Bifurcation, chaotic behavior and soliton solutions to the KP-BBM equation through new Kudryashov and generalized Arnous methods

Chander Bhan; Ravi Karwasra; Sandeep Malik; Sachin Kumar; Ahmed H. Arnous; Nehad Ali Shah; Jae Dong Chung; Chander Bhan; Ravi Karwasra; Sandeep Malik; Sachin Kumar; Ahmed H. Arnous; Nehad Ali Shah; Jae Dong Chung

doi:10.3934/math.2024424

AIMS Mathematics

2024, Volume 9, Issue 4: 8749-8767. doi: 10.3934/math.2024424

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Research article Special Issues

Bifurcation, chaotic behavior and soliton solutions to the KP-BBM equation through new Kudryashov and generalized Arnous methods

1.
Department of Mathematics and Statistics, Central University of Punjab, Bathinda, Punjab 151401, India
2.
Department of Physics and Engineering Mathematics, Higher Institute of Engineering, El Shorouk Academy 11837, Cairo, Egypt
3.
Department of Mechanical Engineering, Sejong University, Seoul 05006, South Korea

Received: 26 November 2023 Revised: 08 January 2024 Accepted: 24 January 2024 Published: 29 February 2024
MSC : 35A09, 35A24, 35C08

This research paper investigates the Kadomtsev-Petviashvii-Benjamin-Bona-Mahony equation. The new Kudryashov and generalized Arnous methods are employed to obtain the generalized solitary wave solution. The phase plane theory examines the bifurcation analysis and illustrates phase portraits. Finally, the external perturbation terms are considered to reveal its chaotic behavior. These findings contribute to a deeper understanding of the dynamics of the Kadomtsev-Petviashvii-Benjamin-Bona-Mahony wave equation and its applications in real-world phenomena.

Keywords:

bifurcation analysis,
new Kudryashov method,
exact solutions,
generalized Arnous method,
Kadomtsev-Petviashvii-Benjamin-Bona-Mahony (KP-BBM) equation,
chaotic behavior

Citation: Chander Bhan, Ravi Karwasra, Sandeep Malik, Sachin Kumar, Ahmed H. Arnous, Nehad Ali Shah, Jae Dong Chung. Bifurcation, chaotic behavior and soliton solutions to the KP-BBM equation through new Kudryashov and generalized Arnous methods[J]. AIMS Mathematics, 2024, 9(4): 8749-8767. doi: 10.3934/math.2024424

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Abstract

1. Introduction

Let $\mathbb{M}_n$ be the set of $n\times n$ complex matrices. $\mathbb{M}_n(\mathbb{M}_k)$ is the set of $n\times n$ block matrices with each block in $\mathbb{M}_k$ . For $A \in \mathbb{M}_n$ , the conjugate transpose of $A$ is denoted by $A^*.$ When $A$ is Hermitian, we denote the eigenvalues of $A$ in nonincreasing order $\lambda_1(A) \geq \lambda_2(A) \geq... \geq \lambda_n(A)$ ; see ^[2,7,8,9]. The singular values of $A$ , denoted by $s_1(A), s_2(A), ..., s_n(A),$ are the eigenvalues of the positive semi-definite matrix $|A| = (A^*A)^{1/2}$ , arranged in nonincreasing order and repeated according to multiplicity as $s_1(A) \geq s_2(A) \geq... \geq s_n(A).$ If $A\in \mathbb{M}_n$ is positive semi-definite (definite), then we write $A \geq 0\; (A > 0)$ . Every $A\in \mathbb{M}_n$ admits what is called the cartesian decomposition $A = {{{\rm{Re\, }}}} A+i{{{\rm{Im\, }}}} A,$ where ${{{\rm{Re\, }}}} A = \frac{A+A^*}{2}, \ {{{\rm{Im\, }}}} A = \frac{A-A^*}{2}.$ A matrix $A\in \mathbb{M}_n$ is called accretive if ${{{\rm{Re\, }}}} A$ is positive definite. Recall that a norm $||\cdot||$ on $\mathbb{M}_n$ is unitarily invariant if $||UAV|| = ||A||$ for any $A\in \mathbb{M}_n$ and unitary matrices $U, V\in \mathbb{M}_n$ . The Hilbert-Schmidt norm is defined as $||A||_2^2 = {{{\rm{tr}}}}(A^*A)$ .

For $A, B > 0$ and $t\in [0, 1]$ , the weighted geometric mean of $A$ and $B$ is defined as follows

$\begin{eqnarray*} A\sharp_{t} B\ = A^{1/2}(A^{-1/2}BA^{-1/2})^{t}A^{1/2}. \end{eqnarray*}$

When $t = \frac{1}{2}$ , $A\sharp_{\frac{1}{2}} B$ is called the geometric mean of $A$ and $B$ , which is often denoted by $A\sharp B$ . It is known that the notion of the (weighted) geometric mean could be extended to cover all positive semi-definite matrices; see [3, Chapter 4].

