Theoretical study of nonlinear magnetosonic waves: Potential applications in plasma diagnostics

Abstract

Nonlinear magnetosonic waves (MSWs) in electron-ion plasmas play a critical role in plasma dynamics, with implications for energy transfer, particle acceleration, and astrophysical phenomena. In this study, deriving a generalized nonlinear Schrödinger equation (NLSE) to describe weakly nonlinear magnetosonic wave packets, accounting for the interplay between magnetic pressure and plasma thermal pressure. Unlike previous works limited to linear or Korteweg−de Vries (KdV) type models, present paper reveals the conditions for the existence of various nonlinear wave structures, including Kuznetsov−Ma (K−M) breather soliton, Akhmediev breather (AB) soliton, bright soliton (BS), and rogue wave (RW). A detailed analysis shows how these waveforms depend on plasma parameters such as external magnetic field strength, electron and ion temperatures, and wave number. Additionally, this work explores a potential application of these findings, using multiple breather solutions as a diagnostic tool to infer challenging to measure plasma parameters from easily observable quantities. The present results not only extend the theoretical understanding of magnetosonic waves but also suggest practical approaches for future plasma diagnostics, providing a foundation for research in controlled fusion, space physics, and astrophysical plasma studies.

Graphical abstract

Keywords

magnetized plasmas / nonlinear phenomena / magnetohydrodynamic waves / perturbative methods / plasma diagnostic techniques

1 Introduction

Plasma is found in phenomena like auroras and stars [1, 2] and in systems like fusion reactors [3, 4], exhibits high conductivity, electromagnetic sensitivity, and nonlinear dynamics, making it vital in nuclear fusion [5, 6], astrophysics [1, 2], material science [7] and other fields [8–10]. Wave phenomena are fundamental in plasma physics, with applications in energy transfer [11], particle acceleration [12, 13], and radiation generation [14, 15]. Plasma waves are categorized by frequency and wavelength into low-frequency waves, such as Alfvén and magnetosonic waves, and high-frequency waves, such as Langmuir waves. Depending on amplitude, these waves may exhibit linear or nonlinear behavior, including solitary waves and shock waves [1, 2]. Understanding their propagation and evolution is crucial for elucidating fundamental plasma processes in laboratory and astrophysical contexts.

Magnetosonic waves play a crucial role in plasma dynamics. When thermal pressure dominates, magnetosonic waves resemble acoustic waves, whereas in strongly magnetized plasmas, their properties approach those of Alfvén wave [1, 2]. These low-frequency modes have been extensively studied in both linear wave theory and weakly nonlinear regimes described by Korteweg−de Vries (KdV)-type equations [16–22]. Furthermore, the nonlinear Schrödinger equation (NLSE) has been widely used to describe nonlinear phenomena, as shown in numerous studies [23–29], particularly in the investigation of excitation and physical mechanisms of various nonlinear localized waves, such as Kuznetsov−Ma (K−M) breather [30–32], Akhmediev breather (AB) [33], bright solitons (BS) [34–36], and rogue waves (RW) [37, 38]. It is generally accepted that, under a plane wave background, the modulation instability (MI) of the system can excite K−M solitons with spatial localization and temporal periodicity, AB with spatial periodicity and temporal localization, and RW with both spatial and temporal localization [39–42].

Recently, scholars have explored various methods to investigate K−M solitons [39, 43–46], AB [39, 46], higher-order RW [47], bright and dark solitons [48–55], and other optical soliton solutions [48–51, 55–58] based on the NLSE. These findings not only provide important theoretical support for nonlinear optics, optical communication systems, plasma physics, fluid dynamics, and mathematical physics, but also offer deeper insights into the complex behaviors of nonlinear wave systems. However, the exploration of fully nonlinear magnetosonic waves, particularly under the influence of MI, remains relatively limited.

