2 d_0 k\right) }}{\sqrt{2 \zeta \varrho +\nu ^2}} \rbrace\) and \(\lbrace f_1=0, d_1=-d_0, \varphi =\frac{\zeta }{\nu }, \varrho =-\frac{d_0 k \nu ^2}{\zeta } \rbrace\) the following explicit solitary wave solutions are extracted as:

$$\begin{aligned} u_8(x, t)=\frac{\frac{\zeta \varrho \left( 1-2 d_0 k\right) \coth \left( \frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }-\frac{\sqrt{\zeta ^2 \left( 1-2 d_0 k\right) } \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\sqrt{2 \zeta \varrho +\nu ^2}}}{\beta }\right) }{k}+d_0 \nu ^2+\frac{\zeta \varrho }{k}}{2 \zeta \varrho +\nu ^2}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(38)
$$\begin{aligned} u_9(x, t)=d_0 \left( -\cosh \left( \frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{\beta }+\frac{\zeta \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta \nu }\right) \right) \text {csch}\left( \frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{\beta }+\frac{\zeta \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta \nu }\right) .~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(39)

\(\bullet\) On selecting \(r=[1, 1, 2, 0]\) and \(s=[i, -i, 0, 0]\), then the Eq. (29), gives \(\Omega (\xi )=\cos (\xi )\), Eqs. (30) and (6) provide \(\lbrace d_0=\frac{1}{2 k}-\frac{d_1^2 k \nu ^2}{2 \varphi ^2 \varrho ^2}, f_1=-d_1, \zeta =\frac{\varphi ^2 \varrho }{d_1 k} \rbrace\) and \(\lbrace d_0=\frac{1}{2} \left( \frac{1}{k}-\frac{d_1 \nu ^2}{\zeta \varrho }\right) , f_1=0, \varphi =\frac{\sqrt{d_1} \sqrt{\zeta } \sqrt{k}}{\sqrt{\varrho }} \rbrace\) the following explicit solitary wave solutions are extracted as:

$$\begin{aligned} u_{10}(x, t)=\frac{1}{2}\Bigg (-\frac{d_1^2 k \nu ^2}{\varphi ^2 \varrho ^2}+\frac{1}{k}+2d_1 \left( \cot ^2\left( \frac{\varphi \left( d_1 k \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }+\varphi \varrho \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }\right) }{\beta d_1 k}\right) -1\right) \nonumber ~~~~~~~~~~~~~~~~~~~~~~~~~~~\\\times \tan \left( \frac{\varphi \left( d_1 k \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }+\varphi \varrho \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }\right) }{\beta d_1 k}\right) \Bigg ). \end{aligned}$$
(40)
$$\begin{aligned} u_{11}(x, t)=\frac{1}{2} \left( \frac{1}{k}-\frac{d_1 \left( 2 \zeta \varrho \tan \left( \frac{\sqrt{d_1} \sqrt{\zeta } \sqrt{k} \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta \sqrt{\varrho }}+\frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{\beta }\right) +\nu ^2\right) }{\zeta \varrho }\right) .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(41)

\(\bullet\) Let \(r=[1, 1, 1, 0]\) and \(s=[3, 2, 0, 0]\), then the Eq. (29), gives \(\Omega (\xi )=e^{2 \xi }+e^{3 \xi }\), Eqs. (30) and (6) provide \(\lbrace d_0=-\frac{f_1 \left( \zeta ^2-\varphi ^2 \left( \nu ^2-5 \zeta \varrho \right) \right) }{12 \zeta \varphi ^2 \varrho }, d_1=0, k=-\frac{6 \varphi ^2 \varrho }{\zeta f_1} \rbrace\) and \(\lbrace f_1=0, d_1=-\frac{d_0}{5}, k=-\frac{5 \zeta \varrho }{d_0 \left( \nu ^2-5 \zeta \varrho \right) }, \varphi =-\frac{\zeta }{\sqrt{\nu ^2-5 \zeta \varrho }} \rbrace\), then we have:

$$\begin{aligned} u_{12}(x, t)=\frac{1}{12} f_1 \left( \frac{\nu ^2-\frac{\zeta ^2}{\varphi ^2}}{\zeta \varrho }+\frac{4}{3 e^{\frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }+\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }}+2}-1\right) .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(42)
$$\begin{aligned} u_{13}(x, t)=\frac{1}{5} d_0 \left( \frac{1}{-\cosh \left( \frac{\zeta \left( \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }-\frac{\left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\sqrt{\nu ^2-5 \zeta \varrho }}\right) }{\beta }\right) +\sinh \left( \frac{\zeta \left( \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }-\frac{\left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\sqrt{\nu ^2-5 \zeta \varrho }}\right) }{\beta }\right) -1}+3\right) ,~~~~~~~~~~~~~~~ \end{aligned}$$
(43)

