Investigating Korteweg-de Vries Dynamics via Laplace Transform Methodology

In this paper, through $\mathbb{R}$ we denote the real field. A function $f:(0, \infty) \rightarrow \mathbb{R}$ is said to be of exponential order if there exist constantats $A, B \in \mathbb{R}$ such that $|f(t)| \leq A e^{B t}$, for all $t>0$. $f(t)$ has its Laplace transform $\mathcal{L}[f(t)]=F(s)$, where

$F(s)=\int_0^{\infty} f(t) e^{-s t} d t$.

In which there exists $\delta \in \mathbb{R}$ such that this integral converges if $\mathfrak{R}(s)>\delta$ and diverges if $\mathfrak{R}(s)<\delta$, where $\mathfrak{R}(s)$ is the real part of $s$. Moreover, $|F(s)| \rightarrow \infty$.

Some properties of Laplace transform can be illustrated as follows

$\begin{gathered}\mathcal{L}\left(\frac{d f}{d t}\right)=\int_0^{\infty} e^{-s t} \frac{d f}{d t} d t =\left.\left(e^{-s t} f(t)\right)\right|_0 ^{\infty}+\int_0^{\infty} e^{-s t} f(t) d t .\end{gathered}$

(By integrating by parts)

From convergence conditions of Laplace transform, we assume that

$\lim _{t \rightarrow \infty} e^{-s t} f(t)=0$.

Applying this result, we obtain the following property

$\mathcal{L}\left(\frac{d f}{d t}\right)=s \mathcal{L}(f)-f(0)$.

The derivative property of a time function with respect to a second variable $x$ has the laplace transform:

$\mathcal{L}\left(\frac{\partial f(t, x)}{\partial x}\right)=\left(\frac{\partial}{\partial x}\right) F(s, x)$.

The integral property of a time function with respect to a second variable $x$ has the laplace transform:

$\mathcal{L}\left(\int_{x_0}^{x_1} f(t, x) d x\right)=\int_{x_0}^{x_1} F(s, x) d x$.

Another property of Laplace transform for a function $f(x, t)$ and $f_x$ is a function of $x$ only, then

$\begin{aligned} & \mathcal{L}\left(f, f_x\right) =\int_0^{\infty} f e^{-s t} \cdot f_x d t \\ & =\left[f e^{-s t} \int f_x d t\right]_0^{\infty}-\int_0^{\infty}\left(f_t-f_x\right) e^{-s t}\left(\int f_x d t\right) d t \\ & =-f(x, 0)\left[\int f_x d t\right]_{t=0}-\mathcal{L}\left(f_t \cdot \int f_x d t\right)+\mathcal{L}\left(f_x \cdot \int f_x d t\right)\end{aligned}$

Let’s examine how the Laplace transform represents solutions to the nonlinear third-order KdV equation with non-zero initial condition

$\frac{\partial u}{\partial t}+a u \frac{\partial u}{\partial x}+b \frac{\partial^3 u}{\partial^3 x}=f(x, t)$,         (4)

$u(x, 0)=\emptyset(x), t \in[0, \infty), x \in R$.          (5)

where, the real-valued function $u(x, t)$ is the average velocity. Waves decay because of the third order term (depressive term), whereas waves steepen because of the nonlinear term.

In the following theorem, we will find the exact solution for the initial-value problem of the nonlinear third-order KdV equation.

Theorem 2.1 The solution $u(x, t)$ of the initial value problem Eqs. (4)-(5) is

$U(x, s)=A(s) e^{\alpha x}+B(s) e^{\beta x}+C(s) e^{\gamma x}+U_p(x)$,           (6)

where, $U_p$ is the particular solution of problem Eqs. (4)-(5) and

$U_p(x)= \begin{cases}\int_x^{\infty}(G(x, \xi)(f(\xi) d \xi, & \xi<x \\ \int_{-\infty}^x G(x, \xi)(f(\xi) d \xi, & \xi>x,\end{cases}$              (7)