Let $A, B, X\in \mathbb{M}_n$ . For $2\times 2$ block matrix $M$ in the form

$\begin{eqnarray*} M = \left( \begin{array}{cc} A & X \\ X^* & B\\ \end{array} \right)\in \mathbb{M}_{2n} \end{eqnarray*}$

with each block in $\mathbb{M}_{n},$ its partial transpose of $M$ is defined by

$\begin{eqnarray*} M^\tau = \left( \begin{array}{cc} A & X^* \\ X & B\\ \end{array} \right). \end{eqnarray*}$

If $M$ and $M^\tau \geq0$ , then we say it is positive partial transpose (PPT). We extend the notion to accretive matrices. If

$\begin{eqnarray*} M = \left( \begin{array}{cc} A & X \\ Y^* & B\\ \end{array} \right)\in \mathbb{M}_{2n}, \end{eqnarray*}$

and

$\begin{eqnarray*} M^{\tau} = \left( \begin{array}{cc} A & Y^* \\ X & C\\ \end{array} \right)\in \mathbb{M}_{2n} \end{eqnarray*}$

are both accretive, then we say that $M$ is APT (i.e., accretive partial transpose). It is easy to see that the class of APT matrices includes the class of PPT matrices; see ^[6,10,13].

Recently, many results involving the off-diagonal block of a PPT matrix and its diagonal blocks were presented; see ^[5,11,12]. In 2023, Alakhrass ^[1] presented the following two results on $2\times2$ block PPT matrices.

Theorem 1.1 (^[1], Theorem 3.1). Let $\left(\begin{array}{cc} A & X \\ X^{*} & B\\ \end{array} \right)$ be PPT and let $X = U|X|$ be the polar decomposition of $X$ , then

$\begin{eqnarray*} |X| \leq ( A\sharp_t B)\sharp (U^*( A \sharp_{1-t} B)U), \quad t \in [0, 1]. \end{eqnarray*}$

Theorem 1.2 (^[1], Theorem 3.2). Let $\left(\begin{array}{cc} A & X \\ X^{*} & B\\ \end{array} \right)$ be PPT, then for $t\in [0, 1],$

$\begin{eqnarray*} {{{\rm{Re\, }}}} X\leq ( A\sharp_t B)\sharp ( A \sharp_{1-t} B)\leq \frac{(A\sharp_t B)+(A\sharp _{1-t} B)}{2}, \end{eqnarray*}$

and

$\begin{eqnarray*} {{{\rm{Im\, }}}} X \leq ( A\sharp_t B)\sharp ( A \sharp_{1-t} B)\leq \frac{(A\sharp_t B)+(A\sharp _{1-t}B)}{2}. \end{eqnarray*}$

By Theorem 1.1 and the fact $s_{i+j-1}(XY)\leq s_i(X)s_j(Y)\; (i+j\leq n+1)$ , the author obtained the following corollary.

Corollary 1.3 (^[1], Corollary 3.5). Let $\left(\begin{array}{cc} A & X \\ X^{*} & B\\ \end{array} \right)$ be PPT, then for $t\in [0, 1],$

$\begin{eqnarray*} s_{i+j-1}(X)\leq s_i(A\sharp_t B)s_j(A\sharp_{1-t} B). \end{eqnarray*}$

Consequently,

$\begin{eqnarray*} s_{2j-1}(X)\leq s_j( A\sharp_t B)s_j(A\sharp_{1-t} B). \end{eqnarray*}$

A careful examination of Alakhrass' proof in Corollary 1.3 actually revealed an error. The right results are $s_{i+j-1}(X)\leq s_i(A\sharp_t B)^{\frac{1}{2}}s_j((A\sharp_{1-t} B)^{\frac{1}{2}})$ and $s_{2j-1}(X)\leq s_j((A\sharp_t B)^{\frac{1}{2}})$ $s_j((A\sharp_{1-t} B)^{\frac{1}{2}}).$ Thus, in this note, we will give a correct proof of Corollary 1.3 and extend the above inequalities to the class of $2\times2$ block APT matrices. At the same time, some relevant results will be obtained.

2. Main results

Before presenting and proving our results, we need the following several lemmas of the weighted geometric mean of two positive matrices.

Lemma 2.1. [, Chapter 4] Let $X, Y\in \mathbb{M}_{n}$ be positive definite, then

$1)$ $X\sharp Y = \max\left\{Z:Z = Z^*, \left(\begin{array}{cc} X & Z \\ Z & Y\\ \end{array} \right)\geq 0\right\}.$

$2)$ $X\sharp Y = X^{\frac{1}{2}}UY^{\frac{1}{2}}$ for some unitary matrix $U$ .

Lemma 2.2. [, Theorem 3] Let $X, Y\in \mathbb{M}_{n}$ be positive definite, then for every unitarily invariant norm,

$\begin{eqnarray*} \begin{aligned} \label{f2} ||X\sharp_tY||&\leq||X^{1-t}Y^t||\\ &\leq||(1-t)X+tY||. \end{aligned} \end{eqnarray*}$

Now, we give a lemma that will play an important role in the later proofs.

Lemma 2.3. Let $M = \left(\begin{array}{cc} A & X \\ Y^{*} & B\\ \end{array} \right)\in \mathbb{M}_{2n}$ be APT, then for $t\in [0, 1]$ ,

$\left( \begin{array}{cc} {{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B & \frac{X+Y}{2} \\ \frac{X^*+Y^*}{2} & {{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B\\ \end{array} \right)$

is PPT.

