1. Introduction
Partial differential equations (PDEs) are used to simulate the majority of complicated processes in real life, such as wave propagation, gas evolution in fluid dynamics, and so on. PDEs may be utilized to produce significantly better predictions and have a far wider range of applications in nature and living. These sophisticated simulations demonstrate why partial differential equations are so crucial to humankind. As a result, the exploration of diverse numerical and exact solution methods illuminates our scientific universe. A variety of effective and strong techniques can be employed to obtain these desired effects, which are advantageous to real-world situations [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17].
PDEs can give a variety of different forms of solutions. The lump solutions are one of the most reliable and well-known solutions of PDEs [18], [19], [20], [21], [22], [23], [24], [25]. Lump solutions are rational function solutions that have been discovered empirically in all directions. For both linear and nonlinear PDEs, lump solutions are among the most important solutions [25], [26], [27], [28], [29]. Lump solutions are well-known for their ability to explain nonlinear interaction solutions. Several academics have looked into lump solutions and various types of integrable equations [28], [29], [30], [31]. There are lump solutions for several non-integrable equations. Furthermore, multiple research have shown that interaction solutions between lumps and other types of exact solutions to nonlinear integrable equations exist [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51].
One of the most powerful tools for studying nonlinear PDEs is the Lie symmetry technique [52]. PDEs may be used to represent almost all problems in science and engineering [53], [54], [55], [56]. One of the most important disciplines of mathematical physics is the application of the Lie symmetry approach for generating explicit solutions to nonlinear PDEs. Many key characteristics of nonlinear PDEs can be studied sequentially, including symmetry reductions, CLs, and explicit solutions by employing symmetries [57], [58], [59]. Furthermore, studying CLs is beneficial in comprehending nonlinear PDEs from a physical standpoint. CLs appear to play a major role in the numerical integration of PDEs, according to several studies. CLs are also useful when using the double reduction approach to solve equations [60], [61], [62], [63]. Various strategies have been proposed to establish CLs [64], [65], [66], [67], [68].
Breather solutions to NPDEs are also interesting examples for examining nonlinear wave interactions, in addition to solitons, due to the significant problems of breather synchronization connected to their self-oscillating features. From this perspective, phase-sensitive breather interactions are now frequently explored. Breather molecules can be generated for co-propagative breathers when the group velocity and temporal phase of breathers are properly synced, however the phase-sensitive collision technique demonstrates different dynamic characteristics for counter propagating breathers. Motivated by this, in this paper, we establish solitons solutions, breather wave solutions by means of an effective techniques extended tanh-coth, sine-cosine and Hiroto binear transformations, respectively. Additionally, by means of the new conservation theorem we establish CLs via Lie-Bäcklund symmetries. The equation under consideration is the (2+1)-dimensional Chaffee-Infante equation and is given by Akbar et al. [69]
${{\nu }_{xt}}+{{\left( \alpha {{\nu }^{3}}-{{\nu }_{xx}}-\alpha \nu \right)}_{x}}+\sigma {{\nu }_{yy}}=0.$
In Eq. (1), α is a coefficient of diffusion and σ degradation coefficient. The diffusion of a gas in a homogeneous medium is an important phenomenon in physical context and the Chaffee-Infante equation provides a useful model to study such phenomena. Many domains, including ocean engineering, physics (particle diffusion), chemistry and biology, utilise the concept of diffusion. The essential concept of diffusion, however, is the same in all of them: a material or collection that is experiencing diffusion extends out from a place or location where it has a larger concentration [69]. Recently, a number of studies on the (2+1)-dimensional Chaffee-Infante equation has been presented, such as Akbar et al. [69] an investigation of the optical solitons to (1), Sulaiman et al. [72] the study of the dynamics of lump solutions to the variable coefficients (2 + 1)-dimensional Chaffee-Infante equation, investigation of the exact solutions to the (2+1)-dimensional Chaffee-Infante equation [73], [74] the study of the exact traveling wave solutions of Chaffee-Infante equation in (2+1)-dimensions, Sakthivel and Chun [75] constructed some new soliton solutions of Chaffee-Infante equation, Demiray and Bayrakci [76] examined the soliton solutions to (1)
For a long time, the interplay of internal waves and ocean topography has been a hot topic of study. The propagation of surface and internal gravity waves is influenced by the driving mechanism of wave solutions. These waves are prevalent in oceans, lakes, and the atmosphere, and the mechanism that causes them could be useful in ocean engineering. A soliton, or solitary wave, is a self-reinforcing wave packet that keeps its structure while propagating at a constant velocity in mathematics, physics and engineering. Solitons are created when nonlinear and dispersive effects in the medium cancel out. Solitons are physical system solutions to a class of weakly nonlinear dispersive partial differential equations. Solitary Waves was first observed Russell [70] during his experiment on efficient design of canal ocean boat. He saw a long wave of water propagating during the experiment without changing its shape. He named this wave as “Great Wave” of translation or “Solitary Waves” and performed further investigations to study the natureof this wave. His description of Solitary Waves dis-confirmed the hypothesis of waves in water according to Airy and Stokes. Breathers are pulsating localized structures arising in coastal engineering with a narrow banded beginning process that have been utilized to imitate extreme waves in a variety of nonlinear dispersive media. Several recent studies, on the other hand, suggest that, despite the associated physical limits, breathers can live in more complex ecosystems, such as random seas.