In this work, we extend the understanding of nonlinear magnetosonic waves by deriving a generalized NLSE and various nonlinear solutions (K−M, AB, BS, RW) for weakly nonlinear magnetosonic wave packets in electron-ion plasmas. We propose an innovative diagnostic application of these nonlinear wave solutions. By exciting multiple breather modes and analyzing measurable properties such as wave velocities, amplitudes, and widths, plasma parameters which are usually difficulty to be measured directly can be inferred. This diagnostic methodology offers a powerful tool for advancing plasma research, with broad applications in controlled fusion, space physics, and astrophysical studies.

The manuscript is organized as follows. Section 2 presents the basic Magnetohydrodynamic (MHD) equations for magnetosonic waves in plasma and explains their scalar form. Section 3 derives the KdV equation through a reductive perturbation method. Section 4 derives the NLSE from the KdV equation. Section 5 explores the MI of magnetosonic waves, while Section 6 discusses the nonlinear wave solutions and their dependence on plasma parameters. Section 7 and Section 8 discusses and summarizes the implications of the results for plasma diagnostics and nonlinear wave dynamics.

2 Model

We consider a non-relativistic, collisionless, magnetized electron-ion plasma system. To simplify the analysis, the plasma is assumed to satisfy the charge neutrality condition n_{e0}=n_{i0}=n_0, where n_{i0} and n_{e0} represent the number densities of electrons and ions at equilibrium state, respectively. The plasma dynamics are governed by the two-fluid MHD equations coupled with Maxwell’s equations

(1)\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{n}_i}{\mathrm{\partial }t}}+\mathrm{\nabla }\cdot (n_i{\mathit{v}}_{\mathit{i}})=0,\\ {\displaystyle \frac{\mathrm{\partial }{\mathit{v}}_{\mathit{i}}}{\mathrm{\partial }t}}+({\mathit{v}}_{\mathit{i}}\cdot \mathrm{\nabla }){\mathit{v}}_{\mathit{i}}={\displaystyle \frac{e}{m_i}}(\mathit{E}+{\mathit{v}}_{\mathit{i}}\times \mathit{B})-{\displaystyle \frac{1}{m_i{n}_i}}\mathrm{\nabla }{p}_i,\\ {\displaystyle \frac{\mathrm{\partial }{n}_e}{\mathrm{\partial }t}}+\mathrm{\nabla }\cdot (n_e{\mathit{v}}_e)=0,\\ {\displaystyle \frac{\mathrm{\partial }{\mathit{v}}_e}{\mathrm{\partial }t}}+({\mathit{v}}_e\cdot \mathrm{\nabla }){\mathit{v}}_e=-{\displaystyle \frac{e}{m_e}}(\mathit{E}+{\mathit{v}}_e\times \mathit{B})-{\displaystyle \frac{1}{m_e{n}_e}}\mathrm{\nabla }{p}_e,\\ \mathrm{\nabla }\times \mathit{E}=-{\displaystyle \frac{\mathrm{\partial }\mathit{B}}{\mathrm{\partial }t}},\\ \mathrm{\nabla }\times \mathit{B}={\mu }_0\mathit{J},\end{array}

where \mathit{J}=e( n_i{\mathit{v}}_i-n_e{\mathit{v}}_e), m_{\alpha }, n_{\alpha }, {\mathit{v}}_{\alpha } and p_{\alpha } are the mass, number density, velocity and the pressure of the particles \alpha, p_{\alpha }=n_{\alpha }{k}_B{T}_{\alpha } is the pressure of the particles \alpha, T_{\alpha } is the temperatures of the particle \alpha, k_B is the Boltzmann constant. \mathrm{\nabla }{p}_{\alpha }=k_B{T}_B\mathrm{\nabla } n_{\alpha }, \alpha is i or e representing the ion fluid or electron fluid. \mathit{E} and \mathit{B} are the electric field and the magnetic field, {\mu }_0 is the permeability of vacuum.

