Open AccessArticle

Superposition and Interaction Dynamics of Complexitons, Breathers, and Rogue Waves in a Landau–Ginzburg–Higgs Model for Drift Cyclotron Waves in Superconductors

Hicham Saber

¹,

Muntasir Suhail

^2,*,

Amer Alsulami

³,

Khaled Aldwoah

^4,*

Alaa Mustafa

⁵

and

Mohammed Hassan

⁶

Department of Mathematics, College of Science, University of Ha’il, Ha’il 55473, Saudi Arabia

Department of Mathematics, College of Science, Qassim University, Buraydah 52571, Saudi Arabia

Department of Mathematics, Turabah University College, Taif University, Taif 21944, Saudi Arabia

⁴

Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia

⁵

Department of Physics, Faculty of Science, Northern Border University, Arar 73213, Saudi Arabia

⁶

Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia

Authors to whom correspondence should be addressed.

Axioms 2024, 13(11), 763; https://doi.org/10.3390/axioms13110763

Submission received: 9 October 2024 / Revised: 28 October 2024 / Accepted: 30 October 2024 / Published: 4 November 2024

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Abstract

This article implements the Hirota bilinear (HB) transformation technique to the Landau–Ginzburg–Higgs (LGH) model to explore the nonlinear evolution behavior of the equation, which describes drift cyclotron waves in superconductivity. Utilizing the Cole–Hopf transform, the HB equation is derived, and symbolic manipulation combined with various auxiliary functions (AFs) are employed to uncover a diverse set of analytical solutions. The study reveals novel results, including multi-wave complexitons, breather waves, rogue waves, periodic lump solutions, and their interaction phenomena. Additionally, a range of traveling wave solutions, such as dark, bright, periodic waves, and kink soliton solutions, are developed using an efficient expansion technique. The nonlinear dynamics of these solutions are illustrated through 3D and contour maps, accompanied by detailed explanations of their physical characteristics.

Keywords:

Landau–Ginzburg–Higgs model; Cole–Hopf transform; Hirota bilinear equation; nonlinear systems; multi-wave complexion

MSC:

35Q55; 35C08; 35B36; 35B40; 35Q60

1. Introduction

Nonlinear evolution equations (NLEEs) are utilized to examine the dynamics of nonlinear physical behaviors. NLEEs have a broad range of practical applications, including engineering [1], physics [2], fluid mechanics [3], and optics [4]. Exact solutions play a crucial role in describing the nonlinear evolution behavior of NLEEs. However, obtaining exact solutions for NLEEs is not always straightforward. Therefore, researchers have developed various analytical methods for deriving exact solutions to NLEEs. Some examples include the F-expansion method [5], the extended tanh-function method [6], the extended direct algebraic method [7], the generalized Kudryashov method [8], and many more [9,10]. The Hirota bilinear method (HB method) is a powerful technique used to find analytical solutions for NLEEs. This method is particularly suitable for extracting multi-soliton solutions to integrable systems, which are a class of nonlinear PDEs with special properties that allow them to be solved exactly [11,12]. In NLEEs, solitons frequently overlap and interact elastically. The HB method can be used to study the collision, superposition, and interaction phenomena of solitons in NLEEs. For instance, Khan et al. [13] studied interaction, bifurcation, lump, and rogue wave solutions of a perturbed generalized KdV equation using the HB technique. The authors in [14] analyzed the interaction phenomena of soliton solutions of the Poisson–Nernst–Planck equation using the HB method. Wang analyzed novel interactions of solitons of NLEEs in shallow water waves [15] with the aid of the HB method. Ma and Li reported transition, bifurcation, and interaction phenomena of a (3 + 1)-dimensional shallow water wave equation [16]. Further notable works on the HB method can be found in [17,18,19,20,21]. The Landau–Ginzburg–Higgs (LGH) equation is a theoretical framework in physics that describes the behavior of certain physical systems within condensed matter physics and field theory. In the context of superconductivity, we consider the LGH equation as follows:

M_{t t} - M_{x x} - a_{1}^{2} M + a_{2}^{2} M^{3} = 0 .

(1)

Here,

M (x, t)

denotes the ion-cyclotron wave for electrostatic potential, and t and x represent the nonlinear temporal and spatial coordinates, respectively, where

a_{1}

and

a_{2}

are non-zero parameters. The LGH equation is a generalization of the Ginzburg–Landau theory, which was initially developed to describe superconductivity, incorporating elements from the Higgs mechanism in particle physics. The LGH equation integrates various concepts from physics and is widely used to describe the dynamics of order parameters in systems undergoing phase transitions, such as the transformation from a disordered to an ordered state, like a liquid becoming a solid. It describes the dynamics of complex scalar fields interacting with gauge fields and has applications across various domains of physics, including condensed matter, cosmology, and high-energy physics. There are several studies on the analysis of the LGH equation in the literature. In [22], the generalized projective Riccati method was used to derive some solitary waves of the LGH equation. The Sardar subequation method was implemented to illustrate some analytical solutions of the LGH equation [23]. The dynamical features and some soliton solutions have been analyzed for the LGH equation in the literature [24]. Further studies can be found in [25,26,27].

The objective of this manuscript is to seek and apply comprehensive, advanced soliton solutions associated with arbitrary parameters to the LGH model using the HB method. This study will also explore the interaction and superposition of various soliton solutions. Additionally, a range of traveling wave solutions, such as dark, bright, periodic waves, and kink soliton solutions, are developed using an efficient expansion technique. The nonlinear dynamics of these solutions are illustrated through 3D and contour maps, accompanied by detailed explanations of their physical characteristics. To the best of our knowledge, the solutions reported in this paper are entirely new and have not been previously studied for the considered LGH equation.

