Bifurcations of traveling wave solutions for the mixed Korteweg-de Vries equation

1.   Introduction

Over the past decades, the study of nonlinear problems has been the interest of many researchers ^{[5,10,11,14,19,24,25,26]}. Also, study of fractional calculus has recently gained great momentum, and has emerged as a significant research area ^{[5,7,15,20,21,30]}. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes; see, for instance, the contribution ^{[2,3,4,6,16,22,23,28,29]} and references therein.

The authors in ^[8] focused on the study of nonlinear jerk problems due to its various physical applications, as form

In 2020, the authors investigated the existence and uniqueness of solutions for the following nonlocal generalized fractional Sturm-Liouville and Langevin equations:

where ${^{c}\mathcal{D}}_{\iota_1^{+}}^{ \alpha}$, ${^{c}\mathcal{D}}_{\iota_1^{+}}^{ \gamma}$ are the Caputo fractional derivatives, $p, q \in C([0, T])$ with $|p|\geq K > 0$, $\varkappa_1, \varkappa_2 : C(J) \to \mathbb{R}$ are continuous functions and $\mathfrak{T} \in C([0, T] \times \mathbb{R})$ ^[27]. The second derivative of the accelaration (fourth derivative of position) is a physical quantity called a snap or jounce, which can be modeled as

It is obvious that the model (1.1) can be reduced to the following equation:

Scientifically, jerk and snap are the third and fourth derivatives of our position with regard to time, respectively. The Eq (1.1) contains a $4^{{\mathrm{th}}}$-order derivative of the variable ${\mathrm{y}}_{1}$, and it describes a $4^{{\mathrm{th}}}$-order dynamical vibration model.

The corresponding fractional model is achieved by using the fractional derivative (of order less than or equal $1$) instead of the standard deivative $\frac{{\mathrm{d}}}{{\mathrm{d}}\mathfrak{t}}$. Many types of fractional derivatives can be used here, such as Riemann-Liouville, Caputo, Hadamard, etc. We prefer to use the generalized fractional derivative ($\mathbb{GFD}$), with respect to differentiable increasing function $\mathbb{G}$. In 2020, Liu {et al.}, developed two iterative algorithms to determine the periods, and then the periodic solutions of nonlinear jerk equations for two possible cases with initial values unknown and initial values given ^[13]. The authors in a recent article ^[23] considered the $\mathbb{G}$-fractional snap model ($\mathbb{GF}$SM) with constant, initial conditions

where the $\mathbb{G}$-Caputo derivatives are illustrated by the symbol

here and the increasing function $\mathbb{G}\in C^{1}([\iota_1, \iota_2])$ is such that $\mathbb{G}^{\prime }(\mathfrak{t})\neq 0$, for each $\mathfrak{t}\in \lbrack \iota_1, \iota_2]$ and continuous function $\mathcal{T}$ belongs to $C(\lbrack \iota_1, \iota_2] \times \mathbb{R}^{4})$ and ${\mathrm{y}}_{0}, {\mathrm{y}}_{1}, {\mathrm{y}}_{2}, {\mathrm{y}}_{3} \in \mathbb{R}$. Abbas {et al.} studied the following coupled system of fractional differential equations:

for $\mathfrak{t} \in [ \iota_1, \iota_2]$ equipped with the generalized fractional integral boundary conditions

where $\varrho \in (0, 1]$, ${^{RL}\mathcal{D}}_{\iota_1^{+}}^{\alpha_i; \varrho}$ denotes the generalized proportional fractional derivatives of Riemann-Liouville type of order $1 < \alpha_i\leq 2$, ${\mathcal{I}}_{\iota_1^{+}}^{\zeta_i; \varrho}$ that denotes the generalized proportional fractional integrals of order $0 < \zeta_i < 1$ and $\tau_i, {\mathtt{η}}_i \in (\iota_1, \iota_2)$ and $\mathfrak{T}_i \in C([\iota_1, \iota_2] \times \mathbb{R}^2)$ ^[1].