2. Soliton solutions
The idea here is to provide explicit soliton solutions to the Chaffee–Infante equation by adapting the extended tanh-coth method and the sine-cosine method. First, by means of the new wave variable $\zeta =ax+by-ct$, the Chaffee–Infante is reduced to the following nonlinear ordinary differential equation
$-{{a}^{3}}{\nu }''+\left( {{b}^{2}}\sigma -ac \right){\nu }'+\alpha a{{\nu }^{3}}-\alpha a\nu =0.$
Now, we solve (2) by the following suggested schemes.
2.1. Extended tanh-coth method
Balancing the orders of the nonlinear term $v^{3}$ against the linear term $v^{\prime \prime}$, leads to the following tanh-coth solution
$\nu \left( \zeta \right)={{\lambda }_{0}}+{{\lambda }_{1}}Y+\frac{{{\lambda }_{2}}}{Y},$
where the function $Y=Y\left( \zeta \right)$ is the solution of the following nonlinear ordinary differential equation
$Y=1-{{Y}^{2}},$
whose solution is Y=tanh(ζ) or Y=coth(ζ). Differentiating (3) implicity, leads to
$\begin{matrix} {Y}'\left( \zeta \right) & = & \left( 1-{{Y}^{2}} \right)\left( {{\lambda }_{1}}-\frac{{{\lambda }_{2}}}{{{Y}^{2}}} \right), \\ {Y}''\left( \zeta \right) & = & \frac{2\left( {{Y}^{2}}-1 \right)\left( {{\lambda }_{1}}{{Y}^{4}}-{{\lambda }_{2}} \right)}{{{Y}^{3}}}. \\\end{matrix}$
Next, by substitution of (3) and (5) in (2), we provide a polynomial in the variable Y. Putting together the coefficients of the same powers of Y and setting each to zero, we arrive the following algebraic system in the unknowns λ0, λ1, λ2, a, b and c,
$\begin{matrix} 0 & = & a\alpha \lambda _{2}^{3}-2{{a}^{3}}{{\lambda }_{2}}, \\ 0 & = & {{\lambda }_{2}}\left( 3a\alpha {{\lambda }_{0}}{{\lambda }_{2}}+ac-\sigma {{b}^{2}} \right), \\ 0 & = & a{{\lambda }_{2}}\left( 2{{a}^{2}}+3\alpha \lambda _{0}^{2}+3\alpha {{\lambda }_{1}}{{\lambda }_{2}}-\alpha \right), \\ 0 & = & a\alpha \lambda _{0}^{3}+a\alpha \left( 6{{\lambda }_{1}}{{\lambda }_{2}}-1 \right){{\lambda }_{0}}-\left( {{\lambda }_{1}}+{{\lambda }_{2}} \right)\left( ac-{{b}^{2}}\sigma \right), \\ 0 & = & a{{\lambda }_{1}}\left( 2{{a}^{2}}+3\alpha \lambda _{0}^{2}+3\alpha {{\lambda }_{1}}{{\lambda }_{2}}-\alpha \right), \\ 0 & = & {{\lambda }_{1}}\left( 3a\alpha {{\lambda }_{0}}{{\lambda }_{1}}+ac-\sigma {{b}^{2}} \right), \\ 0 & = & a\alpha \lambda _{1}^{3}-2{{a}^{3}}{{\lambda }_{1}}. \\ \end{matrix}$
By simplifying the above system, we deduce the following kink and singular-kink solutions to the Chaffee-Infante model:
$\begin{array}{l} \begin{array}{l} v_{1}(x, y, t)=\mp \operatorname{coth}\left(b y \pm\left(\frac{\sqrt{\alpha} x}{\sqrt{2}}-\frac{\sqrt{2} b^{2} \sigma t}{\sqrt{\alpha}}\right)\right) \\ v_{2}(x, y, t)=\mp \tanh \left(b y \pm\left(\frac{\sqrt{\alpha} x}{\sqrt{2}}-\frac{\sqrt{2} b^{2} \sigma t}{\sqrt{\alpha}}\right)\right) \text {, } \\ v_{3}(x, y, t)=\mp \frac{1}{2} \operatorname{coth}\left(b y \pm\left(\frac{\sqrt{\alpha} x}{2 \sqrt{2}}-\frac{t\left(8 \sqrt{2} b^{2} \sigma-3 \alpha^{3 / 2}\right)}{4 \sqrt{\alpha}}\right)\right) \mp \frac{1}{2} \text {, } \\ v_{4}(x, y, t)=\mp \frac{1}{2} \tanh \left(b y \pm\left(\frac{\sqrt{\alpha} x}{2 \sqrt{2}}-\frac{t\left(8 \sqrt{2} b^{2} \sigma-3 \alpha^{3 / 2}\right)}{4 \sqrt{\alpha}}\right)\right) \mp \frac{1}{2} \text {, } \\ \nu_{5}(x, y, t)=\mp\left(\frac{1}{2} \tanh \left(b y \pm\left(\frac{\sqrt{\alpha} x}{2 \sqrt{2}}-\frac{2 \sqrt{2} b^{2} \sigma t}{\sqrt{\alpha}}\right)\right)\right. \\ \left.