2. Cole–Hopf Transformation and Analytical Solutions

In this part, Cole–Hopf transformation is utilized to construct HB form. Then, via the HB equation, various soliton solutions can be achieved by taking specific auxiliary functions. Let

M = \frac{V_{x}}{V} .

(2)

Considering the following function, we substitute it into the linear components of Equation (1) to derive the constraint condition. When

M = \frac{V_{x}}{V}

is substituted into Equation (1) then one can obtain

\begin{matrix} V^{2} V_{x t t} - V V_{x} V_{t t} - 2 V V_{t} V_{x t} - V^{2} V_{x x x} + 3 V V_{x} V_{x x} \\ + 2 {(V_{t})}^{2} V_{x} + V^{2} (- a_{1}^{2}) V_{x} + a_{2}^{2} {(V_{x})}^{3} - 2 {(V_{x})}^{3} = 0 . \end{matrix}

(3)

Here, we use Equation (3), to obtain different types of soliton solutions for the (1 + 1)-dimensional nonlinear Landau–Ginzburg–Higgs Equation (1).

2.1. The Multi-Waves Complexiton Solutions

Consider the test function

V

as follows:

\begin{matrix} \{\begin{matrix} V = m_{1} e^{p_{1}} + m_{2} e^{- p_{1}} + m_{3} s i n (p_{2}) + m_{4} s i n h (p_{3}), \\ p_{1} = x + s_{1} t, \\ p_{2} = x + s_{2} t, \\ p_{3} = x + s_{3} t, \end{matrix} \end{matrix}

(4)

where

s_{j} (j = 0, 1, 2, 3 \dots,)

and

m_{j} (j = 0, 1, 2, 3, 4)

are real parameters that will be determined subsequently. By substituting Equation (4) into Equation (3) and making the necessary modifications, we obtain

m_{1} = 0, s_{1} = s_{3}, a_{1} = a_{2} = \sqrt{2 (1 - s_{3}^{2})} .

(5)

By using the above result in Equation (4), the multiwave complexiton solution can be derived using the transformation outlined in Equation (2), as follows:

M = \frac{m_{2} e^{s_{3} t + x} + m_{3} cos (s_{2} t + x) + m_{4} cosh (s_{3} t + x)}{m_{2} e^{s_{3} t + x} + m_{3} sin (s_{2} t + x) + m_{4} sinh (s_{3} t + x)} .

(6)

Figure 1 visualizes the interaction between a breather and a kink wave, collectively known as a complexiton wave, within the framework of the proposed model. A breather is a localized wave packet that oscillates in time while preserving its shape, representing localized energy fluctuations in a superconducting state. Similarly, a kink is a topological soliton wave that marks a transition between two different phases in the system, such as a phase boundary or domain walls in superconductors. The complexiton refers to the superposition of these different waveforms, resulting in a composite structure that exhibits both oscillatory and topological features. In the left panel, the 3D subplot shows the interaction of the breather and kink in the

(x, t)

-plane. The depicted surface behavior is characterized by periodic oscillations, indicative of the breather, coupled with a sharp transition, signaling the kink wave. The resulting interaction creates a modulated waveform where the oscillations are affected by the kink, leading to a distorted wave profile that deviates from a pure sinusoidal form. The right panel displays a 2D contour plot, presenting a top-down view of the interaction. The contours represent levels of steady field amplitude, portraying the temporal and spatial variation of the solution. Physically, such complexiton waves in superconductors represent the possible interactions between various excitations or defects in the material. The breather represents localized modes of oscillation, while the kink could represent a phase boundary or domain wall.

2.2. The Multi-Wave Solutions

To obtain the multi-wave solution, we assume the following:

\begin{matrix} \{\begin{matrix} V = m_{1} c o s (p_{1}) + m_{2} c o s h (p_{2}) + m_{3} c o s h (p_{3}) + s_{7}, \\ p_{1} = x + s_{1} t + s_{2}, \\ p_{2} = x + s_{3} t + s_{4}, \\ p_{3} = x + s_{5} t + s_{6}, \end{matrix} \end{matrix}

(7)

where

s_{j} (j = 0, 1, 2, 3 \dots,)

and

m_{j} (j = 0, 1, 2, 3, 4) \in R

, which will be defined later. By switching (7) into (3) and making the necessary modifications, we obtain:

m_{1} = 10, s_{1} = s_{3} = - 1, s_{5} = - 1, a_{1} = a_{2} = \sqrt{2 (1 - s_{3}^{2})} .

(8)

By inserting the above parameters into (8) and using (2), we derive:

M = \frac{m_{3} cosh (s_{6} - t + x) - 10 sin (s_{2} - t + x) + m_{2} sinh (s_{4} - t + x)}{s_{7} + 10 cos (s_{2} - t + x) + m_{2} cosh (s_{4} - t + x) + m_{3} sinh (s_{6} - t + x)},

(9)

the multi-wave solution.

Figure 2 illustrates solution (9), showcasing the interaction of multiple wave solutions within the context of the proposed model. This interaction is crucial for describing the behavior of drift cyclotron waves in superconductors, where the inherent nonlinearity of the system results in complex wave structures. In the left panel of Figure 2, the 3D visualization depicts a surface undergoing interaction of bright and dark solitons with kink between distinct states, represented by various colors. The steep gradients in the central region indicate the presence of bright soliton, where the field variable, potentially the superconducting order parameter, undergoes a change. The right panel provides a 2D surface plot, offering a top-down perspective of the interaction of these waves. These multi-wave kinks in superconductors could represent the interaction of multiple phase boundaries or domain walls within the material. Such interactions are crucial, as they can significantly influence the material’s properties.

