Nonlinear analysis of a nonlinear modified KdV equation under Atangana Baleanu Caputo derivative

1.   Introduction

Korteweg and de-Vries developed the classical KdV equation in 1895 as a nonlinear PDE to investigate the waves that occurs on the surfaces of shallow water. Many studies have been conducted on this exactly solvable model. Many scholars have proposed novel applications of the classical KdV equation, such as acoustic waves that produces in a plasma in ion form, and acoustic waves which are produces on a crystal lattice. In ^[1], the classical KdV equation is as follows

Different variations of Eq (1.1) have been published in the literature, including ^[2,3]. In literature ^[2], various efficient methods have been used to get solitons solutions of different kinds of KdV equations. Wang ^[3] used a quadratic function Ansatz to obtain lump solutions for the (2+1)-dimensional KdV equation. In ^[4,5,6], several systematic techniques were utiltized to investigate the different types of KdV equations. Wang and Kara introduced a new 2D-mKdV in 2019 ^[7]. The new (2+1)-dimensional mKdV equation is given by

After 1695, the non-integer order or fractional-order derivative (FOD) was described as a simple academic generalization of the classical derivative. A FOD is an operator that extends the order of differentiation from Natural numbers ($N$) to a set of real numbers ($R$) or even to a set of complex numbers ($C$). Fractional calculus has emerged as one of the most effective methods for describing long-memory processes over the last decade. Engineers and physicists, as well as pure mathematicians, are interested in such models. The models that are represented by differential equations with fractional-order derivatives are the most interesting ^[8,9,10]. Their evolutions are much more complicated than in the classical integer-order case, and deciphering the underlying principle is a difficult task. There are many fractional operators regarding to the kernels involved in the integration. The most popular fractional operator is Caputo-Liouville which is based on power-law kernel, but this kernel has issue about the singularity of the kernel. To tackle the limitation of derivative operators with power-law kernels, new types of nonlocal FOD have been introduced in recent literature. For example, in literature ^[11], Caputo and Fabrizio (CF) introduced a FOD that is focused on the exponential kernel. However, the CF derivative, on the other hand, has some issues with the kernel's locality. In 2016, Atangana and Baleanu constructed an updated version of FOD that is based on the Mittag-Leffler function ^[12]. This derivative solves the issues of locality and singularity. The FOD of Atangana-Baleanu in Caputo sense ($\mathcal{ABC}$) accurately describes the memory. The $\mathcal{ABC}$ operator's most significant applications are available in ^{[13,14,15,16,17]}.

Certain important techniques have been used to solve fractional order differential equations (FODEs). Some of these includes the homotopy perturbation method (HPM), Laplace Adomian decomposition method (LADM) fractional operational matrix method (FOMM), homotopy analysis method (HAM) and many more ^{[18,19,20,21,22]}. In contrast to these techniques, the Laplace-Adomian decomposition method (LADM) is an important tool for solving non linear FODEs. The ADM and the Laplace transformation are two essential methods that are combined in LADM. Furthermore, unlike a Runge-Kutta process, LADM does not require a predefine size declaration. Every numerical or analytical approach has its own set of benefits and drawbacks. For instance, discretization of data is used in collocation techniques that require extra memory and a longer operation. Since the Laplace-Adomian approach has less parameters than all other methods, it is a useful tool that does not necessitate discretion or linearization ^[23]. With the aid of LADM, the smoke model was successfully solved in ^[24]. LADM was utilized by the authors to solve third order dispersive PDE defined by FOD in ^[22].