Proof: Since $M$ is APT, we have that

${{{\rm{Re\, }}}} M = \left( \begin{array}{cc} {{{\rm{Re\, }}}} A & \frac{X+Y}{2}\\ \frac{X^*+Y^*}{2} & {{{\rm{Re\, }}}} B\\ \end{array} \right)$

is PPT.

Therefore, ${{{\rm{Re\, }}}} M \geq 0$ and ${{{\rm{Re\, }}}} M^{\tau} \geq 0$ .

By the Schur complement theorem, we have

$\begin{eqnarray*} {{{\rm{Re\, }}}} B- \frac{X^*+Y^*}{2}({{{\rm{Re\, }}}} A)^{-1}\frac{X+Y}{2}\geq 0, \end{eqnarray*}$

and

$\begin{eqnarray*} {{{\rm{Re\, }}}} A-\frac{X^*+Y^*}{2}({{{\rm{Re\, }}}} B)^{-1}\frac{X+Y}{2} \geq 0. \end{eqnarray*}$

Compute

$\begin{eqnarray*} \begin{aligned} &\frac{X^*+Y^*}{2}({{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B)^{-1}\frac{X+Y}{2}\\ & = \frac{X^*+Y^*}{2}(({{{\rm{Re\, }}}} A)^{-1}\sharp_t ({{{\rm{Re\, }}}} B)^{-1})\frac{X+Y}{2}\\ & = \left(\frac{X^*+Y^*}{2}({{{\rm{Re\, }}}} A)^{-1}\frac{X+Y}{2}\right)\sharp_t \left(\frac{X^*+Y^*}{2}({{{\rm{Re\, }}}} B)^{-1}\frac{X+Y}{2}\right)\\ &\leq {{{\rm{Re\, }}}} B\sharp_t {{{\rm{Re\, }}}} A. \end{aligned} \end{eqnarray*}$

Thus,

$\begin{eqnarray*} ({{{\rm{Re\, }}}} B \sharp_t {{{\rm{Re\, }}}} A)-\frac{X^*+Y^*}{2}({{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B)^{-1}\frac{X+Y}{2} \geq0. \end{eqnarray*}$

By utilizing $({{{\rm{Re\, }}}} B\sharp_t {{{\rm{Re\, }}}} A) = {{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B$ , we have

$\begin{eqnarray*} \left(\begin{array}{cc} {{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B & \frac{X+Y}{2}\\ \frac{X^*+Y^*}{2} & {{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B\\ \end{array} \right)\geq0. \end{eqnarray*}$

Similarly, we have

$\begin{eqnarray*} \left(\begin{array}{cc} {{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B & \frac{X^*+Y^*}{2}\\ \frac{X+Y}{2} & {{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B\\ \end{array} \right)\geq 0. \end{eqnarray*}$

This completes the proof.

First, we give the correct proof of Corollary 1.3.

Proof: By Theorem 1.1, there exists a unitary matrix $U\in \mathbb{M}_{n}$ such that $|X|\leq (A\sharp_t B)\sharp (U^*$ $(A\sharp_{1-t} B)U).$ Moreover, by Lemma 2.1, we have $(A\sharp_t B)\sharp (U^*(A\sharp_{1-t} B)U) = (A\sharp_t B)^{\frac{1}{2}}V(U^*$ $(A\sharp_{1-t} B)^{\frac{1}{2}}U).$ Now, by $s_{i+j-1}(AB)\leq s_i(A)s_j(B)$ , we have

$\begin{eqnarray*} s_{i+j-1}(X) &\leq& s_{i+j-1}((A\sharp_t B)\sharp(U^*(A\sharp_{1-t}B)U)) \\ & = & s_{i+j-1}((A\sharp_t B)^{\frac{1}{2}}VU^*(A\sharp_{1-t} B)^{\frac{1}{2}}U) \\ &\leq& s_{i}((A\sharp_t B)^{\frac{1}{2}})s_j((A\sharp_{1-t} B)^{\frac{1}{2}}), \end{eqnarray*}$

which completes the proof.

Next, we generalize Theorem 1.1 to the class of APT matrices.

Theorem 2.4. Let $M = \left(\begin{array}{cc} A & X \\ Y^{*} & B\\ \end{array} \right)$ be APT, then

$\begin{eqnarray*} \left|\frac{X+Y}{2}\right| \leq ({{{\rm{Re\, }}}} A\sharp_t{{{\rm{Re\, }}}} B)\sharp (U^*({{{\rm{Re\, }}}} A \sharp_{1-t}{{{\rm{Re\, }}}} B)U), \end{eqnarray*}$

where $U\in \mathbb{M}_{n}$ is any unitary matrix such that $\frac{X+Y}{2} = U\left|\frac{X+Y}{2}\right|$ .

Proof: Since $M$ is an APT matrix, we know that

$\left( \begin{array}{cc} {{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B & \frac{X+Y}{2}\\ \frac{X^*+Y^*}{2} & {{{\rm{Re\, }}}} B\sharp_{1-t} {{{\rm{Re\, }}}} A\\ \end{array} \right)$

is PPT.