$\begin{aligned} & G(x, \xi) =\left(\begin{array}{c}(\alpha-\gamma) e^{(\alpha+\gamma) \xi+\beta x} \\ +(\beta-\alpha) e^{(\alpha+\beta) \xi+\gamma x}+(\gamma-\beta) e^{(\gamma+\beta) \xi+\alpha x}\end{array}\right) / \left(\begin{array}{c}\alpha(\gamma-\beta) e^{(\beta+\gamma) \zeta+\alpha x} \\ +\beta(\alpha-\gamma) e^{(\alpha+\gamma) \zeta+\beta x}+\gamma(\beta-\alpha) e^{(\alpha+\beta) \zeta+\gamma x}\end{array}\right)\end{aligned}$,             (8)

α, β and $\gamma$ are roots of the auxiliary equation

$b \lambda^3+\mathrm{a} u \lambda+\mathrm{s}=0$.

Proof. Taking Laplace transform with respect to t on both sides of Eq. (4), we have

$\begin{gathered}s U(x, s)-u(x, 0)+a u(x, t) \frac{\partial U(x, s)}{\partial x}+b \frac{\partial^3 U(x, s)}{\partial x^3} =F(x, s) .\end{gathered}$

Rearranging the equation, we get

$\begin{gathered}b \frac{\partial^3 U(x, s)}{\partial x^3}+a u(x, t) \frac{\partial U(x, s)}{\partial x}+s U(x, s) =\emptyset(x)+F(x, s)\end{gathered}$,            (9)

where, $U(x, s)=\mathcal{L}[u(x, t)]$.

From Eq. (9), yields the characteristic equation of the form

$b \lambda^3+\mathrm{a} u \lambda+\mathrm{s}=0$,           (10)

which has the roots α, β and $\gamma$.

Since the Eq. (9) has a first and third derivatives only with respect to $x$, then its general solution takes the form

$U(x, s)=A(s) e^{\alpha x}+B(s) e^{\beta x}+C(s) e^{\gamma x}+U_p(x)$,                 (11)

where,

$U_p(x)=\int_x^{\infty} G(x, \xi) f(\xi) d \xi+\int_{-\infty}^x G(x, \xi) f(\xi) d \xi$

is a particular solution of the problem Eqs. (4)-(5) with the Green’s function

$G(x, \xi)=\frac{\Delta}{\Delta^{\prime}}$.

where,

$\begin{aligned} & \Delta=\left|\begin{array}{ccc}e^{\alpha \xi} & e^{\beta \xi} & e^{\gamma \xi} \\ \alpha e^{\alpha \xi} & \beta e^{\beta \xi} & \gamma e^{\gamma \xi} \\ e^{\alpha x} & e^{\beta x} & e^{\gamma x}\end{array}\right| \\ & =e^{\alpha \xi}\left(\beta e^{\beta \xi+\gamma x}-\gamma e^{\gamma \xi+\beta x}\right)-e^{\beta \xi}\left(\alpha e^{\alpha \xi+\gamma x}-\gamma e^{\gamma \xi+\alpha x}\right) +e^{\gamma \xi}\left(\alpha e^{\alpha \xi+\beta x}-\beta e^{\beta \xi+\alpha x}\right) \\ & =\beta e^{(\alpha+\beta) \xi+\gamma x}-\gamma e^{(\alpha+\gamma) \xi+\beta x}-\alpha e^{(\alpha+\beta) \xi+\gamma x}+\gamma e^{(\beta+\gamma) \xi+\alpha x} +\alpha e^{(\alpha+\gamma) \xi+\beta x}-\beta e^{(\beta+\gamma) \xi+\alpha x} \\ & =(\alpha-\gamma) e^{(\alpha+\gamma) \xi+\beta x}+(\beta-\alpha) e^{(\alpha+\beta) \xi+\gamma x} +(\gamma-\beta) e^{(\gamma+\beta) \xi+\alpha x}\end{aligned}$