2.3. The Breather Wave Solutions

To find the breather-wave solution, we begin by hypothesizing that the test function

V

is:

\begin{matrix} \{\begin{matrix} p_{1} = s_{0} x + s_{1} t + s_{2}, \\ p_{2} = s_{3} x + s_{4} t, \\ V = m_{1} e^{n_{0} p_{1}} + m_{2} c o s (n_{1} p_{2}) + e^{- n_{0} p_{1}}, \end{matrix} \end{matrix}

where

s_{j} (j = 0, 1, 2, 3 \dots,)

and

m_{j} (j = 0, 1, 2, 3, 4)

represent real constants, which will be defined later. By switching (10) into (3) and making the necessary modifications, we obtain:

\begin{matrix} s_{0} = - \frac{a_{1}}{n_{0} a_{2}}, s_{1} = \frac{1}{2} (\frac{\sqrt{4 a_{1}^{2}}}{\sqrt{n_{0}^{2} a_{2}^{2} (2 - a_{2}^{2})}} - \frac{\sqrt{2 a_{1}^{2} a_{2}^{2}}}{\sqrt{n_{0}^{2} (2 - a_{2}^{2})}}) \\ s_{3} = - \frac{\sqrt{2 s_{4}^{2}}}{\sqrt{2 - a_{2}^{2}}}, n_{1} = - \frac{\sqrt{a_{1}^{2} (a_{2}^{2} - 2)}}{\sqrt{2 s_{4}^{2} a_{2}^{2}}} . \end{matrix}

(10)

By switching the aforementioned results into (10) and using (2), we can deduce the breather-wave solution as follows:

\begin{matrix} M & = \frac{- \frac{a_{1} m_{1} exp (n_{0} (\frac{1}{2} t (\frac{2 \sqrt{2} a_{1}}{a_{2} \sqrt{2 - a_{2}^{2}} n_{0}} - \frac{\sqrt{2} a_{1} a_{2}}{\sqrt{2 - a_{2}^{2}} n_{0}}) - \frac{a_{1} x}{a_{2} n_{0}} + s_{2}))}{a_{2}} + \frac{a_{1} exp (- n_{0} (\frac{1}{2} t (\frac{2 \sqrt{2} a_{1}}{a_{2} \sqrt{2 - a_{2}^{2}} n_{0}} - \frac{\sqrt{2} a_{1} a_{2}}{\sqrt{2 - a_{2}^{2}} n_{0}}) - \frac{a_{1} x}{a_{2} n_{0}} + s_{2}))}{a_{2}} + \frac{\sqrt{a_{2}^{2} - 2} a_{1} m_{2} sin (\frac{a_{1} \sqrt{a_{2}^{2} - 2} (s_{4} t - \frac{\sqrt{2} s_{4} x}{\sqrt{2 - a_{2}^{2}}})}{\sqrt{2} a_{2} s_{4}})}{a_{2} \sqrt{2 - a_{2}^{2}}}}{m_{1} exp (n_{0} (\frac{1}{2} t (\frac{2 \sqrt{2} a_{1}}{a_{2} \sqrt{2 - a_{2}^{2}} n_{0}} - \frac{\sqrt{2} a_{1} a_{2}}{\sqrt{2 - a_{2}^{2}} n_{0}}) - \frac{a_{1} x}{a_{2} n_{0}} + s_{2})) + exp (- n_{0} (\frac{1}{2} t (\frac{2 \sqrt{2} a_{1}}{a_{2} \sqrt{2 - a_{2}^{2}} n_{0}} - \frac{\sqrt{2} a_{1} a_{2}}{\sqrt{2 - a_{2}^{2}} n_{0}}) - \frac{a_{1} x}{a_{2} n_{0}} + s_{2})) + m_{2} cos (\frac{a_{1} \sqrt{a_{2}^{2} - 2} (s_{4} t - \frac{\sqrt{2} s_{4} x}{\sqrt{2 - a_{2}^{2}}})}{\sqrt{2} a_{2} s_{4}})} . \end{matrix}

(11)

The parameter values used for the numerical simulations of the precise solution given in Equation (11) are as follows:

m_{1} = 1, m_{2} = - 1, s_{1} = - 1, s_{2} = \sqrt{3}, s_{4} = \sqrt{2}, n_{0} = 2,

a_{1} = 1.1, a_{2} = 4.5

. The dimensional characteristics of the resulting solution are illustrated in Figure 3. This figure illustrates a breather-wave solution within the context of the proposed model. Breathers are a type of localized, non-dispersive wave solution that oscillates in space and time while maintaining its shape. This characteristic makes them valuable in the study of nonlinear wave phenomena in physical systems. In the left panel, the 3D dynamics depict a series of localized peaks that rise and fall over time, showcasing the oscillatory nature of the breather wave. These peaks are more pronounced near the origin and steadily decrease in amplitude as they move away from the center, highlighting the localized nature of the breather. The wave structure is preserved as time evolves without spreading, a distinctive feature of breather waves. The right panel displays a 2D contour plot, providing a top-down view of the breather wave. Breather waves in superconductors can be harnessed for applications that require localized states of energy or specific modes of excitations that do not disperse over time. For example, in quantum computers, preserving coherence over extended periods is crucial, and breather dynamics could potentially be exploited to enhance the stability of quantum states.