+\frac{1}{2} \operatorname{coth}\left(b y \pm\left(\frac{\sqrt{\alpha} x}{2 \sqrt{2}}-\frac{2 \sqrt{2} b^{2} \sigma t}{\sqrt{\alpha}}\right)\right)\right) \text {, } \\ v_{6}(x, y, t)=\mp \frac{1}{2} \mp \frac{1}{4} \tanh \left(b y \pm\left(\frac{\sqrt{\alpha} x}{4 \sqrt{2}}-\frac{t\left(32 \sqrt{2} b^{2} \sigma-3 \alpha^{3 / 2}\right)}{8 \sqrt{\alpha}}\right)\right) \\ \mp \frac{1}{4} \operatorname{coth}\left(b y \pm\left(\frac{\sqrt{\alpha} x}{4 \sqrt{2}}-\frac{t\left(32 \sqrt{2} b^{2} \sigma-3 \alpha^{3 / 2}\right)}{8 \sqrt{\alpha}}\right)\right). \end{array}\\ v_{1}(x, y, t)=\mp \operatorname{coth}\left(b y \pm\left(\frac{\sqrt{\alpha} x}{\sqrt{2}}-\frac{\sqrt{2} b^{2} \sigma t}{\sqrt{\alpha}}\right)\right)\\ v_{2}(x, y, t)=\mp \tanh \left(b y \pm\left(\frac{\sqrt{\alpha} x}{\sqrt{2}}-\frac{\sqrt{2} b^{2} \sigma t}{\sqrt{\alpha}}\right)\right) \text {, }\\ v_{3}(x, y, t)=\mp \frac{1}{2} \operatorname{coth}\left(b y \pm\left(\frac{\sqrt{\alpha} x}{2 \sqrt{2}}-\frac{t\left(8 \sqrt{2} b^{2} \sigma-3 \alpha^{3 / 2}\right)}{4 \sqrt{\alpha}}\right)\right) \mp \frac{1}{2} \text {, }\\ v_{4}(x, y, t)=\mp \frac{1}{2} \tanh \left(b y \pm\left(\frac{\sqrt{\alpha} x}{2 \sqrt{2}}-\frac{t\left(8 \sqrt{2} b^{2} \sigma-3 \alpha^{3 / 2}\right)}{4 \sqrt{\alpha}}\right)\right) \mp \frac{1}{2} \text {, }\\ \nu_{5}(x, y, t)=\mp\left(\frac{1}{2} \tanh \left(b y \pm\left(\frac{\sqrt{\alpha} x}{2 \sqrt{2}}-\frac{2 \sqrt{2} b^{2} \sigma t}{\sqrt{\alpha}}\right)\right)\right.\\ \left.+\frac{1}{2} \operatorname{coth}\left(b y \pm\left(\frac{\sqrt{\alpha} x}{2 \sqrt{2}}-\frac{2 \sqrt{2} b^{2} \sigma t}{\sqrt{\alpha}}\right)\right)\right) \text {, }\\ v_{6}(x, y, t)=\mp \frac{1}{2} \mp \frac{1}{4} \tanh \left(b y \pm\left(\frac{\sqrt{\alpha} x}{4 \sqrt{2}}-\frac{t\left(32 \sqrt{2} b^{2} \sigma-3 \alpha^{3 / 2}\right)}{8 \sqrt{\alpha}}\right)\right)\\ \mp \frac{1}{4} \operatorname{coth}\left(b y \pm\left(\frac{\sqrt{\alpha} x}{4 \sqrt{2}}-\frac{t\left(32 \sqrt{2} b^{2} \sigma-3 \alpha^{3 / 2}\right)}{8 \sqrt{\alpha}}\right)\right). \end{array}$
Fig. 1, stands for the physical shapes of the provided solutions in (7) and they are of types kink and singular-kink.
Fig. 1. The physical shapes of the obtained solutions to the Chaffee-Infante model as depicted in (7), respectively, ν1,ν2 and ν6. “Extended tanh-coth method”. |
2.2. Sine-cosine function method
Here we seek periodic solutions to the Chaffee-Infante model by proposing the sine-cosine algorithm. The suggested solution is
$\nu \left( \zeta \right)=\lambda {{\sin }^{\beta }}\left( \zeta \right),$
or
$\nu \left( \zeta \right)=\lambda {{\cos }^{\beta }}\left( \zeta \right),$
where $\zeta =ax+by-ct$. Substitution of (8) in (2) results in a finite series with different powers of the sine function whose coefficients are identical to zero. The resulting system with the unknowns a, b, c, λ and β is
$\begin{matrix} 0 & = & 3\beta -\left( \beta -2 \right), \\ 0 & = & a\alpha {{\lambda }^{3}}-{{a}^{3}}\left( \beta -1 \right)\beta \lambda, \\ 0 & = & a\lambda \left( {{a}^{2}}{{\beta }^{2}}-\alpha \right), \\ 0 & = & \beta \lambda \left( {{b}^{2}}\sigma -ac \right). \\\end{matrix}$
Solving the above system results in the following singular-periodic solution
$\nu \left( x,y,t \right)=\pm \sqrt{2}\csc \left( by\pm \left( \sqrt{\alpha }x-\frac{{{b}^{2}}\sigma t}{\sqrt{\alpha }} \right) \right).$
Moreover, if (9) is used instead of (8), another singular-periodic solution will be obtained and defined as
$\nu \left( x,y,t \right)=\pm \sqrt{2}\sec \left( by\pm \left( \sqrt{\alpha }x-\frac{{{b}^{2}}\sigma t}{\sqrt{\alpha }} \right) \right).$
Fig. 2, represents singular-periodic solutions to the Chaffee-Infante model as depicted in (11) and (12).