\(\bullet\)Numerical simulation:

The Fig. 5 represent the physically behaviour for the solution (32) with the parametric selection \(\zeta =0.09, k=1.7, \nu =2.5, \varphi =2.53, \varrho =0.25\), and kink-type soliton behaviour has been depicted in the Fig. 6 for the values \(d_1=1.25, d_0=2.7, \zeta =0.001, \varphi =0.09\) to the solution (34). The solitary periodic wave behavior for the solution (37) is drawn in the Fig. 7 with the suitable values \(\zeta =0.06, \varphi =3.5, \varrho =2.2, d_1=2.2, k=0.2, \nu =2.2\).

Fig. 5
figure 5

Plots of solution (32) for the different values of the fractional parameter.

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Fig. 6
figure 6

Plots of solution (34) for the different values of the fractional parameter.

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Fig. 7
figure 7

Plots of solution (37) for the different values of the fractional parameter.

Full size image

Interaction phenomena

Next, for securing the different form of solutions like breather, lump and other solutions to the studied model, the logarithmic transformation \(\Phi =(\ln f)_{\xi }\) is under consideration and using this relation in Eq. (6), we get the following form as:

$$\begin{aligned} f \left( f_{\xi } \left( \nu ^2 \varphi ^2-\zeta ^2\right) +\zeta \varphi ^2 \varrho f_{\xi \xi }\right) +\zeta f_{\xi }^2 \left( \zeta k-\varphi ^2 \varrho \right) =0, \end{aligned}$$
(44)

Now, by the assistance of the above equation, breather type, lump and other solutions are extracted.

Homoclinic breather approach

For extracting the breather-type solutions, the following test functions are considered

\(\bullet\) Let we start with following assumption

$$\begin{aligned} f=e^{-p \left( \delta _1 \xi +\delta _2\right) }+\kappa _1 e^{p \left( \delta _3 \xi +\delta _4\right) }+\kappa _0 \cos \left( p_1 \left( \delta _5 \xi +\delta _6\right) \right) , \end{aligned}$$
(45)

where, \(\delta _i\)\(( i=1, 2, \cdots , 6)\) and \(\kappa _0,\kappa _1\) are the real constant that are determine later. Substitute Eq. (45) in Eq. (44) with its derivatives and after collecting all the coefficients of same powers exponential trigonometric functions, we get the different sets of parameters as:

Case 1: \(\delta _1=-\delta _3, \varphi =\frac{\zeta }{\sqrt{\nu ^2+\delta _3 \zeta p \varrho }}, k=\frac{\zeta \varrho }{\nu ^2+\delta _3 \zeta p \varrho }, p_1=0\), substitute these values in Eq. (45) and using the defined logarithmic transformation provide the the solution of Eq. (6) and the consequently, the breather solution is expressed as:

$$\begin{aligned} u_1(x, t)=\frac{\delta _3 \kappa _1 p e^{\left( p \left( \delta _4+\delta _3 \left( \frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{\beta }+\frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) \right) \right) }+\delta _3 p e^{\left( -p \left( \delta _2-\delta _3 \left( \frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{\beta }+\frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) \right) \right) }}{\kappa _1 e^{\left( p \left( \delta _4+\delta _3 \left( \frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{\beta }+\frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) \right) \right) }+e^{\left( -p \left( \delta _2-\delta _3 \left( \frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{\beta }+\frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) \right) \right) }+\kappa _0}.~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(46)

Case 2: \(\delta _1=-\delta _3, \delta _5=0, k=\frac{\delta _3 (-p) \varphi ^2 \varrho ^2-\varrho \sqrt{\varphi ^2 \left( 4 \nu ^2+\delta _3^2 p^2 \varphi ^2 \varrho ^2\right) }}{2 \nu ^2}, \zeta =\frac{1}{2} \left( \delta _3 p \varphi ^2 \varrho -\sqrt{4 \nu ^2 \varphi ^2+\delta _3^2 p^2 \varphi ^4 \varrho ^2}\right)\), on using the case-2 and the following breather-type sloution is recovered

$$\begin{aligned} u_2(x, t)=\frac{\delta _3 p \left( \kappa _1 e^{\left( \delta _2+\delta _4\right) p}+1\right) }{\kappa _0 \cos \left( \delta _6 p_1\right) \exp \left( \frac{p \left( \beta \delta _2-\delta _3 \left( \frac{1}{2} \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta } \left( \delta _3 p \varphi ^2 \varrho -\sqrt{4 \nu ^2 \varphi ^2+\delta _3^2 p^2 \varphi ^4 \varrho ^2}\right) +\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }\right) \right) }{\beta }\right) +\kappa _1 e^{\left( \delta _2+\delta _4\right) p}+1}.~~~~~~~~~ \end{aligned}$$
(47)

