Modeling Wave Propagation with Gravity and Surface Tension: Soliton Solutions for the Generalized Hietarinta-Type Equation

Abstract

This study explores analytical and approximate explicit solutions for the generalized Hietarinta-type equation, denoted as (${{\mathcal {G}}}{{\mathcal {H}}}$), in the context of (2+1) dimensions. This investigation utilizes contemporary computational and numerical techniques, including the Khater II technique, He’s variational iteration method, and the septic–B–spline scheme. The ${{\mathcal {G}}}{{\mathcal {H}}}$ equation serves as a mathematical representation of wave propagation on water surfaces, accounting for both gravity and surface tension effects. In fluid dynamics, different wavelengths of waves are associated with distinct phase velocities and frequencies. This research introduces innovative solitary wave solutions, visually depicted through figures, and rigorously assesses their computational accuracy using state-of-the-art numerical methods. The stability of these solutions is scrutinized through the characteristics of the Hamiltonian system. The solutions are reintroduced into the original ${{\mathcal {G}}}{{\mathcal {H}}}$ model using Mathematica 13.1 software to validate the obtained results. This study holds significant importance in mathematical modeling and fluid dynamics because it explores the ${{\mathcal {G}}}{{\mathcal {H}}}$ equation in (2+1) dimensions. The ${{\mathcal {G}}}{{\mathcal {H}}}$ equation is a vital tool for modeling wave propagation on water surfaces, encompassing the interplay of gravity and surface tension, a phenomenon with broad practical applications. Additionally, the study contributes to the field by conducting a stability analysis of these solutions, providing insights into their dynamic behavior and long-term evolution.

Appendix

Here, we explain the highlights of the above–employed analytical (Khater II) method and approximate (variational iteration) method.

1.1 Khater II Method

This section gives the Khater II method’s headlines where it is first time to be used as follows:

Assume the following form for the equation of nonlinear evolution:

$$\begin{aligned} {\mathcal {G}}({\mathcal {U}},\,{\mathcal {U}}_{x},\,{\mathcal {U}}_{t},\,{\mathcal {U}}_{x,\,t},\ldots )=0, \end{aligned}$$

(A1)

where ${\mathcal {G}}={\mathcal {G}}(x,t)$ is a polynomial of ${\mathcal {U}}(x,t)$ and its partial derivatives wherein the highest order derivatives and nonlinear terms are concerned. The main steps of the employed method are as follows

Step 1 The traveling wave transformation

$$\begin{aligned} {\mathcal {U}}(x,\, t)={\mathcal {V}}(\xi ), \quad \xi =x+c\,t, \end{aligned}$$

(A2)

converting Equation (A1) into the following ODE

$$\begin{aligned} {\mathcal {C}}\left( {\mathcal {U}},\, {\mathcal {U}}^{\prime }, {\mathcal {U}}^{\prime \prime },\, \ldots \right) =0, \end{aligned}$$

(A3)

where ${\mathcal {C}}$ is a polynomial in ${\mathcal {U}}(\xi )$ and its total derivatives, wherein $\varphi ^{\prime }(\xi )=\frac{d \varphi }{d \xi }$.

Step 2We suppose the solution of (A3) is of the form

$$\begin{aligned} {\mathcal {V}}(\xi )=\sum \limits _{i=1}^n \left( a_i \, f(\xi )^i+b_i \, \phi (\xi )\, f(\xi )^{i-1}\right) +a_0, \end{aligned}$$

(A4)

where $a_{i},\, b_{i}\,(i=0,1,2,3, \ldots ,n)$ are arbitrary constants to be determined, such that $a_{n} \ne 0$, and $\phi (\xi ), \, f(\xi )$ satisfies the following equation

$$\begin{aligned} {f'(\xi )= -\delta -f(\xi )^2,\, \phi '(\xi )= -f(\xi ) \, \phi (\xi ),} \end{aligned}$$

(A5)

where $\delta$ is arbitrary constant.

Step 3 We determine the positive integer n come out in the suggested general solution by considering the homogeneous balance between the highest order derivatives and the highest order nonlinear terms occurring in (A3) as following

$$\begin{aligned} D\left[ \frac{d^{q} {\mathcal {U}}}{d\xi ^{q}}\right] =n+q,\ \ \ D\left[ {\mathcal {U}}^{p}\left( \frac{d^{q} {\mathcal {U}}}{d\xi ^{q}}\right) ^{s}\right] =np+s(n+q). \end{aligned}$$

(A6)

Step 4 We compute all the required derivatives ${\mathcal {V}}^{\prime },\, {\mathcal {V}}^{\prime \prime },\, \ldots$, and substitute (A4) and the derivatives into (A3) and then we account for the function $\phi (\xi ), \, f(\xi )$. As a result of this substitution, we obtain a polynomial of $\phi (\xi ), \, f(\xi )$ and its derivatives. In this polynomial, we equate all the coefficients to zero. This procedure yields a system of equations whichever can be solved to find $a_k$ and $\phi (\xi ), \, f(\xi )$.