Fig. 2. Singular-periodic solutions to the Chaffee-Infante model as depicted in (11) and (12), respectively. “Sine-cosine method”. |
3. Breather waves solutions
Take the equation
$v\left( x,y,t \right)=\sqrt{\frac{2}{\alpha }}{{\left( \ln \eta \right)}_{x}}=\sqrt{\frac{2}{\alpha }}\frac{{{\eta }_{x}}}{\eta }.$
Putting Eq. (13) into (1), we provide
$\sigma \eta _{y}^{2}+{{\eta }_{x}}\left( {{\eta }_{t}}-3{{\eta }_{xx}} \right)+\eta \left( \alpha {{\eta }_{x}}-\sigma {{\eta }_{yy}}-{{\eta }_{xt}}+{{\eta }_{xxx}} \right)=0.$
Take the following test function:
$\eta \left( x,y,t \right)={{e}^{{{\xi }_{1}}}}+{{\gamma }_{1}}\cos \left( {{\xi }_{2}} \right)+{{\gamma }_{2}}{{e}^{{{\xi }_{3}}}},$
where ${{\xi }_{1}}=-{{\vartheta }_{1}}\left( {{\varepsilon }_{0}}t+x+{{\gamma }_{0}}y \right),\;{{\xi }_{2}}={{\vartheta }_{0}}\left( t{{\varpi }_{0}}+x+y{{\varsigma }_{0}} \right),\;{{\xi }_{3}}={{\vartheta }_{1}}\left( {{\varepsilon }_{0}}t+x+{{\gamma }_{0}}y \right).$
Substituting Eq. (15) into (14), we get a polynomial in powers of exponential and trigonometric functions. Gathering the coefficients of $\sin \left( {{\vartheta }_{0}}\left( t{{\varpi }_{0}}+x+y{{\varsigma }_{0}} \right) \right)\cos \left( {{\vartheta }_{0}}\left( t{{\varpi }_{0}}+x+y{{\varsigma }_{0}} \right) \right),\;{{e}^{-2{{\vartheta }_{1}}\left( {{\varepsilon }_{0}}t+x+{{\gamma }_{0}}y \right)}},\;{{e}^{2{{\vartheta }_{1}}\left( {{\varepsilon }_{0}}t+x+{{\gamma }_{0}}y \right)}}$, ${{e}^{{{\vartheta }_{1}}\left( -\left( {{\varepsilon }_{0}}t+x+{{\gamma }_{0}}y \right) \right)}}\sin \left( {{\vartheta }_{0}}\left( t{{\varpi }_{0}}+x+y{{\varsigma }_{0}} \right) \right),\;{{e}^{{{\vartheta }_{1}}\left( -\left( {{\varepsilon }_{0}}t+x+{{\gamma }_{0}}y \right) \right)}}\cos \left( {{\vartheta }_{0}}\left( t{{\varpi }_{0}}+x+y{{\varsigma }_{0}} \right) \right),\;{{e}^{{{\vartheta }_{1}}\left( {{\varepsilon }_{0}}t+x+{{\gamma }_{0}}y \right)}}\sin \left( {{\vartheta }_{0}}\left( t{{\varpi }_{0}}+x+y{{\varsigma }_{0}} \right) \right)$${{e}^{{{\vartheta }_{1}}\left( {{\varepsilon }_{0}}t+x+{{\gamma }_{0}}y \right)}}\cos \left( {{\vartheta }_{0}}\left( t{{\varpi }_{0}}+x+y{{\varsigma }_{0}} \right) \right)$we gather the following system of algebraic equations:
$\begin{array}{l} -\gamma_{1}^{2} \vartheta_{0}\left(\alpha+2 \vartheta_{0}^{2}\right)=0 \\ -4 \gamma_{2} \varepsilon_{0} \vartheta_{1}^{2}-4 \gamma_{0}^{2} \gamma_{2} \sigma \vartheta_{1}^{2}+\gamma_{1}^{2} \vartheta_{0}^{2}\left(\sigma \varsigma_{0}^{2}+\varpi_{0}\right)=0 \\ 2 \vartheta_{1}^{3}-\alpha \vartheta_{1}=0 \\ \gamma_{2}^{2} \vartheta_{1}\left(\alpha-2 \vartheta_{1}^{2}\right)=0 \\ \gamma_{1} \vartheta_{0}\left(-\alpha+\vartheta_{1}\left(2 \gamma_{0} \sigma \varsigma_{0}+\varepsilon_{0}+3 \vartheta_{1}+\varpi_{0}\right)+\vartheta_{0}^{2}\right)=0 \\ -\gamma_{1}\left(\vartheta_{1}\left(\alpha+3 \vartheta_{0}^{2}\right)+\vartheta_{1}^{2}\left(\gamma_{0}^{2} \sigma+\varepsilon_{0}\right)-\vartheta_{0}^{2}\left(\sigma \varsigma_{0}^{2}+\varpi_{0}\right)+\vartheta_{1}^{3}\right)=0 \\ \gamma_{1} \gamma_{2} \vartheta_{0}\left(-\alpha+\vartheta_{1}\left(-2 \gamma_{0} \sigma \varsigma_{0}-\varepsilon_{0}+3 \vartheta_{1}-\varpi_{0}\right)+\vartheta_{0}^{2}\right)=0 \\ \gamma_{1} \gamma_{2}\left(\vartheta_{1}\left(\alpha+3 \vartheta_{0}^{2}\right)-\vartheta_{1}^{2}\left(\gamma_{0}^{2} \sigma+\varepsilon_{0}\right)+\vartheta_{0}^{2}\left(\sigma \varsigma_{0}^{2}+\varpi_{0}\right)+\vartheta_{1}^{3}\right)=0 \end{array}$