Let $W$ be a unitary matrix defined as $W = \left(\begin{array}{cc} I & 0\\ 0 & U\\ \end{array} \right)$ . Thus,

$\begin{eqnarray*} W^*\left(\begin{array}{cc} {{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B & \frac{X^*+Y^*}{2}\\ \frac{X+Y}{2} & {{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B\\ \end{array} \right)W = \left(\begin{array}{cc} {{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B & |\frac{X+Y}{2}|\\ |\frac{X+Y}{2}| & U^*({{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B)U\\ \end{array} \right)\geq0. \end{eqnarray*}$

By Lemma 2.1, we have

$\begin{eqnarray*} \left|\frac{X+Y}{2}\right|\leq({{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B)\sharp (U^*({{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B)U). \end{eqnarray*}$

Remark 1. When $M = \left(\begin{array}{cc} A & X \\ Y^{*} & B\\ \end{array} \right)$ is PPT in Theorem 2.4, our result is Theorem 1.1. Thus, our result is a generalization of Theorem 1.1.

Using Theorem 2.4 and Lemma 2.2, we have the following.

Corollary 2.5. Let $M = \left(\begin{array}{cc} A & X \\ Y^{*} & B\\ \end{array} \right)$ be APT and let $t\in[0, 1]$ , then for every unitarily invariant norm $|| \cdot ||$ and some unitary matrix $U\in \mathbb{M}_n,$

$\begin{eqnarray*} \begin{aligned} \left|\left|\frac{X+Y}{2}\right|\right|&\leq ||({{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B)\sharp (U^*({{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B)U)||\\ &\leq \left|\left|\frac{({{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B) + U^*({{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B)U}{2}\right|\right|\\ &\leq \frac{||{{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B|| + ||{{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B||}{2}\\ &\leq \frac{||({{{\rm{Re\, }}}} A)^{1-t} ({{{\rm{Re\, }}}} B)^t|| + ||({{{\rm{Re\, }}}} A)^t ({{{\rm{Re\, }}}} B)^{1-t}||}{2}\\ &\leq \frac{||(1-t){{{\rm{Re\, }}}} A+t{{{\rm{Re\, }}}} B|| + ||t{{{\rm{Re\, }}}} A+(1-t) {{{\rm{Re\, }}}} B||}{2}. \end{aligned} \end{eqnarray*}$

Proof: The first inequality follows from Theorem 2.4. The third one is by the triangle inequality. The other conclusions hold by Lemma 2.2.

In particular, when $t = \frac{1}{2},$ we have the following result.

Corollary 2.6. Let $M = \left(\begin{array}{cc} A & X \\ Y^{*} & B\\ \end{array} \right)$ be APT, then for every unitarily invariant norm $|| \cdot ||$ and some unitary matrix $U\in \mathbb{M}_n,$

$\begin{eqnarray*} \begin{aligned} \left|\left|\frac{X+Y}{2}\right|\right|&\leq ||({{{\rm{Re\, }}}} A\sharp {{{\rm{Re\, }}}} B)\sharp (U^*({{{\rm{Re\, }}}} A\sharp {{{\rm{Re\, }}}} B)U)||\\ &\leq \left|\left|\frac{({{{\rm{Re\, }}}} A\sharp {{{\rm{Re\, }}}} B) + U^*({{{\rm{Re\, }}}} A\sharp {{{\rm{Re\, }}}} B)U}{2}\right|\right|\\ &\leq ||{{{\rm{Re\, }}}} A\sharp {{{\rm{Re\, }}}} B||\\ &\leq ||({{{\rm{Re\, }}}} A)^{\frac{1}{2}}({{{\rm{Re\, }}}} B)^{\frac{1}{2}}||\\ &\leq \left|\left|\frac{{{{\rm{Re\, }}}} A+{{{\rm{Re\, }}}} B}{2}\right|\right|. \end{aligned} \end{eqnarray*}$

Squaring the inequalities in Corollary 2.6, we get a quick consequence.

Corollary 2.7. If $M = \left(\begin{array}{cc} A & X \\ Y^{*} & B\\ \end{array} \right)$ is APT, then

$\begin{eqnarray*} \begin{aligned} {{{\rm{tr}}}}\left(\left(\frac{X^*+Y^*}{2}\right)\left(\frac{X+Y}{2}\right)\right)&\leq {{{\rm{tr}}}}(({{{\rm{Re\, }}}} A\sharp {{{\rm{Re\, }}}} B)^2)\\ &\leq {{{\rm{tr}}}}({{{\rm{Re\, }}}} A {{{\rm{Re\, }}}} B)\\ &\leq {{{\rm{tr}}}}\left(\left(\frac{{{{\rm{Re\, }}}} A+{{{\rm{Re\, }}}} B}{2}\right)^2\right). \end{aligned} \end{eqnarray*}$

Proof: Compute

$\begin{eqnarray*} \begin{aligned} {{{\rm{tr}}}}\left(\left(\frac{X^*+Y^*}{2}\right)\left(\frac{X+Y}{2}\right)\right)&\leq {{{\rm{tr}}}}(({{{\rm{Re\, }}}} A\sharp {{{\rm{Re\, }}}} B)^*({{{\rm{Re\, }}}} A\sharp {{{\rm{Re\, }}}} B))\\ & = {{{\rm{tr}}}}(({{{\rm{Re\, }}}} A\sharp {{{\rm{Re\, }}}} B)^2)\\ &\leq {{{\rm{tr}}}}(({{{\rm{Re\, }}}} A)({{{\rm{Re\, }}}} B))\\ &\leq {{{\rm{tr}}}}\left(\left(\frac{{{{\rm{Re\, }}}} A+{{{\rm{Re\, }}}} B}{2}\right)^2\right). \end{aligned} \end{eqnarray*}$

It is known that for any $X, Y\in \mathbb{M}_n$ and any indices $i, j$ such that $i+j\leq n+1$ , we have $s_{i+j-1}(XY)\leq s_i(X)s_j(Y)$ (see [2, Page 75]). By utilizing this fact and Theorem 2.4, we can obtain the following result.