and

$\begin{aligned} & \Delta^{\prime}=\left|\begin{array}{ccc}e^{\alpha \xi} & e^{\beta \xi} & e^{\gamma \xi} \\ \alpha e^{\alpha \xi} & \beta e^{\beta \xi} & \gamma e^{\gamma \xi} \\ \alpha e^{\alpha x} & \beta e^{\beta x} & \gamma e^{\gamma x}\end{array}\right| \\ & =e^{\alpha \xi}\left(\beta \gamma e^{\beta \xi+\gamma x}-\gamma e^{\gamma \xi+\beta x}\right)-e^{\beta \xi}\left(\alpha \gamma e^{\alpha \xi+\gamma x}-\alpha \gamma e^{\gamma \xi+\alpha x}\right) +e^{\gamma \xi}\left(\alpha \beta e^{\alpha \xi+\beta x}-\alpha \beta e^{\beta \xi+\alpha x}\right) \\ & =\beta \gamma e^{(\alpha+\beta) \xi+\gamma x}-\beta \gamma e^{(\alpha+\gamma) \xi+\beta x}-\alpha \gamma e^{(\alpha+\beta) \xi+\gamma x} +\alpha \gamma e^{(\alpha+\beta) \xi+\alpha x}+\alpha \beta e^{(\alpha+\gamma) \xi+\beta x} -\alpha \beta e^{(\beta+\gamma) \xi+\alpha x} \\ & =(\alpha \gamma-\alpha \beta) e^{(\beta+\gamma) \zeta+\alpha x}+(\beta \alpha-\beta \gamma) e^{(\alpha+\gamma) \zeta+\beta x} +(\beta \gamma-\alpha \gamma) e^{(\alpha+\beta) \zeta+\gamma x} \\ & =\alpha(\gamma-\beta) e^{(\beta+\gamma) \zeta+\alpha x}+\beta(\alpha-\gamma) e^{(\alpha+\gamma) \zeta+\beta x} +\gamma(\beta-\alpha) e^{(\alpha+\beta) \zeta+\gamma x}\end{aligned}$

and α, β and $\gamma$ are roots of the Eq. (10). Theorem proved.

We investigate the Galilean transformed version of the homogeneous nonlinear mKdV equation with initial condition:

$\frac{\partial u}{\partial t}+12 \epsilon u_0 u \frac{\partial u}{\partial x}+6 \epsilon u^2 \frac{\partial u}{\partial x}+\frac{\partial^3 u}{\partial^3 x}=0, \epsilon= \pm 1$,            (16)

$u(x, 0)=u_0, \quad t \in[0, \infty), x \in R$            (17)

where, $\epsilon$ indicates whether the equation is focusing or defocusing. This type of equation arises in many physics’ problems, such as propagation of ultrashort few-optical cycle solitons in nonlinear media [22, 23] and Alfven waves equation [24], Schottky barrier transmission lines [25], ion acoustic solitons (by adopting an electron equation of state that corresponds to the observed flat-topped electron distribution functions, it is explored how the asymptotic behavior of small ion-acoustic waves depends on the quantity of resonant electrons [26, 27]), thin ocean jets [28], internal waves [29], traffic jam [30], ion acoustic solitons [26], heat pulses in solids [31]. Additionally, both the positive order and the negative order of the mKdV hierarchy illustrate interesting integrable structures [32].

There are numerous standard methods of solution for the mKdV equation's exact solutions for example: communication method [33], Hirota’s bilinear method [34], and Wronskian method [35].

It is frequently possible to describe the solutions of a soliton equation with bilinear form using a Wronskian by placing certain restrictions on the entry vector of the Wronskian [36]. Following, we may refer to these conditions as the condition equation set (CES). The coefficient matrix typically plays a key part in the CES of a (1+1)-dimensional soliton problem, and this matrix or any similar shape leads to the same solutions for the related soliton equation. In view of the canonical form of the coefficient matrix, it is thus possible to provide a comprehensive categorization (or structure) for the solutions of the soliton equation [36]. It has been established that, by some limiting process, the solutions derived from a coefficient matrix in Jordan form relate to the solutions derived from a diagonal coefficient matrix [33]. As a result, the previous solutions can be considered limit solutions. According to the IST, N solitons are distinguished from one another by N distinct eigenvalues of the relevant spectral problem, or in other words, N different simple poles $\left\{k_j\right\}$ of the transparent coefficient $\frac{1}{a(k)}$. When $k_j$ are multiple-pole solutions, the associated multiple-pole solutions can be derived by restricting the simple-pole solutions using a limiting procedure like $k_2 \rightarrow k_1$. The particular solutions in Wronskian form lend themselves more readily to this limiting procedure [33]. A similar process is useful for comprehending the dynamics of limit solutions.