Inspired by above literature, in this paper, we study Eq (1.2) under $\mathcal{ABC}$-operator. We use LADM to solve the proposed equation. Consider Eq (1.2) under $\mathcal{ABC}$-operator as

2.   Preliminaries

Definition 2.1. ^[12] Let $0 < \varPsi\leq1$ and $f\left(t\right)\in\mathbb{\mathcal{H}}^{1}$. Then ${\mathcal{ABC}}$ FOD of order $\varPsi$ is expressed as

where $c\left(\varPsi\right)$ is the normalization function such that $c\left(0\right) = c\left(1\right) = 1$. The symbol $E_{\varPsi}$ denotes the Mittag-Leffler kernel which is defined as:

Definition 2.2. ^[12] Let $0 < \varPsi\leq1$ and $f\left(t\right)\in\mathbb{\mathcal{H}}^{1}(0, T).$ Then $\mathcal{AB}$ fractional integral of order $\varPsi$ is defined as

Definition 2.3. ^[12] The formula for the Laplace transform of $\mathcal{ABC}$ FOD of $f(t)$ is defined by

Theorem 2.4. ^[25] Let $\mathbb{H}$ be a Banach space and $\bf{X}:\mathbb{H}\rightarrow \mathbb{H}$be a mapping. Then $\bf{X}$ is said to be Picard's $\bf{X}$-stable, if $\forall\; \; \xi, m\in\mathbb{H},$

where $a\geq0,$ and $b\in[0, 1].$ Further, $\bf{X}$ has a fixedpoint.

3.   Lipschitz condition holds for $\mathcal{ABC}$ derivative

Theorem 3.1. ^[12] The $\mathcal{ABC}$ derivative holds the following Lipschitztype condition for $0 < \digamma < \infty$.

Proof. By using definition of ${\mathcal{ABC}},$ wehave

Using the Lipschitz condition for the first order derivative, we can find a small positive constant $\rho_1$ such that

where $\digamma = \frac{c(\varPsi)\rho_1}{\left(1-\varPsi\right)}E_{\varPsi}\left(-\frac{ \varPsi t^{\varPsi}}{\left(1-\varPsi\right)}\right)$. Thus Lipschitz condition holds for $\mathcal{ABC}$ derivative.

4.   Existence theory

Let

Eq (1.3) can be written as

Applying the $\mathcal{ABC}$ FOI to Eq (4.2), we have

First we have to verify that the Lipschitz condition holds for the kernel $\Phi\left(x, y, t;f\right)$. For this, let us take two bounded functions, $f$ and $g$, i.e., $\left\Vert f\right\Vert \leq\varDelta_{1},$ and $\left\Vert g\right\Vert \leq\varDelta_{2}$ where $\varDelta_{1}, \varDelta_{2} > 0,$ and consider

Since $f$ and $g$ are bounded functions. Therefore, their partial derivatives satisfy the Lipschitz conditions and there exists non-negative constants $\mathbb{K}, \mathbb{L}, \mathbb{\mathbb{M}}, \bf{N}, \mathbb{O}, \mathbb{P}$ such that

Let

thus

For further analysis, we make an iterative scheme as

where $f_{0}\left(x, y, t\right) = f\left(x, y, 0\right).$ Now the difference between two consecutive terms can be taken as

Also, we have

with $f_{-1} = 0$.

Theorem 4.1. Assume that $f\left(x, y, t\right)$ is bounded a function. Then

Proof. Consider

The Eq (4.5) gets the form under norm as

To get the required result, we use the concept of mathematical induction. For $\xi = 1$, one can get

Now assume that the result is true for $\xi-1$, i.e.,

Next, we have to show that

To get the result (4.4), consider

This ends the proof.

Theorem 4.2. If the following relation holds at $t = t_{0}\geq0,$ where

Then at least one solution of the new 2D KdV equation under the $\mathcal{ABC}$fractional derivative exists.