Corollary 2.8. Let $M = \left(\begin{array}{cc} A & X \\ Y^{*} & B\\ \end{array} \right)$ be APT, then for any $t\in [0, 1]$ , we have

$\begin{eqnarray*} s_{i+j-1}\left(\frac{X+Y}{2}\right)\leq s_i(({{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B)^{\frac{1}{2}})s_j(({{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B)^{\frac{1}{2}}). \end{eqnarray*}$

Consequently,

$\begin{eqnarray*} s_{2j-1}\left(\frac{X+Y}{2}\right)\leq s_j(({{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B)^{\frac{1}{2}})s_j(({{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B)^{\frac{1}{2}}). \end{eqnarray*}$

Proof: By Lemma 2.1 and Theorem 2.4, observe that

$\begin{eqnarray*} \begin{aligned} s_{i+j-1}\left(\frac{X+Y}{2}\right)& = s_{i+j-1}\left(\left|\frac{X+Y}{2}\right|\right)\\ &\leq s_{i+j-1}(({{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B)\sharp (U^*({{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B)U))\\ & = s_{i+j-1}(({{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B)^{\frac{1}{2}}V(U^*({{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B)U)^{\frac{1}{2}})\\ &\leq s_i(({{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B)^{\frac{1}{2}}V)s_j((U^*({{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B)U)^{\frac{1}{2}})\\ & = s_i(({{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B)^{\frac{1}{2}})s_j(({{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B)^{\frac{1}{2}}). \end{aligned} \end{eqnarray*}$

Finally, we study the relationship between the diagonal blocks and the real part of the off-diagonal blocks of the APT matrix $M$ .

Theorem 2.9. Let $M = \left(\begin{array}{cc} A & X \\ Y^{*} & B\\ \end{array} \right)$ be APT, then for all $t\in [0, 1]$ ,

$\begin{eqnarray*} \begin{aligned} {{{\rm{Re\, }}}}\left(\frac{X+Y}{2}\right)&\leq ({{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B)\sharp({{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B)\\ &\leq \frac{({{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B)+({{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B)}{2}, \end{aligned} \end{eqnarray*}$

and

$\begin{eqnarray*} \begin{aligned} {{{\rm{Im\, }}}}\left(\frac{X+Y}{2}\right)&\leq ({{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B)\sharp({{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B)\\ &\leq \frac{({{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B)+({{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B)}{2}. \end{aligned} \end{eqnarray*}$

Proof: Since $M$ is APT, we have that

${{{\rm{Re\, }}}} M = \left( \begin{array}{cc} {{{\rm{Re\, }}}} A & \frac{X+Y}{2} \\ \frac{X^{*}+Y^{*}}{2} & {{{\rm{Re\, }}}} B\\ \end{array} \right)$

is PPT.

Therefore,

$\begin{eqnarray*} \begin{aligned} \left( \begin{array}{cc} {{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B & {{{\rm{Re\, }}}} \left(\frac{X+Y}{2}\right) \\ {{{\rm{Re\, }}}}\left(\frac{X^{*}+Y^{*}}{2}\right) & {{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B\\ \end{array} \right)& = \frac{1}{2} \left( \begin{array}{cc} {{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B & \frac{X+Y}{2} \\ \frac{X^{*}+Y^{*}}{2} & {{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B\\ \end{array} \right)\\ &+\frac{1}{2} \left( \begin{array}{cc} {{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B & \frac{X^{*}+Y^{*}}{2} \\ \frac{X+Y}{2} & {{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B\\ \end{array} \right)\geq0. \end{aligned} \end{eqnarray*}$

So, by Lemma 2.1, we have

$\begin{eqnarray*} {{{\rm{Re\, }}}} \left(\frac{X+Y}{2}\right)\leq({{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B)\sharp({{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B). \end{eqnarray*}$

This implies the first inequality.

Since ${{{\rm{Re\, }}}} M$ is PPT, we have

$\begin{eqnarray*} \left( \begin{array}{cc} {{{\rm{Re\, }}}} A & -i\frac{X+Y}{2} \\ i\frac{X^{*}+Y^{*}}{2} & {{{\rm{Re\, }}}} B\\ \end{array} \right) = \left( \begin{array}{cc} I & 0 \\ 0 & iI\\ \end{array} \right) ({{{\rm{Re\, }}}} M) \left( \begin{array}{cc} I & 0 \\ 0 & -iI\\ \end{array} \right)\geq0, \\ \left( \begin{array}{cc} {{{\rm{Re\, }}}} A & i\frac{X^{*}+Y^{*}}{2} \\ -i\frac{X+Y}{2} & {{{\rm{Re\, }}}} B\\ \end{array} \right) = \left( \begin{array}{cc} I & 0 \\ 0 & -iI\\ \end{array} \right) (({{{\rm{Re\, }}}} M)^\tau) \left( \begin{array}{cc} I & 0 \\ 0 & iI\\ \end{array} \right)\geq0. \end{eqnarray*}$

Thus,

$\left( \begin{array}{cc} {{{\rm{Re\, }}}} A & -i\frac{X+Y}{2} \\ i\frac{X^{*}+Y^{*}}{2} & {{{\rm{Re\, }}}} B\\ \end{array} \right)$

is PPT.