Solving Eq. (16), we get the following lump collision phenomena to Eq. (1):
Case-1: When
${{\vartheta }_{1}}=\sqrt{\frac{\alpha }{2}},\;{{\vartheta }_{0}}=-\sqrt{\frac{-\alpha }{2}},\;{{\gamma }_{0}}={{\varsigma }_{0}},\;{{\varepsilon }_{0}}=-\sigma \varsigma _{0}^{2},\;{{\varpi }_{0}}=-\sigma \varsigma _{0}^{2},$
we get
$\begin{matrix} {{\eta }_{1}}\left( x,y,t \right) & = & {{\gamma }_{2}}{{e}^{\frac{\sqrt{\alpha }\left( \sigma \left( -t \right)\varsigma _{0}^{2}+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}}}} \\ {} & {} & +{{\gamma }_{1}}\cosh \left( \frac{\sqrt{\alpha }\left( \sigma \left( -t \right)\varsigma _{0}^{2}+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}} \right) \\ {} & {} & +{{e}^{-\frac{\sqrt{\alpha }\left( \sigma \left( -t \right)\varsigma _{0}^{2}+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}}}}. \\ \end{matrix}$
Consequently,
${{v}_{1}}\left( x,y,t \right)=\frac{\sqrt{2}\sqrt{\frac{1}{\alpha }}\left( \frac{\sqrt{\alpha }{{\gamma }_{2}}{{e}^{\frac{\sqrt{\alpha }\left( \sigma \left( -t \right)\varsigma _{0}^{2}+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}}}}}{\sqrt{2}}+\frac{\sqrt{\alpha }{{\gamma }_{1}}\sinh \left( \frac{\sqrt{\alpha }\left( \sigma \left( -t \right)\varsigma _{0}^{2}+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}} \right)}{\sqrt{2}}-\frac{\sqrt{\alpha }{{e}^{-\frac{\sqrt{\alpha }\left( \sigma \left( -t \right)\varsigma _{0}^{2}+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}}}}}{\sqrt{2}} \right)}{{{\gamma }_{2}}{{e}^{\frac{\sqrt{\alpha }\left( \sigma \left( -t \right)\varsigma _{0}^{2}+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}}}}+{{\gamma }_{1}}\cosh \left( \frac{\sqrt{\alpha }\left( \sigma \left( -t \right)\varsigma _{0}^{2}+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}} \right)+{{e}^{-\frac{\sqrt{\alpha }\left( \sigma \left( -t \right)\varsigma _{0}^{2}+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}}}}}.$
Case-2: When
${{\vartheta }_{1}}=\sqrt{\frac{\alpha }{2}},\;{{\vartheta }_{0}}=\sqrt{\frac{-\alpha }{2}},\;{{\gamma }_{0}}={{\varsigma }_{0}},\;{{\varepsilon }_{0}}=-2\sigma \varsigma _{0}^{2}-{{\varpi }_{0}},\;{{\gamma }_{2}}=\frac{\gamma _{1}^{2}}{4},$
we get
$\begin{matrix} {{\eta }_{2}}\left( x,y,t \right) & = & \frac{1}{4}\gamma _{1}^{2}{{e}^{\frac{\sqrt{\alpha }\left( t\left( -2\sigma \varsigma _{0}^{2}-{{\varpi }_{0}} \right)+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}}}}+{{e}^{-\frac{\sqrt{\alpha }\left( t\left( -2\sigma \varsigma _{0}^{2}-{{\varpi }_{0}} \right)+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}}}} \\ {} & {} & +{{\gamma }_{1}}\cosh \left( \frac{\sqrt{\alpha }\left( t{{\varpi }_{0}}+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}} \right). \\\end{matrix}$
