Symmetry reductions and conservation laws of a modified-mixed KdV equation: exploring new interaction solutions

1.   Introduction

Over the previous few decades, researchers have shown considerable interest in the extraction of solutions of nonlinear systems such as traveling wave solutions, including solitary waves, periodic waves, kink and anti-kink solutions, solitons, and some integrability of some intriguing nonlinear partial differential equations (PDEs) ^[1,2,3,4,5], for instance, the nonlinear Schrodinger equation ^[6], Burgers equation ^[7], modified equal width ^[8], modified equal width-Burgers equation ^[9], the KdV model ^[10], combined KdV-mKdV equation ^[11], etc. Solitons are solitary waves that possess elastic scattering properties. These waves emerge as a result of the intricate equilibrium between dispersion, and nonlinearity. The origination of solitons can be attributed to the pioneering formulation of the KdV equation ^[12,13]. The KdV equation is extensively employed for the simulation of the propagation of shallow, thin, and elongated water waves. It provides an explanation for a multitude of physical processes, encompassing acoustic, hydromagnetic, and ion-acoustic waves. The standardized form of the KdV model is expressed as follows:

This equation has a number of essential applications in a variety of scientific domains. Various extensions of KdV equations have been extracted in recent decades. In ^[14], Wang and Kara have introduced KdV and modified KdV equations. They have used the (2+1)-dimensional mKdV equations to derive conservation laws by performing Lie symmetry analysis. In ^[15], the authors studied the bifurcation and established some exact solutions for (2+1)-dimensional KdV equation. Later, Wazwaz ^[16] developed new (3+1)-dimensional KdV and mKdV equations. The author demonstrated their integrability using Painlevé analysis in ^[16] and established several multiple-solution solutions for these equations. Moreover, the combined KdV-mKdV model has been investigated in ^[17] and derive the integrability, stability analysis and soliton structures. Now, we have considered another extension of the KdV equations called the the modified-mixed KdV (mmKdV) model, read as:

where $\Omega = \Omega (x, t)$. The mmKdV equation characterizes an electron distribution with a flat top, wherein the nonlinearity is more pronounced when the width is smaller and the velocity is higher ^[18,19,20].

In ^[20], the following assumption has been proposed

By substituting (1.3) into (1.2), the following equation was obtained

The polynomial function, and rational sinh-cosh methods were used to derive some results.

The Lie symmetry method applied to PDEs, is a powerful approach to obtain reductions and invariant solutions ^[21].

Recently, Tian et al. ^[22] successfully proposed an effective and direct approach to study the symmetry-preserving discretization for a class of generalized higher order equations, and proposed an open problem about symmetries and the multipliers of conservation law.

In this paper we consider Eq (1.4) from the viewpoint of symmetry reductions in PDEs. We obtain the Lie point symmetries admitted by (1.4) for arbitrary constants. We derived conservation laws for Eq (1.4).

Taking into account the relationship between symmetries and conservation laws by using the invariance of two conservation laws under translations, we derive two first integrals. By combining these first integrals, we obtain a triple reduction to a first-order autonomous equation.

2.   Lie point symmetries

A Lie point symmetry ^[23,24,25] for the Eq (1.4) pertains to a set of point transformations that maintain the equations invariance. These transformations can be expressed in infinitesimal form:

where $\varepsilon$ symbolizes the group parameter, and the related vector field is described as follows:

The transformation group will exhibit point symmetry if

The expression "$\Pr^{(3)}{{{\rm{X}}}}$" is the third prolongation of the vector field given by (2.1), while ${\mathcal{E}}$ refers to the solution space of Eq (1.4). The Eq (2.2), which determines the outcome, separates based on the derivatives of $w$. This division results in a linear system that is over-determined, involving the infinitesimals $\tau(t, x, w)$, $\xi(t, x, w)$, and $\eta(t, x, w)$. This system is known as the determining system. By solving the determining system, we are able to derive the subsequent outcomes:

A single-parameter Lie group on the variables $(t, x, w)$ associated with Eq (1.4) is considered as a point symmetry. This Lie group is characterized by a vector field that preserves the solution space of the equation.