For simplicity and generality, we consider a one-dimensional wave propagating perpendicular to the external magnetic field \mathit{B}=B_0\hat{k}. The wave vector is aligned along the x-direction, \mathit{k}=k\hat{i}. The electric field is restricted to the XY-plane, \mathit{E}=E_x \hat{i}+E_y\hat{j}, where \hat{i},\hat{j} and \hat{k} are the unit vector in the positive x, y and z directions respectively. The magnetic field perturbations are confined to the z-direction, \mathit{B}=B_0 \hat{k}+\delta B_z\hat{k}, where \delta B_z is the perturbed magnetic field due to the perturbations (\delta B_z\ll B_0). Under these assumptions, the above set of dynamic equations can be written in the component form as follows:

The z components of Faraday’s law and the x and y components of Ampere’s law are

(2)\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{E}_y}{\mathrm{\partial }x}}=-{\displaystyle \frac{\mathrm{\partial }{B}_z}{\mathrm{\partial }t}},\\ 0=-{\mu }_{0}e(n_i{v}_{ix}-n_e{v}_{ex}),\\ {\displaystyle \frac{\mathrm{\partial }{B}_z}{\mathrm{\partial }x}}={\mu }_{0}e(n_e{v}_{ey}-n_i{v}_{iy}),\end{array}

while the x and y components of continuity equations and the momentum equations for ions and electrons are

(3)\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{n}_i}{\mathrm{\partial }t}}+{\displaystyle \frac{\mathrm{\partial }(n_i{v}_{ix})}{\mathrm{\partial }x=0}},\\ {\displaystyle \frac{\mathrm{\partial }{v}_{ix}}{\mathrm{\partial }t}}+v_{ix}{\displaystyle \frac{\mathrm{\partial }{v}_{ix}}{\mathrm{\partial }x}}={\displaystyle \frac{e}{m_i}}(E_x+v_{iy}{B}_z)-{\displaystyle \frac{k_B{T}_i}{m_i{n}_i}}{\displaystyle \frac{\mathrm{\partial }{n}_i}{\mathrm{\partial }x}},\\ {\displaystyle \frac{\mathrm{\partial }{v}_{iy}}{\mathrm{\partial }t}}+v_{ix}{\displaystyle \frac{\mathrm{\partial }{v}_{iy}}{\mathrm{\partial }x}}={\displaystyle \frac{e}{m_i}}(E_y-v_{ix}{B}_z),\\ {\displaystyle \frac{\mathrm{\partial }{n}_e}{\mathrm{\partial }t}}+{\displaystyle \frac{\mathrm{\partial }(n_e{v}_{ex})}{\mathrm{\partial }x}}=0,\\ {\displaystyle \frac{\mathrm{\partial }{v}_{ex}}{\mathrm{\partial }t}}+v_{ex}{\displaystyle \frac{\mathrm{\partial }{v}_{ex}}{\mathrm{\partial }x}}=-{\displaystyle \frac{e}{m_e}}(E_x+v_{ey}{B}_z)-{\displaystyle \frac{k_B{T}_i}{m_e{n}_e}}{\displaystyle \frac{\mathrm{\partial }{n}_e}{\mathrm{\partial }x}},\\ {\displaystyle \frac{\mathrm{\partial }{v}_{ey}}{\mathrm{\partial }t}}+v_{ex}{\displaystyle \frac{\mathrm{\partial }{v}_{ey}}{\mathrm{\partial }x}}=-{\displaystyle \frac{e}{m_e}}(E_y-v_{ex}{B}_z).\end{array}

3 Nonlinear small-amplitude KdV solitary waves

As well known, the nonlinear waves can be described by a KdV equation under small but finite amplitude and long wavelength approximation for lots of nonlinear systems [59–62]. To analyze nonlinear magnetosonic waves in a plasma, we employ the reductive perturbation method [16–21], which introduces the following stretched space and time coordinates

(4)\xi ={ϵ}_k^{1/2}(x- \lambda t),\tau ={ϵ}_k^{3/2}t,

where {ϵ}_k is a small parameter characterizing the wave number, and \lambda represents the phase velocity of the magnetosonic wave, which will be determined later.