Proof. With the help of Eq (4.3), we have

for $t = t_{0},$ we get

From the above relation, we can say that

Since

this implies that sequence $f_{\xi}\left(x, y, t\right)$ is convergent and therefore the sequence is bounded for each $\xi.$ Further, assume that

Since $f_{\xi}\left(x, y, t\right)$ is bounded. It follows that for $\lambda > 0$, we have $\left\Vert f_{\xi}\left(x, y, t\right)\right\Vert\leq\lambda$. After simple manipulation like we did in Theorems 4.1 and 4.2, we obtain

Using Eq (4.9), one can get

it follows that $\lim_{\xi\rightarrow \infty}f_{\xi}\left(x, y, t\right) = f(x, y, t).$ This finish the proof.

Theorem 4.3. If the inequality (4.8) holds at $t = t_{0}\geq0.$Then the unique solution of proposed equation exists.

Proof. On contrary, suppose that there are two solutions $f$ and $g$ of the proposed equation such that $f\neq g.$ Now

Taking norm both side

but

Using the above inequality, we achieve

thus, our supposition is wrong. Hence, the solution is unique.

5.   Solution of Eq (1.2)

In this section, we briefly discuss the solution of the model by applying Ansatz method. For this purpose, we will consider a test function as

By putting above equation into classical form of the model, we obtain

solution becomes as

solution of classical model becomes as

For $b_{1} = 1$ and $b_{2} = -1,$ the above solution becomes

6.   Solution of Eq (1.3)

In this section, to obtain analytic solution we applying Laplace transform (LT) on both sides of equations $f(x, y, t)$ is the source term. Subject to the initial condition $f(x, y, 0) = f_{_{0}}(x, y, 0).$ On utilizing LT, one can get

The approximate solution is represented by

and the nonlinear term is represented by Adomain polynomials, i.e., $G(f) = f^{2} = \sum_{\xi = 0}^{\infty}\mathscr{A}_{\xi}$, where $\mathscr{A}_{\xi}$ is defined as follows for any $\xi = 0, 1, 2, \cdots$

Using Eq (6.2), we obtain

The following can be obtain by comparing terms

Applying $\mathcal{L}^{-1},$ we get

The required series solution is given as

Here, we present two special cases of the proposed equation.

6.1.   Case A

For the first case, we take the initial condition as

Using the detail procedure as discussed above, we achieve

using Mathematica, we obtain

similarly, other terms can be calculated with the help of Mathematica. The required series solution is given as

Remark 6.1. When we put $\varPsi = 1$ in Eq (6.4), the solution rapidly converges to the exact classical solution, i.e.,

6.2.   Case B

For the first case, we take the initial condition as

Using the detail procedure as discussed above, we get

using Mathematica, we obtain

similarly, other terms can be calculated with the help of Mathematica. The required series solution is given as

Remark 6.2. When we put $\varPsi = 1$ in Eq (6.6), the solution rapidly converges to the exact classical solution, i.e.,

7.   Convergence and stability analysis

Here, we derive some results regarding to the convergence and stability of the proposed scheme with the help of functional analysis. The convergence of the proposed scheme of the is presented in the following theorem.

Theorem 7.1. Let $\mathbb{H}$ be a Banach space and $\mathrm{T}:\mathbb{H}\rightarrow \mathbb{H}$be an operator. Suppose that $f$ be the exact solution of the proposedequation. If $\exists\; \; \varpi$ such that $0\leq\varpi < 1$ and $\left\Vert f_{\xi+1}\right\Vert \leq\varpi\left\Vert f_{\xi}\right\Vert, \, \forall\; \; \xi\in\mathbb{N}\cup\{0\},$ thenthe approximate solution $\sum_{\xi = 0}^{\infty}f_{\xi}$ convergesto the exact solution $f.$

Proof. We construct a series as

We want to prove that the sequence $\left\{ \boldsymbol{S}_{\xi}\right\} _{\xi = 0}^{\infty}$ is a Cauchy sequence in $\mathbb{H}.$ Let us consider

Now for every $\xi, m\in\mathbb{N},$ we have

Now, $\lim_{\xi, m\rightarrow \infty}\left\Vert \boldsymbol{S}_{\xi}-\boldsymbol{S}_{m}\right\Vert = 0.$ This implies that $\left\{ \boldsymbol{S}_{\xi}\right\} _{\xi = 0}^{\infty}$ is Cauchy sequence in $\mathbb{H}.$ So $\exists \; \; f\in\mathbb{H}$ such that $\lim_{\xi\rightarrow \infty}\boldsymbol{S}_{\xi} = f.$ This ends the proof.