\(\bullet\) Consider

$$\begin{aligned} f=\rho _2 \exp \left( 2 \left( -p_0 \left( \delta _3 \xi +\delta _4\right) \right) \right) +\rho _1 \exp \left( -p_0 \left( \delta _3 \xi +\delta _4\right) \right) \cos \left( p_1 \left( \delta _1 \xi +\delta _2\right) \right) +1, \end{aligned}$$
(48)

where, \(\delta _i\) \(( i=1, 2, \cdots , 4)\) and \(\rho _1,\rho _2, p_0\) and \(p_1\) are the real constant that are determine later. Putting the Eq. (48) in Eq. (44) with its derivatives and after collecting all the coefficients of same powers exponential trigonometric functions, we get the different sets of parameters as:

Case 1: \(k=-\frac{i (\zeta -\nu \varphi ) (\zeta +\nu \varphi )}{2 \delta _1 \zeta ^2 p_1}, \varrho =-\frac{i (\zeta -\nu \varphi ) (\zeta +\nu \varphi )}{2 \delta _1 \zeta p_1 \varphi ^2}, \delta _3=-\frac{i \delta _1 p_1}{p_0}\), the breather solution for the case-1 is expressed as:

$$\begin{aligned} u_3(x, t)=\frac{i \delta _1 p_1 e^{i \delta _1 \xi p_1} \left( 2 \rho _2 e^{i \delta _1 \xi p_1}+\rho _1 e^{\delta _4 p_0} \left( \cos \left( p_1 \left( \delta _1 \xi +\delta _2\right) \right) +i \sin \left( p_1 \left( \delta _1 \xi +\delta _2\right) \right) \right) \right) }{\rho _2 e^{2 i \delta _1 \xi p_1}+\rho _1 e^{\delta _4 p_0+i \delta _1 \xi p_1} \cos \left( p_1 \left( \delta _1 \xi +\delta _2\right) \right) +e^{2 \delta _4 p_0}},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(49)

where, \(\xi =\frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{\beta }+\frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\).

Case 2: \(\rho _1=0, \zeta =\frac{\varphi ^2 \varrho }{k}, \delta _3=\frac{\frac{k^2 \nu ^2}{\varphi ^2 \varrho ^2}-1}{2 k p_0}\), then we get:

$$\begin{aligned} u_4(x, t)=-\frac{\rho _2 \left( \frac{k^2 \nu ^2}{\varphi ^2 \varrho ^2}-1\right) \exp \left( -2 p_0 \left( \delta _4+\frac{\left( \frac{k^2 \nu ^2}{\varphi ^2 \varrho ^2}-1\right) \left( \frac{\varphi ^2 \varrho \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{\beta k}+\frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) }{2 k p_0}\right) \right) }{k \left( \rho _2 \exp \left( -2 p_0 \left( \delta _4+\frac{\left( \frac{k^2 \nu ^2}{\varphi ^2 \varrho ^2}-1\right) \left( \frac{\varphi ^2 \varrho \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{\beta k}+\frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) }{2 k p_0}\right) \right) +1\right) }.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(50)

Case 3: \(\rho _2=0, \zeta =\frac{i k \nu ^2}{-2 \delta _1 k p_1 \varrho +i \varrho }, \varphi =\frac{(-1)^{3/4} k \nu }{\varrho \sqrt{2 \delta _1 k p_1-i}}, \delta _3=\frac{i \delta _1 p_1}{p_0}\), then for case-3 the breather solution is written as:

$$\begin{aligned} u_5(x, t)=\frac{(-i) \delta _1 p_1 \rho _1 e^{-p_0 \left( \delta _4+\frac{i \delta _1 \xi p_1}{p_0}\right) } \cos \left( p_1 \left( \delta _1 \xi +\delta _2\right) \right) -\delta _1 p_1 \rho _1 e^{-p_0 \left( \delta _4+\frac{i \delta _1 \xi p_1}{p_0}\right) } \sin \left( p_1 \left( \delta _1 \xi +\delta _2\right) \right) }{1+\rho _1 e^{-p_0 \left( \delta _4+\frac{i \delta _1 \xi p_1}{p_0}\right) } \cos \left( p_1 \left( \delta _1 \xi +\delta _2\right) \right) },~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(51)

where, \(\xi =\frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{\beta }+\frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }, \zeta =\frac{i k \nu ^2}{-2 \delta _1 k p_1 \varrho +i \varrho }, \varphi =\frac{(-1)^{3/4} k \nu }{\varrho \sqrt{2 \delta _1 k p_1-i}}\).