In this study, we consider waves propagating in the x-direction with a magnetic field in the z-direction. In this case, the magnetosonic wave is a longitudinal wave, implying that the wave speed is relatively higher in the x-direction. For nonlinear waves, there are higher-order small quantities in the y-direction (for linear waves, there is no displacement in the y- and z-directions for longitudinal waves). Under this assumption, to ensure that the equations are self-consistent and solvable, the perturbed quantities, such as n_i,n_e ,v_{ix},v_{ex},B_z, and E_y, are expanded in powers of {ϵ}_k as [16, 17]

(5)\mathrm{\Theta }={\mathrm{\Theta }}_0+ {ϵ}_k{\mathrm{\Theta }}^{(1)}+{ϵ}_k^2{\mathrm{\Theta }}^{(2)} ,

while v_{iy},v_{ey} ,E_x are expanded as

(6){\mathrm{\Theta }}^{\prime }={\mathrm{\Theta }}_0^{\prime }+{ϵ}_k^{3/2}{\mathrm{\Theta }}^{\prime (1)}+ {ϵ}_k^{5/2}{\mathrm{\Theta }}^{\prime (2)},

where {\mathrm{\Theta }}_0 and {\mathrm{\Theta }}_0^{\prime } represent equilibrium values, and higher-order terms capture nonlinear corrections.

Substituting these expansions into the hydrodynamic and Maxwell equations, and retaining terms up to lower-order in {ϵ}_k, we derive the following relations: v_{ex}^{(1)}= v_{ix}^{(1)}, n_i^{(1)}=n_e^{(1)}=\frac{n_0}{\lambda }{v}_{ix}^{(1)}, E_y^{(1)}=B_0{v}_{ix}^{(1)}, B^{(1)}=\frac{E_y^{(1)}}{\lambda }, v_{ey}^{(1)}-v_{iy}^{(1)}=\frac{B_0}{n_0{\mu }_{0}e\lambda }\frac{\mathrm{\partial }{v}_{ix}^{(1)}}{\mathrm{\partial }\xi }, v_{ey}^{(1)}-v_{iy}^{(1)}=\left[\frac{\lambda m_i}{e{B}_0}-\frac{k_B(T_e+T_i)}{e\lambda B_0}\right]\frac{\mathrm{\partial }{v}_{ix}^{(1)}}{\mathrm{\partial }\xi }, and the phase velocity in the limited case of the long wave approximation \lambda = \sqrt{\frac{B_0^2}{n_0{\mu }_0{m}_i}+\frac{k_B(T_e+T_i)}{m_i}}. At next lower-order in {ϵ}_k, we have v_{ex}^{(2)}= v_{ix}^{(2)}, v_{iy}^{(1)}=-\frac{m_e}{m_i}{v}_{ey}^{(1)}, v_{ey}^{(1)}=\left[\frac{\lambda m_i}{e{B}_0}-\frac{k_B(T_e+T_i)}{e\lambda B_0}\right]\frac{\mathrm{\partial }{v}_{ix}^{(1)}}{\mathrm{\partial }\xi }, v_{iy}^{(1)}= \left[-\frac{\lambda m_e}{e{B}_0}+\frac{k_B{m}_e(T_e+T_i)}{e\lambda B_0{m}_i}\right]\frac{\mathrm{\partial }{v}_{ix}^{(1)}}{\mathrm{\partial }\xi }. The evolution of the ion velocity component v_{ix}^{(1)} is described by the KdV equation:

(7){\displaystyle \frac{\mathrm{\partial }{v}_{ix}^{(1)}}{\mathrm{\partial }\tau }}+A{v}_{ix}^{(1)} {\displaystyle \frac{\mathrm{\partial }{v}_{ix}^{(1)}}{\mathrm{\partial }\xi }}+B{\displaystyle \frac{{\mathrm{\partial }}^3{v}_{ix}^{(1)}}{\mathrm{\partial }{\xi }^3}}=0,

where A=\frac{B_0}{2{\mu }_0{n}_0{m}_i}[\frac{B_0}{{\lambda }^2}+ \frac{2{\mu }_0{n}_0{m}_i}{B_0}], and B=\frac{B_0}{2{\mu }_0{n}_0{m}_i}[\frac{\lambda m_i{m}_e}{e^2{B}_0}-\frac{k_B{m}_e(T_e+T_i)}{e^2\lambda B_0}].