By Lemma 2.3,

$\left( \begin{array}{cc} {{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B & -i\frac{X+Y}{2} \\ i\frac{X^{*}+Y^{*}}{2} & {{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B\\ \end{array} \right)$

is also PPT.

So,

$\begin{eqnarray*} \frac{1}{2}\left( \begin{array}{cc} {{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B & -i\frac{X+Y}{2} \\ i\frac{X^{*}+Y^{*}}{2} & {{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B\\ \end{array} \right) +\frac{1}{2}\left( \begin{array}{cc} {{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B & i\frac{X^{*}+Y^{*}}{2} \\ -i\frac{X+Y}{2} & {{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B\\ \end{array} \right)\geq0, \end{eqnarray*}$

which means that

$\left( \begin{array}{cc} {{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B & {{{\rm{Im\, }}}}\left(\frac{X+Y}{2}\right) \\ {{{\rm{Im\, }}}}\left(\frac{X+Y}{2}\right) & {{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B\\ \end{array} \right)\geq 0.$

By Lemma 2.1, we have

$\begin{eqnarray*} {{{\rm{Im\, }}}}\left(\frac{X+Y}{2}\right)\leq({{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B)\sharp({{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B). \end{eqnarray*}$

This completes the proof.

Corollary 2.10. Let $\left(\begin{array}{cc} {{{\rm{Re\, }}}} A & \frac{X+Y}{2} \\ \frac{X+Y}{2} & {{{\rm{Re\, }}}} B\\ \end{array} \right)\geq0.$ If $\frac{X+Y}{2}$ is Hermitian and $t\in[0, 1]$ , then,

$\begin{eqnarray*} \begin{aligned} \frac{X+Y}{2}&\leq ({{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B)\sharp({{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B)\\ &\leq \frac{({{{\rm{Re\, }}}} A\sharp_t {{{\rm{Re\, }}}} B)+({{{\rm{Re\, }}}} A\sharp_{1-t} {{{\rm{Re\, }}}} B)}{2}. \end{aligned} \end{eqnarray*}$

Use of AI tools declaration

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

Acknowledgments

The work is supported by National Natural Science Foundation (grant No. 12261030), Hainan Provincial Natural Science Foundation for High-level Talents (grant No. 123RC474), Hainan Provincial Natural Science Foundation of China (grant No. 124RC503), the Hainan Provincial Graduate Innovation Research Program (grant No. Qhys2023-383 and Qhys2023-385), and the Key Laboratory of Computational Science and Application of Hainan Province.

Conflict of interest

The authors declare that they have no conflict of interest.