Lump waves

Consider

$$\begin{aligned} f=\left( b_1 \xi +b_2\right) ^2+\left( b_3 \xi +b_4\right) ^2+b_5, \end{aligned}$$
(52)

where, \(b_i\) \(( i=1, 2, \cdots , 5)\) are the real constant that are determine later. On solving the Eq. (52) and Eq. (44) with its derivatives and after collecting all the coefficients of same powers of \(\xi\), we get

Case 1: \(b_4=\frac{b_1 b_2 b_3+\sqrt{-b_1^2 \left( b_1^2+b_3^2\right) b_5}}{b_1^2}, \zeta =-\nu \varphi , k=-\frac{\varphi \varrho }{2 \nu }\), on tackling the case-1, Eq. (52) and using the defined logarithmic transformation, the lump solution is mentioned as:

$$\begin{aligned} u_6(x, t)=\frac{2 \left( \beta b_2 b_1^3+b_1^4 \varphi \left( \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }-\nu \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }\right) +b_3^2 b_1^2 \varphi \left( \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }-\nu \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }\right) +\Gamma _2\right) }{\beta b_1^2 \left( \frac{\left( \beta b_2+b_1 \varphi \left( \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }-\nu \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }\right) \right) ^2}{\beta ^2}+b_5+\Gamma _1^2\right) },~~~~~~~~~~~ \end{aligned}$$
(53)

where, \(\Gamma _1=\frac{\beta b_1 b_2 b_3+\beta \sqrt{-b_1^2 \left( b_1^2+b_3^2\right) b_5}+b_1^2 b_3 (-\nu ) \varphi \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }+b_1^2 b_3 \varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta b_1^2},\Gamma _2=\beta b_1 b_2 b_3^2+\beta \sqrt{-b_1^2 \left( b_1^2+b_3^2\right) b_5} b_3\)

Case 2: \(b_5=-\frac{\left( b_2 b_3-b_1 b_4\right) ^2}{b_1^2+b_3^2}, \zeta =\frac{\varphi ^2 \varrho }{2 k}, \nu =-\frac{\varphi \varrho }{2 k}\), then the lump solution is mentioned as:

$$\begin{aligned} u_7(x, t)=\frac{2 b_1 \left( b_1 \left( \frac{\varphi ^2 \varrho \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{2 \beta k}+\frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) +b_2\right) +2 b_3 \left( b_3 \left( \frac{\varphi ^2 \varrho \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{2 \beta k}+\frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) +b_4\right) }{\left( b_1 \left( \frac{\varphi ^2 \varrho \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{2 \beta k}+\frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) +b_2\right) ^2+\left( b_3 \left( \frac{\varphi ^2 \varrho \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{2 \beta k}+\frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) +b_4\right) ^2-\frac{\left( b_2 b_3-b_1 b_4\right) ^2}{b_1^2+b_3^2}}. \end{aligned}$$
(54)

Interaction via double exponential form

Let

$$\begin{aligned} f=\kappa _1 \exp \left( \delta _1 \xi +\delta _2\right) +\kappa _2 \exp \left( \delta _3 \xi +\delta _4\right) . \end{aligned}$$
(55)

where \(\delta _{1},\cdots ,\delta _{4}\) are constants to be determined later. Switching Eq. (55) into Eq. (44) and accumulating all coefficients of same powers of exponential function to zero, we get

Case 1: \(\delta _1=\frac{\zeta ^2-\nu ^2 \varphi ^2}{\zeta \varphi ^2 \varrho }, \delta _3=0, k=\frac{\varphi ^2 \varrho }{\zeta }\),