The nonlinearity coefficient A dictates the steepening of the wave, while the dispersion coefficient B governs its spreading tendency. The balance between these effects leads to the formation of stable solitary wave structures.

4 Nonlinear envelope solitary waves

The KdV equation and the NLSE are both fundamental models for describing nonlinear wave phenomena. However, the KdV equation is primarily used to describe one-dimensional pulse-like waves, while the NLSE is mainly applied to describe envelope waves that undergo periodic oscillations (with amplitude modulations varying slowly in time and space), as well as their modulational instability. In order to study the nonlinear envelope magnetosonic waves in a plasma, we try to derive a NLSE from the KdV equation and explores the characteristics of envelope solitary waves in plasmas. The derivation of the NLSE from the KdV equation is a well-established [63–67]. It is widely adopted in plasma physics to explore nonlinear wave phenomena. Recent studies using particle-in-cell (PIC) numerical simulations have further validated the correctness of this method [66], providing robust evidence for its applicability. Therefore, we based on this method and assume that the perturbed ion velocity v_{ix}^{(1)} is

(8)v_{ix}^{(1)}={\displaystyle \sum _{m=1}^{\mathrm{\infty }}{ϵ}_n^m{\displaystyle \sum _{l=-\mathrm{\infty }}^{l=+\mathrm{\infty }}{v}_{ix}^{(m,l)}(X,T){\mathrm{e}}^{\mathrm{i}l(k\xi -\omega \tau )},}}

where X={ϵ}_n(\xi -V\tau ) and T={ϵ}_n^2\tau are the slow spatial and temporal coordinates, respectively. Here, {ϵ}_n represents a small parameter.

Substituting Eq. (8) into KdV equation, and collecting terms to order in {ϵ}_n, the dispersion relation is obtained \omega = -B{k}^3. At higher orders, the group velocity of the envelope wave is V=-3B{k}^2. Further analysis at the next order yields the NLSE

(9)i{\displaystyle \frac{\mathrm{\partial }{v}_{ix}^{(1,1)}}{\mathrm{\partial }T}}+P {\displaystyle \frac{{\mathrm{\partial }}^2{v}_{ix}^{(1,1)}}{\mathrm{\partial }{X}^2}}+Q|v_{ix}^{(1,1)}{|}^2{v}_{ix}^{(1,1)}=0,

where P=-3Bk, Q=-\frac{A^2}{6Bk}. Notably, the coefficient PQ>0 indicates modulational instability, leading to the formation of localized nonlinear structures.

Letting v_{ix}^{(1,1)}= \sqrt{\frac{2P}{Q}}{v}^{\prime }, T=\frac{1}{2P}{T}^{\prime }, X=X^{\prime }, Eq. (9) becomes

(10)i{\displaystyle \frac{\mathrm{\partial }{v}^{\prime }}{\mathrm{\partial }{T}^{\prime }}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\mathrm{\partial }}^2{v}^{\prime }}{\mathrm{\partial }{X}^{\prime 2}}}+|v^{\prime }{|}^2{v}^{\prime } =0.

Eq. (10) is a standard NLSE, which has been well studied previously [23–26]. In addition, the NLSE admits a rich variety of nonlinear solutions, including solitary waves, breathers, and rogue waves, depending on the plasma parameters and initial conditions. These solutions highlight the role of MI in the formation and evolution of nonlinear waveforms, bridging the gap between theory and observed phenomena in plasmas.