2.4. Interaction of a Rouge Wave with Kink Soliton

By representing

V

as a function.

\begin{matrix} \{\begin{matrix} V = p_{1}^{2} + p_{2}^{2} + s_{9} + c_{0} e^{p_{3}}, \\ p_{1} = s_{1} x + s_{2} t + s_{3}, \\ p_{2} = s_{4} x + s_{5} t + s_{6}, \\ p_{3} = s_{7} x + s_{8} t, \end{matrix} \end{matrix}

(12)

we investigate the interaction between a rogue wave and a kink soliton. Let

p_{w}, (j = 1, 2, \dots, 9)

, and

c_{0}

represent 10 real coefficients, which will be defined later. By substituting Equation (12) into Equation (3) and making the necessary modifications, we obtain the following results. Using Mathematica 13.0 software, we determine the values of the unknown parameters as follows:

\begin{matrix} s_{1} = \frac{\sqrt{2} s_{5}}{\sqrt{- 2 + s_{6}^{2}}}, s_{2} = \frac{2 s_{5}}{\sqrt{2 - s_{6}^{2}} \sqrt{- 2 + s_{6}^{2}}} - \frac{s_{5} s_{6}^{2}}{\sqrt{2 - s_{6}^{2}} \sqrt{- 2 + s_{6}^{2}}}, \\ s_{3} = \frac{- (2 \sqrt{2}) / \sqrt{- 2 + s_{6}^{2}} + (\sqrt{2} s_{6}^{2}) / \sqrt{- 2 + s_{6}^{2}} + \frac{8 s_{5} s_{6}}{\sqrt{2 - s_{6}^{2}} \sqrt{- 2 + s_{6}^{2}}} - \frac{4 s_{5} s_{6} s_{6}^{2}}{\sqrt{2 - s_{6}^{2}} \sqrt{- 2 + s_{6}^{2}}}}{4 s_{5}}, \\ s_{4} = \frac{\sqrt{2} s_{5}}{\sqrt{2 - s_{6}^{2}}}, c_{0} = - 1 \end{matrix}

(13)

In every direction, the solutions will be confined to regions where the parameters meet the condition

c_{0} > 0 .

By substituting Equation (13) into Equation (12) and then using Equation (2), we derive the precise solution as follows:

\begin{matrix} M = \frac{2 (\frac{\sqrt{2} s_{5}}{\sqrt{- 2 + s_{6}^{2}}} x + s_{2} t + s_{3}) \frac{\sqrt{2} s_{5}}{\sqrt{- 2 + s_{6}^{2}}} + 2 (\frac{\sqrt{2} s_{5}}{\sqrt{2 - s_{6}^{2}}} x + s_{5} t + s_{6}) \frac{\sqrt{2} s_{5}}{\sqrt{2 - s_{6}^{2}}} - s_{7} e^{s_{7} x + s_{8} t}}{{(\frac{\sqrt{2} s_{5}}{\sqrt{- 2 + s_{6}^{2}}} x + s_{2} t + s_{3})}^{2} + {(\frac{\sqrt{2} s_{5}}{\sqrt{2 - s_{6}^{2}}} x + s_{5} t + s_{6})}^{2} + s_{9} - e^{s_{7} x + s_{8} t}} . \end{matrix}

(14)

where

s_{2}

and

s_{3}

are available in Equation (13). The values of the parameters employed in the numerical simulations of the exact solution, as described by Equation (14), are chosen as follows

s_{5} = 1, s_{6} = 1, s_{7} = 1, s_{8} = 1, s_{9} = 1

. The dimensional characteristics of the derived solution are illustrated in Figure 4.

This figure illustrates the interaction of a rogue wave and a kink wave. This interaction is particularly interesting because it reveals the complex behavior that arises due to the interplay and superposition of different waves in nonlinear systems. In the left panel (a), a 3D visualization is presented, where the x-axis denotes the spatial dimension x, the y-axis denotes time t, and the z-axis represents the wave amplitude. The rogue wave, characterized by its localized nature and significant amplitude, is evident as a peak-like structure standing out from the surrounding waveforms. The kink is visible in the simulated figure as an elongated characteristic along the x-axis, suggesting a continuous but sharp transition in amplitude. The interaction between the rogue wave and the kink wave occurs where the peak of the rogue wave intersects with the kink wave. Panel (b) displays the contour behavior more clearly, with the color gradient representing the wave amplitude.

2.5. Lump-2-Kinks Interaction

Here, we obtain the lump–kink solution by assuming that

V

is a combination of hyperbolic and exponential functions, as follows:

\begin{matrix} \{\begin{matrix} p_{1} = x + A_{1} t, \\ p_{2} = x + A_{2} t, \\ p_{3} = x + A_{3} t, \\ p_{w} = x + A_{w} t, w = 1, 2, 3 \\ V = 1 + sech (p_{1}) + e^{p_{2}} + e^{p_{3}} . \end{matrix} \end{matrix}

(15)

Here,

A_{1}, A_{2}, A_{3}

are the real constants to be determined. Substituting Equation (15) into Equation (3), and comparing the coefficients of hyperbolic and exponential functions, we determine the values of the unknown as follows.

\begin{matrix} A_{1} = \frac{3 - 2 a_{1}^{2} - A_{3}^{2}}{2 A_{3}}, A_{2} = \frac{9 - 12 a_{1}^{2} + 4 a_{1}^{4} - 12 A_{3}^{2} - A_{3}^{4}}{2 A_{3} (- 3 + 2 a_{1}^{2} + A_{3}^{2}} . \end{matrix}

(16)

The solutions will be localized in all directions when the parameters meet the condition

A_{3} > 0 .