We center our consideration on the problem of the existence and uniqueness along with the Hyers-Ulam stability $({\mathrm{U}}$-${\mathrm{H}}$-$\mathbb{S})$ of solutions for fractional nonlinear couple snap system $(\mathbb{C}{\mathrm{S}}\mathbb{S})$ in the $\mathbb{G}$-Caputo sense $(\mathbb{G}{\mathrm{C}})$ with initial conditions

subject to the following integral boundary conditions

where the $\mathbb{G}{\mathrm{C}}$ derivatives are illustrated by symbol

here the function $\mathbb{G}\in C^{1}(\Sigma)$ is increasing with $\mathbb{G}^{\prime }(\mathfrak{t}) \neq 0$, for all $\mathfrak{t}\in \Sigma = \lbrack \iota_1, \iota_2]$ and the functions $h_{k}\in C(\Sigma\times \mathbb{R}^{8}), (k = 1, 2)$ and ${\mathrm{g}}_{kj}\in C(\Sigma, \mathbb{R}), (j = 0, 1, 2, 3;\, k = 1, 2)$ are continuous functions. It is obvious that the $\mathbb{C}{\mathrm{S}}\mathbb{S}$ (1.4) and (1.5) can be rewritten as

where

The main novelty of this work is that we establish our results with the help of the technique of fixed point theorems for a fractional nonlinear $\mathbb{C}{\mathrm{S}}\mathbb{S}$ furnished with generalized operators, which leads to some general theoretical findings involving the following special cases: $\mathbb{G}$ as $\mathbb{G}_1(\iota) = 2^{\iota}$, $\mathbb{G}_2(\iota) = \iota$ (Caputo derivative), $\mathbb{G}_3(\iota) = \ln \iota$ (Caputo-Hadamard derivative), $\mathbb{G}_4(\iota) = \sqrt{\iota}$ (Katugampola derivative).

This paper is organized as follows: In Section 2, we present some necessary definitions and lemmas that are needed in the subsequent sections. In Section 3, we adopt some fixed point theorems to prove the existence and uniqueness of solutions for problem (1.4). The stability results are extensively discussed in Section 3.2. An illustrative example is presented in Section 4.

2.   Preliminaries

Some primitive notions, definitions and notations, which will be utilized throughout the manuscript, are recalled here. Consider the function $\mathbb{G}$ with assumptions in system (1.4). We start this part by defining $\mathbb{G}$-Riemann-Liouville fractional ($\mathbb{GF}$-${\mathrm{RL}}$) integrals and derivatives ^[17]. For $\eta > 0$, the $\eta^{{\mathrm{th}} }$-$\mathbb{GF}$-${\mathrm{RL}}$ integral for an integrable function ${\mathrm{v}}:\Sigma \to \mathbb{R}$ w.r.t $\mathbb{G}$ is illustrated as follows

where $\Gamma (\eta) = \int_{0}^{+\infty } e^{-\mathfrak{t}} \mathfrak{t}^{ \eta -1}\, {\mathrm{d}}\mathfrak{t}, \, \eta > 0$. Let $n\in \mathbb{N}$ and $\mathbb{G}, {\mathrm{v}}\in C^{n}(\Sigma)$ be such that $\mathbb{G}$ has the same properties mentioned above. The $\eta^{{\mathrm{th}}}$-$\mathbb{GF}$-${\mathrm{RL}}$ derivative of ${\mathrm{v}}$ is defined by

in which $n = [\eta ]+1$, where $A = \frac{1}{\mathbb{G}^{\prime }(\mathfrak{t})} \frac{ {\mathrm{d}}}{ {\mathrm{d}}\mathfrak{t}}$. The $\eta ^{{\mathrm{th}}}$-$\mathbb{G}$-fractional-Caputo derivative of ${\mathrm{v}}$ is defined by $^{c}\mathcal{D}_{\iota_1^{+}}^{\eta; \mathbb{G}} {\mathrm{v}}(\mathfrak{t}) = \mathcal{I}_{\iota_1^{+}}^{n-\eta; \mathbb{G}} A^{(n)}{\mathrm{v}}(\mathfrak{t})$, in which $n = [\eta ]+1$, $(\eta \notin \mathbb{N})$, $n = \eta$ for $\eta \in \mathbb{N}$ ^[17]. In other words,

This extension (2.2) gives the Caputo derivative when $\mathbb{G}(\mathfrak{t}) = \mathfrak{t}$ ^[17]. Also, in the case $\mathbb{G}(\mathfrak{t}) = \ln \mathfrak{t}$, it yields the Caputo-Hadamard derivative. If ${\mathrm{v}}\in C^{n}(\Sigma)$, the $\eta^{{\mathrm{th}}}$-$\mathbb{G}$-fractional-Caputo derivative of ${\mathrm{v}}$ is specified as ^[18]

The composition rules for above $\mathbb{G}$-operators are recalled in this lemma.