5 Modulational instability

Modulational instability is a fundamental nonlinear mechanism that governs the stability of wave packets under perturbations. Its analysis provides critical insights into the transition from linear stability to the emergence of localized nonlinear structures. To study MI, we consider a small perturbation superimposed on a plane wave background of the NLSE [68]:

(11)v_{ix}^{\left(1,1\right)} =\left[v_0^{\left(1,1\right)}+\delta v(X,T)\right]{\mathrm{e}}^{-\mathrm{i}\mathrm{\Delta }T},

where v_0^{(1,1)} is the background wave amplitude, \delta v is a small perturbation, and \mathrm{\Delta }=- Q|v_0^{\left(1,1\right)}{|}^2 is a frequency shift.

Substituting this expression into the NLSE and linearizing, we obtain the dispersion relation for the perturbation

(12){\mathrm{\Omega }}^2=P{K}^2\left[P{K}^2-2Q{\left|v_0^{\left(1,1\right)}\right|}^2\right],

where K is the perturbation wave number, \mathrm{\Omega } is its frequency, P and Q are the NLSE coefficients.

The stability of the wave packet depends on the sign of PQ. If , {\mathrm{\Omega }}^2>0 for all K, and the wave packet is stable. If PQ>0, for small K, leading to exponential growth of the perturbation and MI, wave number K satisfying

(13)

The maximum growth rate of the instability is

(14){\gamma }_{max}=\left|Q\right|{\left|v_0^{(1,1)}\right|}^2,

indicating that stronger background amplitudes amplify the instability.

In addition, MI has been observed and studied in many branches of science such as energy localization in the system (anharmonic lattices at finite energy density, discrete lattices, time-delay-memristive neural network, dissipative physical systems) [69–72] and the formation of different wave structures (plasma waves, matter waves, electromagnetic waves and optical waves) [73–78]. These are crucial for understanding energy concentration and extreme events in plasmas, with broad implications for fusion research, space plasmas, and astrophysics.

6 Several kinds of magnetosonic nonlinear waves

Lot of researchers focus on recent NLSE due to its critical role in describing nonlinear phenomena and its broad range of applications across various fields [79–83]. To address the complexities of the NLSE, numerous analytical and numerical techniques have been developed, including the Hirota method [84], Darboux transformation [85], inverse scattering transform [86], and the split-step Fourier method [87]. These methods provide powerful tools for exploring the rich dynamics and solutions of the NLSE.

The NLSE admits both linear and nonlinear wave solutions. The linear solution represents a uniform background wave [35, 88, 89], expressed as v_0^{\prime }(X^{\prime },T^{\prime }) = a{\mathrm{e}}^{\mathrm{i}{a}^2{T}^{\prime }}, where a is the background amplitude. However, Nonlinear excitations are arise from perturbations of the linear background, evolving into a variety of distinct waveforms. Depending on the interaction between nonlinearity and dispersion, the NLSE supports a range of nonlinear solutions with different frequencies and characteristics.

Recent studies have extensively explored the analytical solutions of the NLSE. When the condition PQ>0 is satisfied, the NLSE describes envelope solitary waves, including K−M [30–32], AB [33], BS [34–36], and RW [37, 38]. These waveforms have been validated in experiments, such as nonlinear optical fibers, where the K−M, AB, and RW have been observed [90–92]. High-order rogue waves have also been successfully generated in controlled water wave tank experiments [93–95], providing further evidence for their physical relevance. Conversely, when , the NLSE supports other types of solutions, such as dark solitary waves [96, 97], which exhibit distinct spatial and temporal profiles.

6.1 K−M breather solution

K−M breather solution is periodic structures that amplify and decay in time while remaining localized in space. Their amplitude and frequency are determined by plasma parameters such as the magnetic field, wave number, and thermal properties. The breather solution of Eq. (10) can be expressed as

(15)v{}^{\prime }(X^{\prime },T^{\prime })=[a-2{\displaystyle \frac{(b^2-a^2)\mathrm{c}\mathrm{o}\mathrm{s}(\zeta )+\mathrm{i}b\sqrt{b^2-a^2}\mathrm{s}\mathrm{i}\mathrm{n}(\zeta )}{b\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(2X^{\prime }\sqrt{b^2-a^2})-a\mathrm{c}\mathrm{o}\mathrm{s}(\zeta )}}]{\mathrm{e}}^{\mathrm{i}{a}^2{T}^{\prime }},

where \zeta =2b{T}^{\prime }\sqrt{b^2-a^2}, T^{\prime }=2PT, and .