1.2 He’s Variational Iteration Method

Here is a general nonlinear differential equation, which illustrates the basic concept of He’s variational iteration method:

$$\begin{aligned} {\mathcal {L}}\, {\mathcal {U}}(x,t)+{\mathcal {N}}\, {\mathcal {U}}(x,t)=g(x,t), \end{aligned}$$

(A7)

where ${\mathcal {L}},\, {\mathcal {N}},\, g(x,t)$ are respectively describe a linear operator, a nonlinear operator, and a known analytical function. According to the variational method, a correction functional can be constructed as follows:

$$\begin{aligned} {\mathcal {U}}_{\rho +1}(x,t)={\mathcal {U}}_{\rho }(x,t)+\int _{0}^{t} \varsigma \, \left( L {\mathcal {U}}_{\rho }(x,\, \xi )+N \tilde{{\mathcal {U}}}_{\rho }(x,\, \xi )-g(x,\, \xi )\right) d \xi . \end{aligned}$$

(A8)

In this case, $\varsigma$ denotes a general Lagrange multiplier that can be identified optimally using variational theory, and $\tilde{{\mathcal {U}}}_{\rho }$ represents a restricted variation, i.e. $\delta \, \tilde{{\mathcal {U}}}_{\rho }=0$. The stationary conditions

$$\begin{aligned} \left\{ \begin{array}{c} 1+\varsigma =0, \\ \\ \varsigma ^{\prime }=0. \end{array} \right. \end{aligned}$$

(A9)

This in turn gives

$$\begin{aligned} \varsigma =-1. \end{aligned}$$

(A10)

Using He’s variational iteration technique to the investigated model for investigating the model’s approximate solution, gets the following values of the analytical, numerical, and absolute difference between these two values.

1.3 Septic–B–Spline Scheme

Based on the septic B–spline, the suggested solution of the obtained ordinary differential equation (3) is given as follow

$$\begin{aligned} {\mathcal {H}}(\eth )=\sum _{i=-1} ^{n+1} \mho _{i}\,\aleph _{i}, \end{aligned}$$

(A11)

where $\mho _{i},\, \aleph _{i}$ satisfies the next conditions

$$\begin{aligned} L\,{\mathcal {H}}(\eth )=\emptyset (\eth _{i},{\mathcal {H}}(x_{i}))\,\,\text {where}\,\, (i=0,1,\ldots ,n), \end{aligned}$$

(A12)

and

$$\begin{aligned} \aleph _{i}(\eth )=\frac{1}{h^{5}}\left\{ \begin{array}{cc} (\eth -\eth _{i-4})^{7},&{}\eth \in [\eth _{i-4},\eth _{i-3}],\\ (\eth -\eth _{i-4})^{7}-8(\eth -\eth _{i-3})^{7},&{}\eth \in [\eth _{i-3},\eth _{i-2}],\\ (\eth -\eth _{i-4})^{7}-8(\eth -\eth _{i-3})^{7}+28\varsigma (\eth -\eth _{i-2})^{7},&{}\eth \in [\eth _{i-2},\eth _{i-1}], \\ (\eth -\eth _{i-4})^{7}-8(\eth -\eth _{i-3})^{7}+28 (\eth -\eth _{i-2})^{7}+56 (\eth -\eth _{i-1})^{7},&{}\eth \in [\eth _{i-1},\eth _{i}], \\ (\eth _{i+4}-\eth )^{7}-8(\eth _{i+3}-\eth )^{7}+28 (\eth _{i+2}-\eth )^{7}+56 (\eth _{i+1}-\eth )^{7},&{}\eth \in [\eth _{i},\eth _{i+1}], \\ (\eth _{i+4}-\eth )^{7}-8(\eth _{i+3}-\eth )^{7}+28 (\eth _{i+2}-\eth )^{7},&{}\eth \in [\eth _{i+1},\eth _{i+2}], \\ (\eth _{i+4}-\eth )^{7}-8(\eth _{i+3}-\eth )^{7},&{}\eth \in [\eth _{i+2},\eth _{i+3}],\\ (\eth _{i+4}-\eth )^{7},&{}\eth \in [\eth _{i+3},\eth _{i+4}], \\ 0,&{}\text {otherwise,} \end{array} \right. \end{aligned}$$

(A13)

where L is a linear operator, $i \in [-3,n+3]$. Thus, the approximate solution is given by

$$\begin{aligned} v_{i}(\eth )=\mho _{i-3}+120\,\mho _{i-2}+1191\,\mho _{i-1}+2416\,\mho _{i}+1191\,\mho _{i+1}+120\,\mho _{i+2}+\mho _{i+3}. \end{aligned}$$

(A14)

Substituting Eq. (A14) into Eq. (3), obtains a system of equations.