Lemma 2.1. ^[18] Let $n-1 < \eta < n$ and ${\mathrm{v}}\in C^{n}(\Sigma)$. Then the following holds

for all $\mathfrak{t}\in \Sigma$. Moreover, if $m\in \mathbb{N}$ and ${\mathrm{v}}\in C^{n+m}(\Sigma)$, then, the following holds

Observe that from Eq (2.3) if $A^{(j)}{\mathrm{v}}(\iota_1) = 0$, for $j = n, n+1, \dots, n+m-1$, we can get the following relation

Lemma 2.2. ^[12] Let $\eta, \nu > 0,$ and ${\mathrm{v}}\in C(\Sigma)$. Then for each $\mathfrak{t}\in \Sigma$ and by assuming

we have

(1) $\mathcal{I}_{\iota_1^{+}}^{\eta; \mathbb{G}}\left(\mathcal{I}_{\iota_1^{+}}^{\nu; \mathbb{G}}{\mathrm{v}}\right) (\mathfrak{t}) = \mathcal{I}_{\iota_1^{+}}^{\eta +\nu; \mathbb{G}} {\mathrm{v}}(\mathfrak{t})$;

(2) $^{c}\mathcal{D}_{\iota_1^{+}}^{\eta; \mathbb{G}}\left(\mathcal{I}_{\iota_1^{+}}^{\eta; \mathbb{G}}{\mathrm{v}}\right) (\mathfrak{t}) = {\mathrm{v}}(\mathfrak{t})$;

(3) $\mathcal{I}_{\iota_1^{+}}^{\eta; \mathbb{G}}(F_{\iota_1}(\mathfrak{t}))^{\nu -1} = \dfrac{\Gamma (\nu)}{\Gamma (\nu +\eta)}(F_{\iota_1}(\mathfrak{t}))^{\nu +\eta -1}$;

(4) $^{c}\mathcal{D}_{\iota_1^{+}}^{\eta; \mathbb{G}} (F_{a}(\mathfrak{t}))^{\nu -1} = \dfrac{\Gamma (\nu)}{\Gamma (\nu - \eta)}(F_{ \iota_1}(\mathfrak{t}))^{\nu -\eta -1}$;

(5) $^{c}\mathcal{D}_{\iota_1^{+}}^{\eta; \mathbb{G}}(F_{\iota_1}(\mathfrak{t}))^{j} = 0, \; n - 1 \leq \eta \leq n, n\in \mathbb{N}, \, j = 0, 1, \dots, n-1$.

Theorem 2.3. ^[9] (Banach's fixed point theorem) Consider $\Pi:\mathcal{Y}\to \mathcal{Y}$ to be a contraction operator, such that $\mathcal{Y}$ is a Banach space. Then, there are only one ${\mathrm{y}}^\ast \in \mathcal{Y}$, such that $\Pi({\mathrm{y}}^\ast) = {\mathrm{y}}^\ast$.

Lemma 2.4. ^[9] (Krasnoselskii's fixed point theorem) Assume that $\mathcal{B}\subset X$ is a closed convex and nonempty, and $\mathfrak{L}_1$, $\mathfrak{L}_2 : \mathcal{B} \to X$ nonlinear operators, such that:

(ⅰ) $\mathfrak{L}_1u + \mathfrak{L}_2v \in \mathcal{B}$ whenever $u, v \in \mathcal{B}$;

(ⅱ) $\mathfrak{L}_1$ is a contraction mapping;

(ⅲ) $\mathfrak{L}_2$ is compact and continuous.