This solution comprises two components, the first term represents a plane wave background (PWB), while the second term corresponds to a BS related term. The parameter b primarily determines the shape of the localized wave, while a represents the amplitude of the background wave [35]. Together, a and b influence the overall structure of the nonlinear wave, dictating its shape and dynamic behavior.

The K−M breather solution of Eq. (15) in the experimental coordinates can be expressed as

(16)\begin{array}{rl}{v}_{ix}^{\mathrm{K}-\mathrm{M}}=& {ϵ}_k{ϵ}_n\sqrt{\frac{2P}{Q}}\left[a-2\frac{(b^2-a^2)\mathrm{c}\mathrm{o}\mathrm{s}(\zeta )+\mathrm{i}b\sqrt{b^2-a^2}\mathrm{s}\mathrm{i}\mathrm{n}(\zeta )}{b\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(2X^{\prime }\sqrt{b^2-a^2})-a\mathrm{c}\mathrm{o}\mathrm{s}(\zeta )}\right]\\ & \times {\mathrm{e}}^{\mathrm{i}[k{ϵ}_k^{1/2}x-(k\lambda {ϵ}_k^{1/2}+\omega {ϵ}_k^{3/2}-2P{a}^2{ϵ}_n^2{ϵ}_k^{3/2})t]},\end{array}

where \zeta =4bP{ϵ}_n^2{ϵ}_k^{3/2}\sqrt{b^2-a^2}t, X^{\prime }={ϵ}_n [{ϵ}_k^{1/2}x+( 3B{k}^2{ϵ}_k^{3/2}- {ϵ}_k^{1/2}\lambda )t]. The phase velocity of the K−M breather solution is

(17)V_p^{\mathrm{K}-\mathrm{M}} =\lambda +{\displaystyle \frac{\omega }{k}}{ϵ}_k-{\displaystyle \frac{2P{a}^2}{k}}{ϵ}_n^2{ϵ}_k,

or

(18)V_p^{\mathrm{K}-\mathrm{M}} =V_p^{K-M}\left(B_0,T_e,T_i,{ϵ}_k,{ϵ}_n,a,k\right),

where {\lambda }^2=\frac{B_0^2}{n_0{\mu }_0{m}_i}+\frac{k_B(T_e+T_i)}{m_i}.

The spatial profile of the ion velocity for the K−M breather solution v_{ix}^{\mathrm{K}-\mathrm{M}} [Eq. (16)] at a fixed time t=0, is shown in Fig.1. The modulation highlights energy localization and the impact of modulational instability under specific plasma conditions provides insights into nonlinear plasma wave dynamics.

To simplify the analysis, the phase velocity and group velocity of the K−M breather solution are normalized using the Alfvén wave velocity v_{\alpha }, which defined as v_{\alpha }=\sqrt{\frac{B_0^2}{{\mu }_0{m}_i{n}_{i0}}}=1.5423\times {10}^4 m/s, where the parameters are taken from the previous studies [98], B_0=2.00\times {10}^{-5} T, {\mu }_0=1.2566370614\times {10}^{-6} N/A², m_i=1.67261 \times {10}^{-27} kg, n_{i0}=8.0\times {10}^{14} m⁻³.

The dependence of the phase velocity of the K−M breather solution on plasma parameters can be given from Eq. (17). This equation reveals that the first term corresponds to the phase velocity of the magnetosonic wave, the second term represents a high order terms, while the third term denotes the higher order term. Notably, the phase velocity of the K−M breather solution is slightly larger than the Alfvén wave velocity due to these higher-order effects, indicating the influence of nonlinearity and dispersion.