References

[1]	J. D. Logan, An introduction to nonlinear partial differential equations, John Wiley & Sons, 2008. https://doi.org/10.1002/9780470287095
[2]	S. Fucik, A. Kufner, Nonlinear differential equations, Elsevier, 1980.
[3]	S. Larsson, V. Thomée, Partial differential equations with numerical methods, Berlin, Heidelberg: Springer, 2003. https://doi.org/10.1007/978-3-540-88706-5
[4]	E. L. Ince, Ordinary differential equations, New York: Dover Publications, 2012.
[5]	Z. Ji, Y. F. Nie, L. F. Li, Y. Y. Xie, M. C. Wang, Rational solutions of an extended (2+1)-dimensional Camassa-Holm-Kadomtsev-Petviashvili equation in liquid drop, AIMS Math., 8 (2023), 3163–3184. https://doi.org/10.3934/math.2023162 doi: 10.3934/math.2023162
[6]	M. B. Almatrafi, Construction of closed form soliton solutions to the space-time fractional symmetric regularized long wave equation using two reliable methods, Fractals, 31 (2023), 2340160. https://doi.org/10.1142/S0218348X23401606 doi: 10.1142/S0218348X23401606
[7]	A. R. Alharbi, M. B. Almatrafi, Exact solitary wave and numerical solutions for geophysical KdV equation, J. King Saud Univ. Sci., 34 (2022), 102087. https://doi.org/10.1016/j.jksus.2022.102087 doi: 10.1016/j.jksus.2022.102087
[8]	M. B. Almatrafi, A. Alharbi, New soliton wave solutions to a nonlinear equation arising in plasma physics, Comput. Model. Eng. Sci., 137 (2023), 827–841. https://doi.org/10.32604/cmes.2023.027344 doi: 10.32604/cmes.2023.027344
[9]	M. A. E. Abdelrahman, M. B. Almatrafi, A. Alharbi, Fundamental solutions for the coupled KdV system and its stability, Symmetry, 12 (2020), 1–13. https://doi.org/10.3390/sym12030429 doi: 10.3390/sym12030429
[10]	A. R. Alharbi, Traveling-wave and numerical solutions to a Novikov-Veselov system via the modified mathematical methods, AIMS Math., 8 (2023), 1230–1250. https://doi.org/10.3934/math.2023062 doi: 10.3934/math.2023062
[11]	A. R. Alharbi, Numerical solutions to two-dimensional fourth order parabolic thin film equations using the Parabolic Monge-Ampere method, AIMS Math., 8 (2023), 16463–16478. https://doi.org/10.3934/math.2023841 doi: 10.3934/math.2023841
[12]	N. A. Shah, Y. S. Hamed, K. M. Abualnaja, J. D. Chung, R. Shah, A. Khan, A comparative analysis of fractional-order Kaup-Kupershmidt equation within different operators, Symmetry, 14 (2022), 1–23. https://doi.org/10.3390/sym14050986 doi: 10.3390/sym14050986
[13]	A. H. Arnous, M. S. Hashemi, K. S. Nisar, M. Shakeel, J. Ahmad, I. Ahmad, et al., Investigating solitary wave solutions with enhanced algebraic method for new extended Sakovich equations in fluid dynamics, Results Phys., 57 (2024), 107369. https://doi.org/10.1016/j.rinp.2024.107369 doi: 10.1016/j.rinp.2024.107369
[14]	A. H. Arnous, A. Biswas, Y. Yildirim, L. Moraru, M. Aphane, S. Moshokoa, et al., Quiescent optical solitons with Kudryashov's generalized quintuple-power and nonlocal nonlinearity having nonlinear chromatic dispersion: generalized temporal evolution, Ukr. J. Phys. Optics, 24 (2023), 105–113. https://doi.org/10.3116/16091833/24/2/105/2023 doi: 10.3116/16091833/24/2/105/2023
[15]	J. Vega-Guzman, M. F. Mahmood, Q. Zhou, H. Triki, A. H. Arnous, A. Biswas, et al., Solitons in nonlinear directional couplers with optical metamaterials, Nonlinear Dyn., 87 (2017), 427–458. https://doi.org/10.1007/s11071-016-3052-2 doi: 10.1007/s11071-016-3052-2
[16]	A. R. Seadawy, A. H. Arnous, A. Biswas, M. R. Belic, Optical solitons with Sasa-Satsuma equation by F-expansion scheme, Optoelectron. Adv. Mat., 13 (2019), 31–36.
[17]	N. Mahak, G. Akram, Exact solitary wave solutions by extended rational sine-cosine and extended rational sinh-cosh techniques, Phys. Scr., 94 (2019), 115212. https://doi.org/10.1088/1402-4896/ab20f3 doi: 10.1088/1402-4896/ab20f3
[18]	M. L. Wang, X. Z. Li, Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations, Phys. Lett. A, 343 (2005), 48–54. https://doi.org/10.1016/j.physleta.2005.05.085 doi: 10.1016/j.physleta.2005.05.085
[19]	A. M. Wazwaz, The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations, Appl. Math. Comput., 167 (2005), 1196–1210. https://doi.org/10.1016/j.amc.2004.08.005 doi: 10.1016/j.amc.2004.08.005
[20]	A. M. Wazwaz, The tanh-coth and the sech methods for exact solutions of the Jaulent-Miodek equation, Phys. Lett. A, 366 (2007), 85–90. https://doi.org/10.1016/j.physleta.2007.02.011 doi: 10.1016/j.physleta.2007.02.011
[21]	A. S. A. Rady, E. S. Osman, M. Khalfallah, The homogeneous balance method and its application to the Benjamin-Bona-Mahoney (BBM) equation, Appl. Math. Comput., 217 (2010), 1385–1390. https://doi.org/10.1016/j.amc.2009.05.027 doi: 10.1016/j.amc.2009.05.027
[22]	W. X. Ma, T. W. Huang, Y. Zhang, A multiple exp-function method for nonlinear differential equations and its application, Phys. Scr., 82 (2010), 065003. https://doi.org/10.1088/0031-8949/82/06/065003 doi: 10.1088/0031-8949/82/06/065003
[23]	M. J. Ablowitz, P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Cambridge University Press, 1991. https://doi.org/10.1017/CBO9780511623998