Then, there exists $w\in \mathcal{B}$, such that $w = \mathfrak{L}_1w + \mathfrak{L}_2 w.$

Definition 2.5. ^[29] Let $X_1, X_2$ be Banach spaces and $\Lambda_1, \Lambda_2: X_1\times X_2\rightarrow X_1\times X_2$ be two operators. Then, the operational equations system provided by

is called ${\mathrm{U}}$-${\mathrm{H}}$-$\mathbb{S}$, if there exist $\alpha_i > 0, \, (i = 1, \dots, 4)$, such that, $\forall \, \rho_1, \rho_2 > 0$, and each solution $({\mathrm{u_1^\ast}}, {\mathrm{u_2^\ast}}) \in X_1 \times X_2$ of the identities

there exists $({\mathrm{v_1^\ast}}, {\mathrm{v_2^\ast}}) \in X_1\times X_2$ a solution of system (2.5), such that

Theorem 2.6. ^[29] Let $X_1, X_2$ be Banach spaces and $\Lambda_1, \Lambda_2: X_1\times X_2\to X_1\times X_2$ be two operators that satisfy

for each $({\mathrm{u_1}}, {\mathrm{u_2}}), ({\mathrm{u_1^\ast}}, {\mathrm{u_2^\ast}})\in X_1\times X_2$ and if the matrix

it converges to zero. Then, the system (2.6) is ${\mathrm{U}}$-${\mathrm{H}}$-$\mathbb{S}$.

3.   Main results

Here, we analyze the existence properties of solutions, and their uniqueness for the proposed fractional $\mathbb{G}$-$\mathbb{C}{\mathrm{S}}\mathbb{S}$ (1.6) using Krasnoselskii and Banach fixed point theorems. We need after lemma, which indicate the corresponding integral equation.

Lemma 3.1. For given continuous mappings $h, g_{k}\, (k = 0, 1, 2, 3)$ belongs to $C(\Sigma)$, and the solution of the linear $\mathbb{G}$-snap problem is

where $q, p, r, s\in (0, 1]$, are formulated by

Define the vector space

Then, $X_{k}, \, k = 1, 2,$ are Banach spaces via the norm

Hence, the product space $X_{1}\times X_{2}$ is a Banach space with the norm

3.1.   Existence and uniqueness

In view of Lemma 3.1, the solution of the coupled system (1.6) can be given as

Define the functional $\Lambda _{k} : X_{k} \to \mathbb{R}$, such that

Under some conditions, we show next that the functional $\Lambda :X_{1} \times X_{2} \to \mathbb{R}^{2}$ is a contraction, where $\Lambda$ is given as

Theorem 3.2. Let $h_{k}\in C(\Sigma\times\mathbb{R}^8), \, (k = 1, 2)$ be continuous functions. Moreover, assume that

(H1) there exist real constants $\ell_k > 0, \, (k = 1, 2)$, so that

for any $\mathfrak{t}\in \Sigma$, ${\mathrm{v}}_i, {\mathrm{v}}_i^\ast\in C([a, b])$ and $i = 1, 2, \dots, 8$.

Then, the fractional $\mathbb{G}$-$\mathbb{C}{\mathrm{S}}\mathbb{S}$ (1.6) admits a unique solution on $\Sigma$ if $\Phi\ell < 1$, whenever $\ell = \max\{ \ell_1, \ell_2\}$, $\Phi = \max\{\Phi_1, \Phi_2\}$ and

with $\Phi_k\ell_k < 1$.

Proof. First of all, we define a closed bounded ball

satisfying

where

and

Now, define the operator

where $\Lambda_k$ is given in (3.2). To show that $\Lambda(\mathbb{B}_\varepsilon) \subset \mathbb{B}_\varepsilon$, by using hypotheses (H1), for $({\mathrm{v}}_1, {\mathrm{v}}_2) \in\mathbb{B}_\varepsilon$ and $\mathfrak{t} \in \Sigma$, we get

Also,

and

Thus, due to (3.8)–(3.11) and (3.5), we obtain

Hence, we deduce that $\Vert \Lambda ({\mathrm{v}}_{1}, {\mathrm{v}}_{2})\Vert\leq\varepsilon,$ for $({\mathrm{v}}_1, {\mathrm{v}}_2) \in\mathbb{B}_\varepsilon$, so $\Lambda(\mathbb{B}_\varepsilon)\subset\mathbb{B}_\varepsilon.$ Next, we prove that $\Lambda$ is a contraction operator, by using ${\mathrm{(H1)}}$, for $({\mathrm{v}}_1, {\mathrm{v}}_2), ({\mathrm{u}}_1, {\mathrm{u}}_2) \in\mathbb{B}_\varepsilon$ and $\mathfrak{t}\in\Sigma$, we have

and

Therefore, due to (3.12)–(3.15), we get

Consequently,

Since $\Phi\ell < 1$, therefore, $\Lambda$ is a contraction operator. Thus, by Banach's fixed point Theorem 2.3, the operator $\Lambda$ has a unique fixed point, which is the unique solution of fractional G-snap system (1.6) and the proof is finished.