Consequently,
${{v}_{2}}\left( x,y,t \right)=\frac{\sqrt{2}\sqrt{\frac{1}{\alpha }}\left( \frac{\sqrt{\alpha }\gamma _{1}^{2}{{e}^{\frac{\sqrt{\alpha }\left( t\left( -2\sigma \varsigma _{0}^{2}-{{\varpi }_{0}} \right)+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}}}}}{4\sqrt{2}}-\frac{\sqrt{\alpha }{{e}^{-\frac{\sqrt{\alpha }\left( t\left( -2\sigma \varsigma _{0}^{2}-{{\varpi }_{0}} \right)+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}}}}}{\sqrt{2}}+\frac{\sqrt{\alpha }{{\gamma }_{1}}\sinh \left( \frac{\sqrt{\alpha }\left( t{{\varpi }_{0}}+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}} \right)}{\sqrt{2}} \right)}{\frac{1}{4}\gamma _{1}^{2}\frac{\sqrt{\alpha }\left( t\left( -2\sigma \varsigma _{0}^{2}-{{\varpi }_{0}} \right)+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}}+{{e}^{-\frac{\sqrt{\alpha }\left( t\left( -2\sigma \varsigma _{0}^{2}-{{\varpi }_{0}} \right)+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}}}}+{{\gamma }_{1}}\cosh \left( \frac{\sqrt{\alpha }\left( t{{\varpi }_{0}}+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}} \right)}.$
Case-3: When
$\begin{matrix} {{\vartheta }_{1}} & = & -\sqrt{\frac{\alpha }{2}},\;{{\vartheta }_{0}}=-\sqrt{\frac{-\alpha }{2}},\;{{\gamma }_{0}}={{\varsigma }_{0}},\;{{\varepsilon }_{0}}=-\sigma \varsigma _{0}^{2}, \\ {{\varpi }_{0}} & = & -\sigma \varsigma _{0}^{2},\;{{\gamma }_{2}}=0, \\\end{matrix}$
we get
$\begin{matrix} {{\eta }_{3}}\left( x,y,t \right) & = & {{\gamma }_{1}}\cosh \left( \frac{\sqrt{\alpha }\left( \sigma \left( -t \right)\varsigma _{0}^{2}+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}} \right) \\ {} & {} & +{{e}^{\frac{\sqrt{\alpha }\left( \sigma \left( -t \right)\varsigma _{0}^{2}+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}}}}. \\\end{matrix}$
Consequently,
${{v}_{3}}\left( x,y,t \right)=\frac{\sqrt{2}\sqrt{\frac{1}{\alpha }}\left( \frac{\sqrt{\alpha }{{\gamma }_{1}}\sinh \left( \frac{\sqrt{\alpha }\left( \sigma \left( -t \right)\varsigma _{0}^{2}+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}} \right)}{\sqrt{2}}+\frac{\sqrt{\alpha }{{e}^{\frac{\sqrt{\alpha }\left( \sigma \left( -t \right)\varsigma _{0}^{2}+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}}}}}{\sqrt{2}} \right)}{{{\gamma }_{1}}\cosh \left( \frac{\sqrt{\alpha }\left( \sigma \left( -t \right)\varsigma _{0}^{2}+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}} \right)+{{e}^{\frac{\sqrt{\alpha }\left( \sigma \left( -t \right)\varsigma _{0}^{2}+x+y{{\varsigma }_{0}} \right)}{\sqrt{2}}}}}.$
Case-4: When
${{\vartheta }_{0}}=-i{{\vartheta }_{1}},\;{{\gamma }_{0}}={{\varsigma }_{0}},\;{{\varepsilon }_{0}}=-2\sigma \varsigma _{0}^{2}-{{\varpi }_{0}},\;{{\gamma }_{2}}=\frac{\gamma _{1}^{2}}{4},\;\alpha =2\vartheta _{1}^{2},$
we get