[24]	N. J. Zabusky, M. D. Kruskal, Interaction of "solitons" in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15 (1965), 240. https://doi.org/10.1103/PhysRevLett.15.240 doi: 10.1103/PhysRevLett.15.240
[25]	W. X. Ma, Lump solutions to the Kadomtsev-Petviashvili equation, Phys. Lett. A, 379 (2015), 1975–1978. https://doi.org/10.1016/j.physleta.2015.06.061 doi: 10.1016/j.physleta.2015.06.061
[26]	T. B. Benjamin, J. L. Bona, J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci., 272 (1972), 47–78. https://doi.org/10.1098/rsta.1972.0032 doi: 10.1098/rsta.1972.0032
[27]	A. M. Wazwaz, Exact solutions of compact and noncompact structures for the KP-BBM equation, Appl. Math. Comput., 169 (2005), 700–712. https://doi.org/10.1016/j.amc.2004.09.061 doi: 10.1016/j.amc.2004.09.061
[28]	D. V. Tanwar, A. M. Wazwaz, Lie symmetries, optimal system and dynamics of exact solutions of (2+1)-dimensional KP-BBM equation, Phys. Scr., 95 (2020), 065220. https://doi.org/10.1088/1402-4896/ab8651 doi: 10.1088/1402-4896/ab8651
[29]	D. V. Tanwar, A. K. Ray, A. Chauhan, Lie symmetries and dynamical behavior of soliton solutions of KP-BBM equation, Qual. Theory Dyn. Syst., 21 (2022), 24. https://doi.org/10.1007/s12346-021-00557-8 doi: 10.1007/s12346-021-00557-8
[30]	Y. D. Yu, H. C. Ma, Explicit solutions of (2+1)-dimensional nonlinear KP-BBM equation by using exp-function method, Appl. Math. Comput., 217 (2010), 1391–1397. https://doi.org/10.1016/j.amc.2009.05.035 doi: 10.1016/j.amc.2009.05.035
[31]	J. J. Li, J. Manafian, N. T. Hang, D. T. N. Huy, A. Davidyants, Interaction among a lump, periodic waves, and kink solutions to the KP-BBM equation, Int. J. Nonlinear Sci. Numer. Simul., 24 (2023), 227–243. https://doi.org/10.1515/ijnsns-2020-0156 doi: 10.1515/ijnsns-2020-0156
[32]	M. A. Abdou, Exact periodic wave solutions to some nonlinear evolution equations, Int. J. Nonlinear Sci., 6 (2008), 145–153.
[33]	Y. Y. Xie, L. F. Li, Multiple-order breathers for a generalized (3+1)-dimensional Kadomtsev-Petviashvili Benjamin-Bona-Mahony equation near the offshore structure, Math. Comput. Simul., 193 (2022), 19–31. https://doi.org/10.1016/j.matcom.2021.08.021 doi: 10.1016/j.matcom.2021.08.021
[34]	B. L. Feng, J. Manafian, O. A. Ilhan, A. M. Rao, A. H. Agadi, Cross-kink wave, solitary, dark, and periodic wave solutions by bilinear and He's variational direct methods for the KP-BBM equation, Int. J. Mod. Phys. B, 35 (2021), 2150275. https://doi.org/10.1142/S0217979221502751 doi: 10.1142/S0217979221502751
[35]	S. Malik, M. S. Hashemi, S. Kumar, H. Rezazadeh, W. Mahmoud, M. S. Osman, Application of new Kudryashov method to various nonlinear partial differential equations, Opt. Quantum Electronics, 55 (2023), 8. https://doi.org/10.1007/s11082-022-04261-y doi: 10.1007/s11082-022-04261-y
[36]	A. H. Arnous, M. Mirzazadeh, M. S. Hashemi, N. A. Shah, J. D. Chung, Three different integration schemes for finding soliton solutions in the (1+1)-dimensional Van der Waals gas system, Results Phys., 55 (2023), 107178. https://doi.org/10.1016/j.rinp.2023.107178 doi: 10.1016/j.rinp.2023.107178
[37]	E. M. E. Zayed, A. H. Arnous, A. Secer, M. Ozisik, M. Bayram, N. A. Shah, et al., Highly dispersive optical solitons in fiber Bragg gratings for stochastic Lakshmanan-Porsezian-Daniel equation with spatio-temporal dispersion and multiplicative white noise, Results Phys., 55 (2023), 107177. https://doi.org/10.1016/j.rinp.2023.107177 doi: 10.1016/j.rinp.2023.107177
[38]	S. Malik, S. Kumar, Pure-cubic optical soliton perturbation with full nonlinearity by a new generalized approach, Optik, 258 (2022), 168865. https://doi.org/10.1016/j.ijleo.2022.168865 doi: 10.1016/j.ijleo.2022.168865
[39]	A. H. Arnous, Optical solitons to the cubic quartic Bragg gratings with anti-cubic nonlinearity using new approach, Optik, 251 (2022), 168356. https://doi.org/10.1016/j.ijleo.2021.168356 doi: 10.1016/j.ijleo.2021.168356
[40]	M. Kumar, R. K. Gupta, A new generalized approach for soliton solutions and generalized symmetries of time-fractional partial differential equation, Int. J. Appl. Comput. Math., 8 (2022), 200. https://doi.org/10.1007/s40819-022-01420-3 doi: 10.1007/s40819-022-01420-3
[41]	S. Malik, S. Kumar, A. Biswas, Y. Yıldırım, L. Moraru, S. Moldovanu, et al., Cubic-quartic optical solitons in fiber Bragg gratings with dispersive reflectivity having parabolic law of nonlinear refractive index by Lie symmetry, Symmetry, 14 (2022), 1–17. https://doi.org/10.3390/sym14112370 doi: 10.3390/sym14112370
[42]	Y. Kai, L. K. Huang, Dynamic properties, Gaussian soliton and chaotic behaviors of general Degasperis-Procesi model, Nonlinear Dyn., 111 (2023), 8687–8700. https://doi.org/10.1007/s11071-023-08290-4 doi: 10.1007/s11071-023-08290-4
[43]	S. Malik, S. Kumar, A. Das, A (2+1) dimensional combined KdV-mKdV equation: integrability, stability analysis and soliton solutions, Nonlinear Dyn., 107 (2022), 2689–2701. https://doi.org/10.1007/s11071-021-07075-x doi: 10.1007/s11071-021-07075-x

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