Next, we are ready to study the existence of solution of fractional $\mathbb{G}$-$(\mathbb{C}{\mathrm{S}}\mathbb{S})$ (1.6). For this regaed, we define the operators $\Omega, \Pi :X_{1}\times X_{2}\rightarrow \mathbb{R}^2, \Omega = (\Omega_1, \Omega_2), \Pi = (\Pi_1, \Pi_2)$, such that $\Lambda_k = \Omega_k+\Pi_k$, where

and

Theorem 3.3. Let $h_{k}\in C(\Sigma \times\mathbb{R}^8), (k = 1, 2)$ be continuous functions. Moreover, assume that

(H2) there exist real constants $\lambda_k > 0, \, (k = 1, 2)$, so that

Then, the fractional $\mathbb{G}$-$\mathbb{C}{\mathrm{S}}\mathbb{S}$ (1.6) has at least one solution on $\Sigma$.

Proof. At the beginning, we define a closed bounded ball

which satisfying

where

Firstly, we will prove $\Omega{\mathrm{v}}+\Pi{\mathrm{u}}\in\mathbb{B}_r$. By using ${\mathrm{(H2)}}$, for ${\mathrm{v}} = ({\mathrm{v}}_1, {\mathrm{v}}_2), {\mathrm{u}} = ({\mathrm{u}}_1, {\mathrm{u}}_2) \in\mathbb{B}_r$ and $\mathfrak{t}\in[a, b]$, we have

and

Hence, from (3.19) and (3.20), we have

Then,

this implying that $\Omega{\mathrm{v}}+\Pi{\mathrm{u}}\in\mathbb{B}_r$.

Secondly, we will prove that the operator $\Pi$ is a contraction mapping. It is clearly that $\Pi_k$ is a contraction with the constant zero. Thus, $\Pi$ is a contraction operator.

Third, we will prove that the operator $\Omega$ is a continuous. Let $({\mathrm{v}}_{n, 1}, {\mathrm{v}}_{n, 2})$ be a sequence of a bounded ball $\mathbb{B}_r$, such that $({\mathrm{v}}_{n, 1}, {\mathrm{v}}_{n, 2})\rightarrow ({\mathrm{v}}_{1}, {\mathrm{v}}_{2})$ as $n\rightarrow \infty$ in $\mathbb{B}_r$, we find that

By continuity of $h_{{\mathrm{v}}_1, {\mathrm{v}}_2, k}$, we have

as $n\rightarrow \infty$. So, $\Omega$ is a continuous operator.

Fourth, we will prove that the operator $\Omega$ is a compact operator. By using ${\mathrm{(H2)}}$, for ${\mathrm{v}} = ({\mathrm{v}}_1, {\mathrm{v}}_2), \in\mathbb{B}_r$ and $\mathfrak{t_1}, \mathfrak{t_2}\in \Sigma$ with $\mathfrak{t_1} < \mathfrak{t_2}$, we have

As $\mathfrak{t_1} \to \mathfrak{t_2}$, we obtain

impling that $\Omega$ is equicontinuous. Furthermore, in view of (3.19), $\Omega$ is uniformly bounded. Hence, due to the Arzelá-Ascoli theorem, we deduce that $\Omega$ is a compact operator. Then, all the conditions of Theorem 2.4 are holding. Thus, fractional $\mathbb{G}$-$\mathbb{C}{\mathrm{S}}\mathbb{S}$ (1.6) has at least one solution $({\mathrm{v}}_1, {\mathrm{v}}_2) \in\mathbb{B}_r$. The proof is completed.