Case 2: \(\delta _3=0, \varrho =\frac{k \nu }{\frac{\delta _1 k \varphi }{\sqrt{1-\delta _1 k}}-\frac{\varphi }{\sqrt{1-\delta _1 k}}}, \zeta =-\frac{\nu \varphi }{\sqrt{1-\delta _1 k}}\), and the solutions corresponding the case-1 and case-2 are expressed as:

$$\begin{aligned} u_8(x, t)=\frac{\kappa _1 \left( \zeta ^2-\nu ^2 \varphi ^2\right) \exp \left( \delta _2+\frac{\left( \zeta ^2-\nu ^2 \varphi ^2\right) \left( \frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{\beta }+\frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) }{\zeta \varphi ^2 \varrho }\right) }{\zeta \varphi ^2 \varrho \left( e^{\delta _4} \kappa _2+\kappa _1 \exp \left( \delta _2+\frac{\left( \zeta ^2-\nu ^2 \varphi ^2\right) \left( \frac{\zeta \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{\beta }+\frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }\right) }{\zeta \varphi ^2 \varrho }\right) \right) }.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(56)
$$\begin{aligned} u_9(x, t)=\frac{\delta _1 \kappa _1 \exp \left( \delta _2+\delta _1 \left( \frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }-\frac{\nu \varphi \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{\beta \sqrt{1-\delta _1 k}}\right) \right) }{e^{\delta _4} \kappa _2+\kappa _1 \exp \left( \delta _2+\delta _1 \left( \frac{\varphi \left( \frac{1}{\Gamma (\beta )}+x\right) ^{\beta }}{\beta }-\frac{\nu \varphi \left( \frac{1}{\Gamma (\beta )}+t\right) ^{\beta }}{\beta \sqrt{1-\delta _1 k}}\right) \right) }.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{aligned}$$
(57)

\(\bullet\)Numerical simulation:

Fig. 8
figure 8

Plots of solution (53) for the different values of the fractional parameter.

Full size image
Fig. 9
figure 9

Plots of solution (56) for the different values of the fractional parameter.

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The different forms of the lump type with the effect of fractional parameters for the solution (53) has been shown in the Fig. 8, while the Fig. 9 shows the dynamics of the solution (56) for the different values of the fractional parameter.

Discussion and concluding remarks

Regarding the nonlinear propagation of ultrasonic waves through diverse media, such as human tissue, the Westervelt equation governs as a mathematical framework in ultrasound imaging. This Westervelt equation expands to fractional derivatives in order to accurately show the nonlinearity and dispersion that happen when ultrasound waves hit biological tissues. By providing an advanced comprehension of wave behavior, it is especially beneficial in situations where conventional linear models fail to adequately represent the complex interplay between ultrasonic waves and biological structures. By presenting a more precise illustration of wave interactions within complex media, this equation substantially improves modeling and analysis in ultrasound imaging projects.

In this work, we have investigated different types of solitary wave structures of the \(\beta\)- fractional Westervelt equation arising in ultrasound imaging by applying new designed integration methods. Different type of solutions like, bright, dark, singular, combined, kink type solitons as well as hyperbolic, periodic and exponential solutions have been recovered. Moreover, we have secured the breather, lump solutions and other solutions. A variety of Figs. 1, 2, 3, 4, 5, 6, 7, 8, 9 by the assistance of parameters have been sketched for observing the physical mechanism of the obtained results in the imaging process. Moreover, the period and frequency are defined by the time required to complete a cycle of the waveform and the number of cycles per second, respectively, that define a periodic wave solution, which characterizes a wave with a continuous pattern that repeats. Dark soliton describes the solitary waves with lower intensity than the background. Dark solitons are more difficult to handle than standard solitons, but they have shown to be more stable and robust to losses and the bright soliton causes a temporary increase in an associated wave amplitude. Bright soliton describes the solitary waves whose peak intensity is larger than the background. Furthermore, a lump solution is a rational function solution which is real analytic and decays in all directions of space variables. Breather waves refer to solitary waves that exhibit both partial localization and periodic structure in either space or time. Breathers serve crucial functions in nonlinear physics and have been observed in various physical domains, including optics, hydrodynamics, and quantized superfluidity. Hence, the obtained solutions differ significantly from previously studied solutions. These results demonstrate the practicality of the proposed approaches for extracting solitary wave solutions of nonlinear systems and its capacity to advance the field of soliton theory. The accuracy of these results have been verified using the software applications Mathematica by performing back-substitution of the solutions in the governing model. The obtained outcomes in this study may be useful for the transmission of ultrasound in tissue, underwater acoustics, the formation of bubbles in a liquid by sound waves, the suspension of objects in mid-air using sound waves, and other related applications. Using the imaging process produced by ultrasound imaging, medical researchers are able to examine the body’s internal tissues to gain more information about internal problems. It has many uses both industrially and medically. Additionally, without diagnostic imaging tools, physicians’ ability to identify and assess medical problems will certainly be diminished. Like other areas of clinical examination and pathology, imaging is often considered an integral part of modern medical diagnostic methods. The outcomes of this investigation are expected to greatly improve our comprehension of the underlying physical principles that propel evolutionary processes. Furthermore, this work has the potential to be extended by implementing the application of additive noise or multiplicative color noise.