After substituting Equation (15) using Equation (16), into Equation (2), we obtain the exact solution as

\begin{matrix} M = \frac{sech (x + \frac{3 - 2 a_{1}^{2} - A_{3}^{2}}{2 A_{3}} t) \tanh (x + \frac{3 - 2 a_{1}^{2} - A_{3}^{2}}{2 A_{3}} t) + e^{x + \frac{9 - 12 a_{1}^{2} + 4 a_{1}^{4} - 12 A_{3}^{2} - A_{3}^{4}}{2 A_{3} (- 3 + 2 a_{1}^{2} + A_{3}^{2})} t} + e^{x + A_{3} t}}{1 + sech (x + \frac{3 - 2 a_{1}^{2} - A_{3}^{2}}{2 A_{3}} t) + e^{x + \frac{9 - 12 a_{1}^{2} + 4 a_{1}^{4} - 12 A_{3}^{2} - A_{3}^{4}}{2 A_{3} (- 3 + 2 a_{2}^{2} + A_{3}^{2})} t} + e^{x + A_{3} t}} . \end{matrix}

(17)

The surface and contour behaviors of the analytical solution (17), showcasing two kinks and a single lump wave, are visualized in Figure 5. The left panel (a) presents a 3D view of the interaction, where the spatial dimension x, time t, and wave amplitude are represented on the axes. The lump wave is identifiable due to its localized, solitary peak that diminishes rapidly away from the center, appearing as a distinct peak. The interaction between the lump and the kink waves is visible where the lump intersects with the kink wave, leading to a localized deformation in the kink. This interaction demonstrates how concentrated energy, represented by the lump, can modify and temporarily alter the shape of the kink wave, highlighting the interplay between localized and extended waveforms in such systems. The right panel (b) displays a 2D contour plot, visualizing the distribution of wave amplitudes over time and space. The lump wave appears as a concentrated, bright, and dark region, while the kink wave is represented as a continuous transition across the spatial coordinate. In superconductors, such interactions could have significant implications. Lump waves, representing localized energy, might introduce temporary fluctuations in the superconducting states. When lumps interact with kinks, they could potentially represent transitions within the superconducting material.

3. The Traveling Wave Solutions

The leading steps of the improved

\frac{Φ^{'}}{Φ^{'} + Φ + υ}

-expansion method are presented in this section. We begin by applying the following transformation to derive the traveling wave solutions:

M (x, t) = M (μ), μ = x - w t .

(18)

Inserting Equation (18), into Equation (1), then:

- (1 - w^{2}) M^{″} - a_{1}^{2} M + a_{2}^{2} M^{3} = 0 .

(19)

Using the improved

\frac{Φ^{'}}{Φ^{'} + Φ + υ}

-expansion approach, we can assume that the solution to Equation (19) is

M = \sum_{w = 0}^{r} j_{w} {(\frac{Φ^{'}}{Φ^{'} + Φ + υ})}^{w} + \sum_{p = 1}^{r} k_{p} {(\frac{Φ^{'}}{Φ^{'} + Φ + υ})}^{- p},

(20)

where

j_{w}

are the coefficients of the polynomial

{(\frac{Φ^{'}}{Φ^{'} + Φ + υ})}^{w}, w = 0, 1, 2 \dots \dots r,

and

{(\frac{Φ^{'}}{Φ^{'} + Φ + υ})}^{- p}, p = 1, 2 \dots \dots r .

Suppose

\frac{Φ^{'}}{Φ^{'} + Φ + υ}

is a function that satisfies the given equation.

Φ^{″} + Φ^{'} a_{0} + h_{1} Φ + h_{1} p_{0} = 0 .

(21)

Now, by balancing

M^{″}

and

M^{3}

in (19), the value of

r = 1,

in (20) examined thoroughly and individually, can be identified as:

M = j_{0} + j_{1} (\frac{Φ^{'}}{Φ^{'} + Φ + υ}) + \sum_{p = 1}^{r} k_{1} {(\frac{Φ^{'}}{Φ^{'} + Φ + υ})}^{- 1} .

(22)

Now, replace (22) with (21) in (19) and set the coefficients of the various powers of

Φ^{'}

to zero. This yields:

Set: 1

j_{0} = - \frac{a_{1} (a_{0} - 2 h_{1})}{\sqrt{a_{2}^{2} u}}, j_{1} = \frac{2 a_{1} (a_{0} - h_{1} - 1)}{a_{2} \sqrt{u}}, k_{1} = 0, s_{1} = \frac{\sqrt{u - 2 a_{1}^{2}}}{\sqrt{u}},

(23)

Set: 2

j_{0} = - \frac{a_{1} (a_{0} - 2 h_{1})}{\sqrt{a_{2}^{2} u}}, j_{1} = 0, k_{1} = \frac{2 a_{1} (a_{0} - h_{1} - 1)}{u 2 \sqrt{u}}, s_{1} = \frac{\sqrt{u - 2 a_{1}^{2}}}{\sqrt{u}},

(24)

To find the traveling wave solutions of (1) for the above set of solutions, we consider the following cases:

Case: 1 For the set

j_{0} = - \frac{a_{1} (a_{0} - 2 h_{1})}{\sqrt{a_{2}^{2} u}}, j_{1} = \frac{2 a_{1} (a_{0} - h_{1} - 1)}{a_{2} \sqrt{u}}, k_{1} = 0, s_{1} = \frac{\sqrt{u - 2 a_{1}^{2}}}{\sqrt{u}},

u = a_{0}^{2} - 4 h_{1} > 0,

we obtain the following solution to Equation (1)

\begin{matrix} M = - \frac{a_{1} (a_{0} - 2 h_{1})}{\sqrt{a_{2}^{2} u}} + \frac{(2 a_{1} (a_{0} - h_{1} - 1))}{\sqrt{u a_{2}^{2}}} (\frac{l_{1} (\sqrt{u} + a_{0}) + l_{2} (a_{0} - \sqrt{u}) e^{μ \sqrt{u}}}{l_{1} (\sqrt{u} + a_{0} - 2) + l_{2} (- \sqrt{u} + a_{0} - 2) e^{μ \sqrt{u}}}), \end{matrix}

(25)

Case: 2 For the set

j_{0} = - \frac{a_{1} (a_{0} - 2 h_{1})}{\sqrt{a_{2}^{2} u}}, j_{1} = \frac{2 a_{1} (a_{0} - h_{1} - 1)}{a_{2} \sqrt{u}}, k_{1} = 0, s_{1} = \frac{\sqrt{u - 2 a_{1}^{2}}}{\sqrt{u}},

u = a_{0}^{2} - 4 h_{1} < 0,

Now, let us consider the following solution to Equation (1) in detail.

\begin{matrix} M = - \frac{a_{1} (a_{0} - 2 h_{1})}{\sqrt{a_{2}^{2} (u)}} + \frac{2 a_{1} (a_{0} - h_{1} - 1)}{u 2 \sqrt{u}} (\frac{\begin{matrix} (l_{1} \sqrt{- (u)} + a_{0}) s i n (\frac{1}{2} μ \sqrt{- (u)}) \\ + c o s (\frac{1}{2} μ \sqrt{- (u)}) (a_{0} l_{1} - l_{2} \sqrt{- (u)}) \end{matrix}}{\begin{matrix} c o s (\frac{1}{2} μ \sqrt{- (u)}) (l_{2} \sqrt{- (u)} + (a_{0} - 2) l_{1}) \\ + s i n (\frac{1}{2} μ \sqrt{- (u)}) (l_{1} \sqrt{- (u)} + (a_{0} - 2) l_{2}) \end{matrix}}) . \end{matrix}

(26)

Figure 6 illustrates the dynamics of the analytical solution (25). The two subplots provide a detailed simulation of the periodic wave dynamics. In panel (a), the 3D surface depicts the wave amplitude as a function of (x) and (t). The figure showcases sharp, regularly spaced peaks and troughs, indicating a highly structured periodic wave structure. This periodicity suggests that the system exhibits regular oscillatory dynamics, a characteristic feature of wave dynamics in superconductors. The variation in the heights of the peaks may represent amplitude modulation effects resulting from the superposition of various wave modes. In panel (b), the surface plot of the wave solution in the x-t plane is presented, further clarifying the wave’s behavior.

In Figure 7a, the 3D behavior illustrates the amplitude of the soliton as a function of space (x) and time (t). The figure demonstrates that the soliton’s amplitude reaches its highest point at the peak and then rapidly decays to nearly zero in the surrounding regions. This behavior highlights the stability and robustness of the solitons, as they propagate through space and time without experiencing dispersion or spreading. Figure 7b displays a surface plot that captures the soliton’s behavior in the x-t plane. The diagonal stripe represents the soliton’s trajectory. In drift cyclotron waves within superconductors, the appearance of stable, localized wave packets, such as solitons, is possible when nonlinear effects and dispersion balance each other.

For the Set: 2

j_{0} = - \frac{a_{1} (a_{0} - 2 h_{1})}{\sqrt{a_{2}^{2} u}}, j_{1} = 0, k_{1} = \frac{2 a_{1} (a_{0} - h_{1} - 1)}{u 2 \sqrt{u}}, s_{1} = \frac{\sqrt{u - 2 a_{1}^{2}}}{\sqrt{u}},

(27)

we have the following cases.

Case: 1 If

u = a_{0}^{2} - 4 h_{1} > 0,

and

j_{0} = - \frac{a_{1} (a_{0} - 2 h_{1})}{\sqrt{a_{2}^{2} u}}, j_{1} = 0, k_{1} = \frac{2 a_{1} (a_{0} - h_{1} - 1)}{u 2 \sqrt{u}}, s_{1} = \frac{\sqrt{u - 2 a_{1}^{2}}}{\sqrt{u}},

(28)

then, we have the following solution to Equation (1)

M = - \frac{a_{1} (a_{0} - 2 h_{1})}{\sqrt{u a_{2}^{2}}} + \frac{2 a_{1} (a_{0} - h_{1} - 1)}{u 2 \sqrt{u} (\frac{(\sqrt{u} a_{2}) (l_{2} e^{μ \sqrt{u}} (a_{0} - \sqrt{u}) + l_{1} (a_{0} + \sqrt{u}))}{l_{2} e^{μ \sqrt{u}} (a_{0} - \sqrt{u} - 2) + l_{1} (a_{0} + \sqrt{u} - 2)})} .