3.2.   Ulam-Hyers stability

In this part, we review the stability criterion in the context of the ${\mathrm{U}}$-${\mathrm{H}}$-$\mathbb{S}$ for solutions of the fractional $\mathbb{G}$-$\mathbb{C}{\mathrm{S}}\mathbb{S}$ (1.6).

Theorem 3.4. Let ${\mathrm{(H1)}}$ and $\Phi_k\ell_k < 1, (k = 1, 2)$ hold. Then, the fractional $\mathbb{G}$-$\mathbb{C}{\mathrm{S}}\mathbb{S}$ (1.6) is ${\mathrm{U}}$-${\mathrm{H}}$-$\mathbb{S}$.

Proof. According to Theorem 3.2, we have

which yields that

where

Since $\Phi_k \ell_k < 1$ and each geometric sequences $(\Phi_k \ell_k)^n\rightarrow 0,$ hence $\varXi^n\rightarrow 0$ as $n\rightarrow \infty.$ Therefore, due to Theorem 2.6, the fractional $\mathbb{G}$-$\mathbb{C}{\mathrm{S}}\mathbb{S}$ (1.6) is ${\mathrm{U}}$-${\mathrm{H}}-\mathbb{S}$.

4.   Application

We allow here a few illustrations of the fractional $\mathbb{G}$-$\mathbb{C}{\mathrm{S}}\mathbb{S}$, based on numerical recreation to analyze their solutions. In these cases, we consider distinctive cases of the function $\mathbb{G}$ to cover the Caputo, Caputo-Hadamard and Katugampola adaptations.

Example 4.1. Based on the system (1.6), by assuming $\Sigma = [0.05, 0.95]$,

we consider a fractional $\mathbb{C}{\mathrm{S}}\mathbb{S}$ as

for $\mathfrak{t} \in \Sigma$ and

Clearly,

and

Thus, we can rewrite the above system as Eq (1.6). At present, we will have

with $\ell_1 = \frac{5}{36}$ and

with $\ell_2 = \frac{1}{2\sqrt{3}}$. So $\ell = \frac{1}{2\sqrt{3}}$. Now, from (3.6), we consider four cases for $\mathbb{G}$ as:

● $\mathbb{G}_1(\iota) = 2^{\iota}$,

● $\mathbb{G}_2(\iota) = \iota$ (Caputo derivative),

● $\mathbb{G}_3(\iota) = \ln \iota$ (Caputo-Hadamard derivative),

● $\mathbb{G}_4(\iota) = \sqrt{\iota}$ (Katugampola derivative).

Thus,

and

Hence,

and we have

On the other hand, by using equations in (3.7), we get

for $j = 0, 1, 2, 3$ and

for $k = 1, 2$. By employing Eq (3.6), we obtain

and so we can choose

We define the Algorithm 1 for obtaining the values of $\Phi\ell$, $\Delta_i$ and $\varepsilon$, which is shown in the MATLAB commands. One can check numerical results of $\Phi\ell$, $\Delta_i$ and $\varepsilon$ in Tables 1 and 2 for $\iota\in \lbrack 0.05, 0.95]$, and in Figure 1. Accordingly, all requirements of Theorem 3.2 hold, and so the fractional nonlinear couple snap system $(\mathbb{C}{\mathrm{S}}\mathbb{S})$ in the $\mathbb{G}$-Caputo sense $(\mathbb{G}{\mathrm{C}})$ with initial conditions (4.1) has one unique solution on the $\lbrack 0.05, 0.95]$.

5.   Conclusions

In this paper, we defined a new fractional mathematical model of a BVP consisting of a coupled snap equation with integral boundary conditions in the framework of the generalized sequential $\mathbb{G}$-operators, and turned to the investigation of the qualitative behaviors of its solutions, including existence, uniqueness, stability and inclusion version. To confirm the existence criterion, we used the Krasnoselskii theorem, and to confirm the uniqueness criterion, we utilized the Banach theorem. Different kinds of stability criteria were studied based on the standard definitions of these notions. In the final step, we designed examples, and, by assuming different cases for the function $\mathbb{G}$ and order $q$, we obtained numerical results of these two suggested fractional coupled snap systems in some versions, such as Caputo, Caputo-Hadamard and Katugampola.

Conflict of interest

We declare that no competing interests.

Appendix