To find the exact solution of problem Eqs. (16)-(17), we use Laplace transforms on Eq. (16), yields

$\begin{aligned} & s U(x, s)-u_0+12 \epsilon u_0 u \frac{\partial U(x, s)}{\partial x} +6 \epsilon u^2 \frac{\partial U(x, s)}{\partial x}+\frac{\partial^3 U(x, s)}{\partial x^3}=0\end{aligned}$,

Reorganizing the equation, we get

$\begin{aligned} & \frac{\partial^3 U(x, s)}{\partial x^3}+12 \epsilon u_0 u \frac{\partial U(x, s)}{\partial x}+ 6 \epsilon u^2 \frac{\partial U(x, s)}{\partial x}+s U(x, s)=u_0\end{aligned}$         (18)

The auxiliary equation of Eq. (18) takes the following form:

$\lambda^3+12 \epsilon u_0 u \lambda+6 \epsilon u^2 \lambda+s=0$,               (19)

this equation has three roots three roots α, β and $\gamma$.

By virtue of the discriminate

$32 \epsilon^3\left(2 u_0 u+u^2\right)^3+s^2$,

there are three cases for the roots

Case i. If $32 \epsilon^3\left(2 u_0 u+u^2\right)^3+s^2>0$. Then the unique real solution of Eq. (19) has a unique real solution of the form

$\begin{aligned} & \alpha=\sqrt[3]{-\frac{s}{2}+\sqrt{\frac{s^2}{4}+8 \epsilon^3\left(2 u_0 u+u^2\right)^3}} +\sqrt[3]{-\frac{s}{2}-\sqrt{\frac{s^2}{4}+8 \epsilon^3\left(2 u_0 u+u^2\right)^3}} .\end{aligned}$

Case ii. If $32 \epsilon^3\left(2 u_0 u+u^2\right)^3+s^2<0$.  Then Eq. (19) has three real roots as follows:

$\begin{aligned} & \alpha=2 \sqrt{-2 \in\left(2 u_0 u+u^2\right)} \\ & \cos \left(\frac{\operatorname{Arccos}\left(\frac{s}{4 \in\left(2 u_0 u+u^2\right)} \sqrt{-\frac{1}{2 \in\left(2 u_0 u+u^2\right)}}\right)+2 \pi}{3}\right) \\ & \beta=2 \sqrt{-2 \in\left(2 u_0 u+u^2\right)} \\ & \cos \left(\frac{\operatorname{Arccos}\left(\frac{s}{4 \in\left(2 u_0 u+u^2\right)} \sqrt{-\frac{1}{2 \in\left(2 u_0 u+u^2\right)}}\right)+4 \pi}{3}\right) \\ & \gamma=2 \sqrt{-2 \in\left(2 u_0 u+u^2\right)} \\ & \cos \left(\frac{\operatorname{Arccos}\left(\frac{s}{4 \in\left(2 u_0 u+u^2\right)} \sqrt{-\frac{1}{2 \in\left(2 u_0 u+u^2\right)}}\right)}{3}\right) \\ & \end{aligned}$

Case iii. If $32 u^3+s^2=0$, then Eq. (19) has three real roots with two the same.

Hence, the general solution of problem Eqs. (16)-(17) is

$\begin{aligned} U(x, s) & =A(s) e^{\alpha x}+B(s) e^{\beta x}+C(s) e^{\gamma x} +\int_{-\infty}^{+\infty} G(x, \xi) f(\xi) d \xi\end{aligned}$

where,

α, β and $\gamma$ are the roots of the auxillary Eq. (19), $G(x, \xi)$ is also calculated in Theorem 2.1.