(29)

Case: 2 If

u = a_{0}^{2} - 4 h_{1} < 0,

and

j_{0} = - \frac{a_{1} (a_{0} - 2 h_{1})}{\sqrt{a_{2}^{2} u}}, j_{1} = 0, k_{1} = \frac{2 a_{1} (a_{0} - h_{1} - 1)}{u 2 \sqrt{u}}, s_{1} = \frac{\sqrt{u - 2 a_{1}^{2}}}{\sqrt{u}},

(30)

then, we have the following solution to Equation (1)

M = - \frac{a_{1} (a_{0} - 2 h_{1})}{\sqrt{u a_{2}^{2}}} + \frac{2 a_{1} (a_{0} - h_{1} - 1)}{u 2 \sqrt{u} (\frac{(l_{1} \sqrt{- (u)} + a_{0}) s i n (\frac{1}{2} μ \sqrt{- (u)}) + c o s (\frac{1}{2} μ \sqrt{- (u)}) (a_{0} l_{1} - l_{2} \sqrt{- (u)})}{c o s (\frac{1}{2} μ \sqrt{- (u)}) (l_{2} \sqrt{- (u)} + (a_{0} - 2) l_{1}) + s i n (\frac{1}{2} μ \sqrt{- (u)}) (l_{1} \sqrt{- (u)} + (a_{0} - 2) l_{2})})} .

(31)

Figure 8 depicts the dynamics of the exact solution (29), showcasing a kink wave as described earlier. In panel (a), the 3D plot illustrates the amplitude of the kink wave as a function of space (x) and time (t). Panel (b) provides a surface plot that captures the kink wave’s propagation in the x-t plane.

Figure 9 depicts the dynamics of the exact solution (31), showcasing a dark soliton wave. In panel (a), the 3D plot illustrates the amplitude of the dark soliton as a function of x and t, revealing that the wave’s amplitude is lower than its surroundings. Panel (b) provides a surface plot that captures the dark soliton’s propagation in the x-t plane.

4. Conclusions

This study successfully employs the HB transformation technique to explore the nonlinear evolution behavior of the LGH model, which describes drift cyclotron waves in superconductors. Through the Cole–Hopf transformation, several wave solutions, including complexitons, breathers, rogue waves, and periodic lumps, among others, are systematically derived and analyzed. The interactions of different waveforms, such as breathers and kinks, and lumps with kinks, are particularly noteworthy. These interactions are visualized through 3D and 2D contour plots, revealing the complex behavior that emerges within the nonlinear LGH model. The results provide insights into the oscillatory and topological features characterizing these interactions, underscoring their importance in understanding the behavior of superconductors. Furthermore, the study highlights the potential implications of these waves, especially in the field of superconductivity. The stable and localized nature of breather waves, for instance, holds promise for applications that require localized energy states, such as quantum computing. Similarly, the interactions between rogue waves and kink solitons, as well as lump–kink interactions, demonstrate the rich dynamics that can influence the properties of superconducting materials. In summary, the solutions and interactions presented in this work not only advance the theoretical understanding of nonlinear waves in superconductors, but also pave the way for future research, including the development of advanced superconducting technologies. The novel results presented herein, including the discovery of new exact solutions, contribute significantly to the ongoing exploration of nonlinear evolution equations and their applications in various physical systems. In the future, this work can be further extended by considering the coefficients in the proposed model as variable coefficients, as given in the literature [28,29,30].

Author Contributions

Conceptualization, M.H.; Software, A.M.; Formal analysis, A.A.; Investigation, A.A.; Resources, M.H.; Writing—original draft, M.S.; Writing – review & editing, H.S. and K.A.; Project administration, K.A. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

All data are included in this paper.

Acknowledgments

The Researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2024-9/1).

Conflicts of Interest

The authors claim no conflict of interest.

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Figure 1. The visualization of solution (6) with assumed parameters

m_{2} = 2, m_{3} = - 2, m_{4} = - 2,

s_{2} = 3,

s_{3} = 2

. (a) 3D behavior in spatial and temporal coordinates. (b) Contour plot in 2D.

Figure 1. The visualization of solution (6) with assumed parameters

m_{2} = 2, m_{3} = - 2, m_{4} = - 2,

s_{2} = 3,

s_{3} = 2

. (a) 3D behavior in spatial and temporal coordinates. (b) Contour plot in 2D.

Figure 2. The visualization of solution (9) with assumed parameters

m_{2} = 4, m_{3} = 1, s_{1} = - 1,

s_{2} = 1, s_{3} = 3, s_{4} = 1, s_{5} = - 1, s_{6} = 1 .

(a) 3D behavior in spatial and temporal coordinates. (b) Contour plot in 2D.

Figure 2. The visualization of solution (9) with assumed parameters

m_{2} = 4, m_{3} = 1, s_{1} = - 1,

s_{2} = 1, s_{3} = 3, s_{4} = 1, s_{5} = - 1, s_{6} = 1 .

(a) 3D behavior in spatial and temporal coordinates. (b) Contour plot in 2D.

Figure 3. The visualization of solution (11) with the parameters

m_{1} = 0.2, m_{2} = - 1, s_{1} = - 1,

s_{2} = \sqrt{3}, s_{4} = \sqrt{2}, n_{0} = 2, a_{1} = 1.1, a_{2} = 5.18 .

(a) 3D behavior in spatial and temporal coordinates. (b) Contour plot in 2D.

Figure 3. The visualization of solution (11) with the parameters

m_{1} = 0.2, m_{2} = - 1, s_{1} = - 1,

s_{2} = \sqrt{3}, s_{4} = \sqrt{2}, n_{0} = 2, a_{1} = 1.1, a_{2} = 5.18 .

(a) 3D behavior in spatial and temporal coordinates. (b) Contour plot in 2D.

Figure 4. The visualization of solution (14) with parameters,

s_{5} = 1, s_{6} = 1, s_{7} = 1, s_{8} = 1, s_{9} = 1

. (a) 3D behavior in spatial and temporal coordinates. (b) Contour plot in 2D.