The incorporation of each identified symmetry enables a reduction in the number of independent variables within Eq (1.4). Specifically, this reduction leads to the transformation of PDEs into ordinary differential equations (ODEs). Moreover, these ODEs may possess additional symmetries that facilitate a further reduction in the equation's order. The solutions of these reduced ODEs correspond to invariant solutions denoted as $u(t, x, w)$ in relation to Eq (1.4).

Theorem 2.1. (i) The point symmetries for Eq (1.4), with $\alpha \neq 0$, $\beta \neq 0$, $\delta \neq 0$, are generated by:

(ii) For some particular parameters $\alpha$ and $\beta$, there are additional generators given below. For $\alpha = 0$,

(iii) For $\beta = 0$,

(i) Their commutator is given for $\alpha = 0$ by

(ii) Their commutator is given for $\beta = 0$ by

The corresponding optimal systems are:

(i) For $\alpha = 0,$

(ii) For $\beta = 0,$

2.1.   Conservation laws

The conservation law for the PDE denoted by ^{[26,27,28,29,30]}: $G(t, x, w, w_t, w_x, w_{xx}, \ldots) = 0,$ can be expressed as $D_t T + D_x \Phi = 0$. Here, $T$ represents the conserved density and $\Phi$ is the flux vector, both functions of $t, x, w$. The conserved current is denoted as $(T, \Phi)$.

Each conservation law for the PDE $G = 0$ is linked to a corresponding multiplier. An injective relationship exists between nontrivial conserved currents $(T, \Phi)|{\mathcal{E}}$ modulo trivial ones and nonzero multipliers $Q|{\mathcal{E}}$. This relationship is characterized by $QG = D_t T + D_x\Phi$ holding as an identity, where $Q$ is a function of $t, x, w$, and derivatives of $w$, ensuring that $Q|{\mathcal{E}}$ is non-singular. Various explicit methods can be employed to obtain a conserved current for each solution $Q$.

We will focus on examining low-order multipliers of the form $Q(t, x, w, w_x, w_{xx}).$ The determining equations can be decomposed into an over-determined linear system of equations. Solving this system for $Q$ is straightforward, subject to the conditions $\alpha \neq 0$, $\beta \neq 0$, $\delta \neq 0$.

Proposition 2.1. All multipliers ^[31] admitted by the mmKdV equation (1.4), with $\alpha \neq 0$, $\beta \neq 0$, $\delta \neq 0$, are given by:

These multipliers yield all the nontrivial conservation laws, summarized as below.

Theorem 2.2. The conservation laws for the mmKdV equation (1.4), with $\alpha \neq 0$, $\beta \neq 0$, $\delta \neq 0$ are given by:

2.2.   Traveling wave reduction, scaling reduction, and first integrals from symmetry reduction

Symmetry reduction ^[32] is frequently employed, with one prevalent application being the simplification to ODEs.

A traveling wave has the form

where $c$ is the velocity of the traveling wave.

Inserting Eq (2.20) into the Eq (1.2) results in a following ODE:

In ^[33], a comprehensive multi-reduction approach was presented, demonstrating that symmetry-based conservation laws give rise to first integrals. This method directly utilizes the inherent symmetry to identify all first integrals. As a result of this reduction, two distinct first integrals are obtained. The functionally independent first integrals of the ODE given by (2.21) are subsequently derived from the corresponding symmetry-invariant multipliers. $Q_1 = 1, \; Q_2 = \frac{w^2}{2}$ are given by

and eliminating $U''$ yields an autonomous nonlinear first-order ODE

with $C_1$, $C_2$ arbitrary constants.

We point out that, for special values of the constants, the general solution of Eq (2.23) can be written in terms of the Jacobi elliptic functions ^[34].

Now, we consider the scaling symmetry

which gives similarity solutions

where $z$ and $U$ are the scaling invariants.