Next, we present the Picard's $\bf{X}$-stability of the proposed scheme in the following theorem.

Theorem 7.2. Let $\bf{X}$ be a self-mapping which is definedas

The iteration is $\bf{X}$-stable in $\mathbb{L}^{1}(a, b),$ ifthe condition

is satisfied.

Proof. With the help of Banach contraction theorem, first we show that the mapping $\bf{X}$ possesses a unique fixed point. For this, assume that the bounded iterations for $(\xi, m)\in\mathbb{N}\times\mathbb{N}.$ Let $\Phi_{1}, \Phi_{2} > 0$ such that $\left\Vert f_{\xi}\right\Vert \leq\Phi_{1},$ and $\left\Vert f_{m}\right\Vert \leq\Phi_{2}.$ Consider

Now, using triangle inequality, we have

Using boundedness of $f_{\xi}$ and $f_{m},$ we have

where $\aleph_{i}, \:i = 1, 2, 3, 4, 5$, are functions obtained from $\mathcal{L}^{-1}\left[\frac{s^{\varPsi}(1-\varPsi)+\varPsi)}{s^{\varPsi}c(\varPsi)}\mathcal{L}\left[*\right]\right].$ Using assumption (7.2), the mapping $\bf{X}$ fulfills the contraction condition. Hence by Banach fixed point result, $\bf{X}$ has a unique fixed point. Also, the mapping $\bf{X}$ satisfies the condition of Theorem 2.4 with

Thus, the mapping $\bf{X}$ fulfills all conditions of Picard's $\bf{X}$-stable. Hence our proposed scheme is Picard's $\bf{X}$-stable.

8.   Discussion and conclusions

Thanks to the Mittag-Leffler kernel, which solves the singularity and locality problems with the Caputo and Caputo-Fabrizio FOD kernels. Since $\mathcal{ABC}$-derivative is based on the Mittag-Leffler kernel, it has recently become popular for investigating the dynamics of a mathematical model that governs a physical process. We use $\mathcal{ABC}$-derivative to investigate the soliton solution of the new modified KdV equation in (2+1) dimension in the current paper. Since the presence of a solution is essential for the study of a model, we have deduced some results using fixed point theory that guarantees at least one solution and the unique solution of the proposed equation. There are several techniques for solving FDEs, but among the analytical methods, LADM is the most effective and accurate. Under $\mathcal{ABC}$-derivative, we used the LADM to obtain the solution of the proposed equation. Graphs of the solution are used to observe the method's convergence. The exact solution and the approximate solution obtained with the aid of LADM are in good agreement (See Figures 1 and 2). The dynamics of the solution under the $\mathcal{ABC}$-derivative have been investigated. We can see from Figures 3 and 4 that the fractional-order solution curves are approaching the integer-order curve when fractional-order equals 1. Figure 5 shows that the $\mathcal{ABC}$-derivative solution curve is much closer to the exact solution than the Caputo-Fabrizio derivative curve. As a result, the proposed model is better than Caputo-Fabrizio's. Figures 6–9 denote the graphical representation of solution obtained in the Case B. The considered equation will be studied under more generalized fractional operators in the next paper.

Acknowledgments

The authors wish to express their sincere thanks to the honorable referees for their valuable comments and suggestions to improve the quality of the paper. In addition, authors would like to express their gratitude to the United Arab Emirates University, Al Ain, UAE for providing the financial support with Grant No. 12S005-UPAR 2020.

Conflict of interest

In this research, there are no conflicts of interest.