Figure 4. The visualization of solution (14) with parameters,

s_{5} = 1, s_{6} = 1, s_{7} = 1, s_{8} = 1, s_{9} = 1

. (a) 3D behavior in spatial and temporal coordinates. (b) Contour plot in 2D.

Figure 5. The visualization of solution (17) with parameters,

a_{1} = 3, a_{2} = 2, A_{3} = 2

. (a) 3D behavior in spatial and temporal coordinates. (b) Contour plot in 2D.

Figure 5. The visualization of solution (17) with parameters,

a_{1} = 3, a_{2} = 2, A_{3} = 2

. (a) 3D behavior in spatial and temporal coordinates. (b) Contour plot in 2D.

Figure 6. The visualization of solution (25) with parameters,

a_{0}

= 0.3,

h_{1}

= 0.2,

l_{1}

= −2,

l_{2}

= 1,

a_{1}

= 0.3,

a_{2}

= 3. (a) 3D behavior in spatial and temporal coordinates. (b) Contour plot in 2D.

Figure 6. The visualization of solution (25) with parameters,

a_{0}

= 0.3,

h_{1}

= 0.2,

l_{1}

= −2,

l_{2}

= 1,

a_{1}

= 0.3,

a_{2}

= 3. (a) 3D behavior in spatial and temporal coordinates. (b) Contour plot in 2D.

Figure 7. The visualization of solution (26) with parameters,

a_{0}

= −6.5,

h_{1}

= 2,

l_{1}

= −2,

l_{2}

= 1,

a_{1}

= 3,

a_{2}

= 0.3. (a) 3D behavior in spatial and temporal coordinates. (b) Contour plot in 2D.

Figure 7. The visualization of solution (26) with parameters,

a_{0}

= −6.5,

h_{1}

= 2,

l_{1}

= −2,

l_{2}

= 1,

a_{1}

= 3,

a_{2}

= 0.3. (a) 3D behavior in spatial and temporal coordinates. (b) Contour plot in 2D.

Figure 8. The visualization of solution (29) with parameters,

a_{0} = 4.3, h_{1} = 2, l_{1} = - 2, l_{2} = 1,

a_{1}

= 0.3,

a_{2} = 3 .

(a) 3D behavior in spatial and temporal coordinates. (b) Contour plot in 2D.

Figure 8. The visualization of solution (29) with parameters,

a_{0} = 4.3, h_{1} = 2, l_{1} = - 2, l_{2} = 1,

a_{1}

= 0.3,

a_{2} = 3 .

(a) 3D behavior in spatial and temporal coordinates. (b) Contour plot in 2D.

Figure 9. The visualization of solution (31) with the parameters

a_{0} = 6.5, h_{1} = 2, l_{1}

= 0.5,

l_{2} = \sqrt{3},

a_{1}

= 0.1,

a_{2}

= 0.1. (a) 3D behavior in spatial and temporal coordinates. (b) Contour plot in 2D.

Figure 9. The visualization of solution (31) with the parameters

a_{0} = 6.5, h_{1} = 2, l_{1}

= 0.5,

l_{2} = \sqrt{3},

a_{1}

= 0.1,

a_{2}

= 0.1. (a) 3D behavior in spatial and temporal coordinates. (b) Contour plot in 2D.

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Saber, H.; Suhail, M.; Alsulami, A.; Aldwoah, K.; Mustafa, A.; Hassan, M. Superposition and Interaction Dynamics of Complexitons, Breathers, and Rogue Waves in a Landau–Ginzburg–Higgs Model for Drift Cyclotron Waves in Superconductors. Axioms 2024, 13, 763. https://doi.org/10.3390/axioms13110763

AMA Style

Saber H, Suhail M, Alsulami A, Aldwoah K, Mustafa A, Hassan M. Superposition and Interaction Dynamics of Complexitons, Breathers, and Rogue Waves in a Landau–Ginzburg–Higgs Model for Drift Cyclotron Waves in Superconductors. Axioms. 2024; 13(11):763. https://doi.org/10.3390/axioms13110763

Chicago/Turabian Style

Saber, Hicham, Muntasir Suhail, Amer Alsulami, Khaled Aldwoah, Alaa Mustafa, and Mohammed Hassan. 2024. "Superposition and Interaction Dynamics of Complexitons, Breathers, and Rogue Waves in a Landau–Ginzburg–Higgs Model for Drift Cyclotron Waves in Superconductors" Axioms 13, no. 11: 763. https://doi.org/10.3390/axioms13110763

APA Style

Saber, H., Suhail, M., Alsulami, A., Aldwoah, K., Mustafa, A., & Hassan, M. (2024). Superposition and Interaction Dynamics of Complexitons, Breathers, and Rogue Waves in a Landau–Ginzburg–Higgs Model for Drift Cyclotron Waves in Superconductors. Axioms, 13(11), 763. https://doi.org/10.3390/axioms13110763

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Superposition and Interaction Dynamics of Complexitons, Breathers, and Rogue Waves in a Landau–Ginzburg–Higgs Model for Drift Cyclotron Waves in Superconductors

Abstract

1. Introduction

2. Cole–Hopf Transformation and Analytical Solutions

2.1. The Multi-Waves Complexiton Solutions

2.2. The Multi-Wave Solutions

2.3. The Breather Wave Solutions

2.4. Interaction of a Rouge Wave with Kink Soliton

2.5. Lump-2-Kinks Interaction

3. The Traveling Wave Solutions

4. Conclusions

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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