$\begin{matrix} {} & {} & {{\eta }_{4}}\left( x,y,t \right)=\frac{1}{4}\gamma _{1}^{2}{{e}^{{{\vartheta }_{1}}\left( t\left( -2\sigma \varsigma _{0}^{2}-{{\varpi }_{0}} \right)+x+y{{\varsigma }_{0}} \right)}} \\ {} & {} & +{{e}^{-{{\vartheta }_{1}}\left( t\left( -2\sigma \varsigma _{0}^{2}-{{\varpi }_{0}} \right)+x+y{{\varsigma }_{0}} \right)}}+{{\gamma }_{1}}\cosh \left( {{\vartheta }_{1}}\left( t{{\varpi }_{0}}+x+y{{\varsigma }_{0}} \right) \right). \\\end{matrix}$
Consequently,
${{v}_{4}}\left( x,y,t \right)=\frac{\sqrt{\frac{1}{\vartheta _{1}^{2}}}\left( \frac{1}{4}\gamma _{1}^{2}{{\vartheta }_{1}}{{e}^{{{\vartheta }_{1}}\left( t\left( -2\sigma \varsigma _{0}^{2}-{{\varpi }_{0}} \right)+x+y{{\varsigma }_{0}} \right)}}-{{\vartheta }_{1}}{{e}^{-{{\vartheta }_{1}}\left( t\left( -2\sigma \varsigma _{0}^{2}-{{\varpi }_{0}} \right)+x+y{{\varsigma }_{0}} \right)}}+{{\gamma }_{1}}{{\vartheta }_{1}}\sinh \left( {{\vartheta }_{1}}\left( t{{\varpi }_{0}}+x+y{{\varsigma }_{0}} \right) \right) \right)}{\frac{1}{4}\gamma _{1}^{2}{{e}^{{{\vartheta }_{1}}\left( t\left( -2\sigma \varsigma _{0}^{2}-{{\varpi }_{0}} \right)+x+y{{\varsigma }_{0}} \right)}}+{{e}^{-{{\vartheta }_{1}}\left( t\left( -2\sigma \varsigma _{0}^{2}-{{\varpi }_{0}} \right)+x+y{{\varsigma }_{0}} \right)}}+{{\gamma }_{1}}\cosh \left( {{\vartheta }_{1}}\left( t{{\varpi }_{0}}+x+y{{\varsigma }_{0}} \right) \right)}.$
4. Conservation laws using Lie-Bäcklund symmetries
The Lie-Bäcklund transformation group is an example of a tangent transformation group. It is thought to be the extension of the one-parameter group of continuous symmetry transformations. The form of the vector field below can be used to determine the Lie-Bäcklund symmetry generator of the governing equation:
$Q={{\eta }^{\nu }}\left( x,t,y,\nu,{{\nu }_{x}},{{\nu }_{xx}},{{\nu }_{xxx}},{{\nu }_{xxxx}} \right).$
Over-determined systems are discovered using a fourth prolongation to the governing equation. The over-determined system’s general acquired solutions are:
$\begin{matrix} {} & {} & {{\eta }^{\nu }}(x,t,y,\nu,{{\nu }_{x}},{{\nu }_{t}},{{\nu }_{y}},{{\nu }_{xx}},{{\nu }_{xt}},{{\nu }_{xy}},{{\nu }_{tt}},{{\nu }_{ty}},{{\nu }_{xxx}},{{\nu }_{xxt}}, \\ {} & {} & {{\nu }_{xxy}},{{\nu }_{xtt}},{{\nu }_{xty}},{{\nu }_{ttt}},{{\nu }_{tty}})= \\ {} & {} & 2{{c}_{4}}\sigma t{{u}_{y}}-{{c}_{4}}{{u}_{x}}y+{{c}_{1}}{{u}_{t}}+{{c}_{2}}{{u}_{x}}+{{c}_{3}}{{u}_{y}}, \\\end{matrix}$
where a2, a3, a4 and a5 are arbitrary constants. Thus we reach:
$\begin{matrix} {{Q}_{1}} & = & {{\nu }_{t}}\frac{\partial }{\partial \nu }, \\ {{Q}_{2}} & = & {{\nu }_{x}}\frac{\partial }{\partial \nu }, \\ {{Q}_{3}} & = & {{\nu }_{y}}\frac{\partial }{\partial \nu }, \\ {{Q}_{4}} & = & \left( 2\sigma t{{u}_{y}}-{{u}_{x}}y \right)\frac{\partial }{\partial \nu }, \\\end{matrix}$
The lagrangian is obtained as
$L=\chi \left( 3\alpha {{\nu }_{x}}{{\nu }^{2}}-\alpha {{\nu }_{x}}+\sigma {{\nu }_{yy}}+{{\nu }_{xt}}-{{\nu }_{xxx}} \right),$
and the adjoint system is given by
${{\chi }_{x}}\left( \alpha -3\alpha {{\nu }^{2}} \right)+\sigma {{\chi }_{yy}}+{{\chi }_{xt}}+{{\chi }_{xxx}}=0,$
where χ is the new dependent variable.