The similarity reduced ODE for this solution is a third-order nonlinear ODE

In ^[35,36,37], it is observed that when considering ODEs resulting from a symmetry reduction under the scaling of the Eq (1.2), the conservation laws of the equation that are invariant under scalings reduce to a first integral of the ODE. Additionally, the work in ^[33] introduced a general multi-reduction method, demonstrating that all first integrals arising from conservation laws can be directly obtained using the symmetry. This reduction process results in two first integrals. The derived first integral of the ODE (2.25) is obtained from the corresponding symmetry-invariant multiplier. $Q_3 = \beta t u^2-x$ is given by

which is a nonlinear second-order ODE.

3.   Exact solution through unified Riccati equation expansion method

Here, we will gain the exact solution of Eq (2.21) by using the unified Riccati equation approach ^[38]. According to this, the proposed method has the following finite form of the analytic solution:

where $B_k, (k = 1, 2, 3, ..., n)$ are arbitrary parameters such that $B_n\not = 0$ and $n$ is a positive integer. Further, the function $G(\tau)$ must satisfy the following equation:

On solving Eq (3.2) and taking constant of integration as zero, yields the following solution:

(1) If $\Delta > 0,$

(2) If $\Delta < 0,$

(3) If $\Delta = 0,$

where $\Delta = p_{1}^{2}-4 p_{0} p_{2}$.

3.1.   Solution for Eq (2.21)

Apply the homogeneous balance principle to calculate the balancing number $n$, which will be used to define the degree of the analytic solution. Balancing the dispersive and highest nonlinear terms, we get $n = 1$. Therefore, the method of solution takes the linear form as

By inserting Eqs (3.8) and (3.2) into Eq (2.21), collecting terms having identical powers of $G^{k}(\tau)$, and equating the coefficients to zero leads to the derivation of the subsequent outcomes.

Family 1.

Family 2.

Use Family 1:

Case 1: If $\Delta > 0$, then

Case 2: If $\Delta < 0$, then

Case 3: If $\Delta = 0$, then

Use Family 2:

Case 1: If $\Delta > 0$, then

Case 2: If $\Delta < 0$, then

Case 3: If $\Delta = 0$, then

3.2.   Solution for Eq (2.26)

By applying the homogeneous balance principle to Eq (2.26), we get $n = 1$ and Eq (3.1) yields to the following:

On substituting Eqs (3.2) and (3.21) into Eq (2.26) with $C_{1} = 0$, collecting all terms having identical powers of $G^{k}(\tau)$ and setting each coefficient to zero, and solving the algebraic equations by Maple, we obtain the following result:

As $\Delta = p_{1}^{2}-4 p_{0} p_{1} = 0$ for arbitrary values of $p_{0}, p_{1}$, and $p_{2}$, we have

where $z = \frac{x^{3}}{t}$ and $c_{1}$ is constant of integration.

4.   Conclusions

In this study, we have successfully obtained the ODE by utilizing all available Lie symmetry generators. Through the utilization of these symmetry generators, the proposed equation has been reduced to an ODE via symmetry reductions. The unified Riccati equation expansion method has been effectively employed to obtain the ODE, as well as new interaction solutions such as kink solitons and other solitons associated with hyperbolic functions. These solutions have been retrieved with great success. Additionally, in order to demonstrate the visual representation of several wave patterns with different system features and to validate the accuracy of our findings as shown in Figures 1–7. The outcomes of this endeavor will serve as a source of inspiration and motivation for future discussions in the realm of nonlinear physical sciences. Through the analysis of the computations, we are able to ascertain the significant value of this method in terms of its ability to locate precise wave solutions in a more comprehensive manner. In subsequent endeavors, we can expand upon the methods provided to incorporate various other nonlinear models. The resulting solutions present themselves as innovative, captivating, and potentially instrumental in enhancing our comprehension of energy transfer and diffusion processes within mathematical models of diverse fields that are pertinent, wherein nonlinear challenging matters are encountered.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

Prof. Maria Luz Gandarias is the Guest Editor of special issue "Lie Symmetry Analysis and Conservation Laws for Nonlinear Differential Equations and Applications" for AIMS Mathematics. Prof. Maria Luz Gandarias was not involved in the editorial review and the decision to publish this article.

All authors declare no conflicts of interest in this paper.