$\begin{aligned} C^{x}= & \frac{1}{2} v\left(2 \alpha\left(3 \nu^{2}-1\right) \nu_{t}+\nu_{t t}-2 \nu_{x x t}\right)+\nu_{x t} v_{x}-\frac{1}{2} \nu_{t}\left(v_{t}+2 v_{x x}\right) \\ C^{t}= & \frac{1}{2}\left(\nu_{x t} v-\nu_{t} v_{x}\right) \\ C^{y}= & \sigma \nu_{t y} v-\sigma \nu_{t} v_{y} \\ C^{x}= & \nu_{x}\left(\alpha\left(3 \nu^{2}-1\right) v-v_{x x}-\frac{1}{2} v_{t}\right)+\nu_{x x} v_{x}+\frac{1}{2} \nu_{x, t} v-\nu_{x x x} v \\ C^{t}= & \frac{1}{2}\left(\nu_{x x} v-\nu_{x} v_{x}\right) \\ C^{y}= & \sigma \nu_{x y} v-\sigma \nu_{x} v_{y} \\ C^{x}= & \frac{1}{2} v\left(2 \alpha\left(3 \nu^{2}-1\right) \nu_{y}+\nu_{t y}-2 \nu_{x x y}\right)+\nu_{x y} v_{x}-\frac{1}{2} \nu_{y}\left(v_{t}+2 v_{x x}\right) \\ C^{t}= & \frac{1}{2}\left(\nu_{x y} v-\nu_{y} v_{x}\right) \\ C^{y}= & \sigma \nu_{y y} v-\sigma \nu_{y} v_{y} \\ C^{x}= & \left(2 \sigma t \nu_{y}-y \nu_{x}\right)\left(\alpha\left(3 \nu^{2}-1\right) v-v_{x x}-\frac{1}{2} v_{t}\right) \\ & +v_{x}\left(2 \sigma t \nu_{x y}-y \nu_{x x}\right) \\ & +\frac{1}{2} v\left(2 \sigma \nu_{y}+2 \sigma t \nu_{t y}-y \nu_{x t}\right)-v\left(2 \sigma t \nu_{x x y}-y \nu_{x x x}\right) \\ C^{t}= & \frac{1}{2}\left(v\left(2 \sigma t \nu_{x y}-y \nu_{x x}\right)-v_{x}\left(2 \sigma t \nu_{y}-y \nu_{x}\right)\right) \\ C^{y}= & \sigma v\left(2 \sigma t \nu_{y y}-\nu_{x}-y \nu_{x y}\right)-\sigma v_{y}\left(2 \sigma t \nu_{y}-y \nu_{x}\right) \end{aligned}$
5. Results and discussion
The Chaffee-Infante equation in two dimensions is investigated in this paper. The studied equation is successfully provided with certain essential wave solutions, lump interaction phenomena, and new conservation laws. Topological and solitary soliton solutions were found using the extended tanh-coth approach. Singular periodic wave solutions were found using the sine-cosine function approach. Breather wave solutions were reported using the Hirota bilinear technique. Solitons are self-reinforcing wave packets that keep their structure while propagating at a constant speed. Breathers are pulsating localized structures that have been used to simulate extreme waves in a variety of nonlinear dispersive media. They start with a thin banded process. Furthermore, the hyperbolic sine function appears in the gravitational potential of a cylinder and the calculation of the Roche limit, the hyperbolic cosine function appears in the shape of a hanging cable (the so-called CATENARY), the hyperbolic tangent appears in the calculation of magnetic moment and special relativity rapidity, and the hyperbolic cotangent appears in the Langevin function for magnetic porosity [71].
Akbar et al. [69] presented topological, singular solitons, singular periodic wave and exponential function solutions to the (2+1)-dimensional Chaffee-Infante equation. Sulaiman et al. studied the variable coefficient version of the studied equation and lump solutions were reported [72]. Mao [73] reported some exact solutions to Eq. (1) in terms of elliptic functions via the canonical-like transformation method and trial equation method. Tahir et al. [74] secured some important exact travelling wave solutions to the generalized Kudryashov method. Sakthivel and Chun [75] reported some travelling wave solutions to Eq. (1) using the exp-function method. Demiray and Bayrakci [76] constructed some important soliton solutions to the studied model by using the sine-Gordon equation expansion method. This study expanded the search to breather wave solutions and some other soliton solutions with important physical features. On the other hand, it develops new conservation laws he Lie-Bäcklund symmetries and a novel conservation theorem (Fig. 3, Fig. 4, Fig. 5, Fig. 6).
Fig. 3. The graphs of (18) under ${{\varsigma }_{0}}=-0.6,\;\alpha =7.9,\;{{\gamma }_{2}}=0.4,\;{{\gamma }_{1}}=6.7,\;\sigma =-0.45$ |
Fig. 4. The graphs of (20) under ${{\varsigma }_{0}}=-3,\;\alpha =2,\;{{\varpi }_{0}}=3.9,\;{{\gamma }_{1}}=-3,\;\sigma =-3$ |
Fig. 5. The graphs of (22) under ${{\varsigma }_{0}}=0.5,\;\alpha =8.84,\;{{\gamma }_{1}}=-0.7,\;\sigma =-3$ |
Fig. 6. The graphs of (24) under ${{\varsigma }_{0}}=-2,\;{{\vartheta }_{1}}=-2,\;{{\varpi }_{0}}=-2,\;{{\gamma }_{1}}=-1,\;\sigma =-1$ |
6. Conclusion
This research looked at the Chaffee-Infante equation in two dimensions. Two strong analytical methodologies, the extended tanh-coth and sine-cosine approaches, were employed to come up with some novel solutions to the equation. The breather wave solutions were calculated using the Cole-Hopf transformation. Lie-Bäcklund symmetries and a novel conservation theorem were used to create the CLs. The collected solutions’ physical features are displayed to provide a clear view of them. The discoveries can be applied to a variety of disciplines to better understand complex physical processes. It’s also worth noting that all of the reported solutions have met the requirements of the original equation. The methods presented are simple to use and may be used to confidently find solutions for a variety of nonlinear PDEs. Furthermore, this study’s findings are wholly new and have never been published before. More lump interaction phenomena to the (2+1)-dimensional Chaffee-Infante equation will be published in future studies.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