To further investigate this relationship, Fig.2 illustrates the dependence of the phase velocity of the K−M solution V_p^{\mathrm{K}-\mathrm{M}} on the wave number k for different external magnetic field B_0 in (a), different parameters a in (b), different temperature of electrons T_e in (c), different temperature of ions T_i in (d). The results show that V_p^{\mathrm{K}-\mathrm{M}} decreases as k increases, indicating that shorter wavelengths correspond to higher phase velocities. Furthermore, it is noted from Fig.2(a)−(d) that V_p^{\mathrm{K}-\mathrm{M}} increases as B_0, a, T_e and T_i increase. However, it can be seen from Fig.2(d) that the ion temperature has little effect on the phase velocity of the K−M breather solution.

The group velocity of the K−M breather solution is expressed as

(19)V_g^{\mathrm{K}-\mathrm{M}} =\lambda -3 B{k}^2{ϵ}_k,

or

(20)V_g^{\mathrm{K}-\mathrm{M}} =V_g^{\mathrm{K}-\mathrm{M}}\left(B_0,T_e,T_i,{ϵ}_k,k\right) .

This relation demonstrates that the group velocity is also close to the phase velocity of the magnetosonic wave.

Fig.3 illustrates the dependence of group velocity V_g^{\mathrm{K}-\mathrm{M}} of the K−M breather solution on the wave number k for different plasma parameters. Fig.3(a)−(c) specifically show the influence of the external magnetic field B_0, the electron temperature T_e, and the ion temperature T_i, respectively. The other parameters are chosen from the previous work [88]: a=0.2,b=0.6, {ϵ}_k=0.10, {ϵ}_n=0.08, n_0=8.0 \times {10}^{14} m⁻³. It is evident that V_g^{\mathrm{K}-\mathrm{M}} decreases with increasing k, indicating that shorter wavelengths correspond to lower group velocities, consistent with the dispersive nature of the system. Moreover, V_g^{\mathrm{K}-\mathrm{M}} increases with increasing B_0, T_e, and T_i, as observed in Fig.3(a)−(c). However, the influence of T_i on V_g^{\mathrm{K}-\mathrm{M}} is relatively minor compared to that of B_0 and T_e, suggesting that the group velocity is more sensitive to the external magnetic field and electron temperature.

Eq. (17) and Eq. (19) reveal that both the phase velocity and group velocity of the K−M breather solution depend on the parameters {ϵ}_n and {ϵ}_k. To explore this dependence in detail, Fig.4 presents the spatial profile of v_{ix}^{\mathrm{K}-\mathrm{M}} at t=0 s for various values of {ϵ}_n and {ϵ}_k. The results indicate that both parameters influence the K−M breather solution. Specifically, {ϵ}_k has a significant impact on both V_p^{\mathrm{K}-\mathrm{M}} and V_g^{\mathrm{K}-\mathrm{M}}, directly modifying the dispersive and nonlinear characteristics of the wave. In contrast, the effect of {ϵ}_n on these velocities is minimal, suggesting that {ϵ}_n primarily affects other aspects of the wave structure, such as amplitude or localization. This distinction highlights the differing roles of {ϵ}_n and {ϵ}_k in shaping the dynamics of magnetosonic waves.

For further understanding, Fig.5 shows the dependence of the ratio of phase velocity to the group velocity of the K−M breather solution on the wave number k. The effects of different parameters are illustrated in Fig.5(a)−(d) for {ϵ}_n, {ϵ}_k, B_0 and T_e, respectively. The ratio is given by the following expression

(21){\displaystyle \frac{V_p^{\mathrm{K}-\mathrm{M}}}{V_g^{\mathrm{K}-\mathrm{M}}}}={\displaystyle \frac{\lambda +(\omega /k){ϵ}_k-(2P{a}^2/k){ϵ}_n^2{ϵ}_k}{\lambda -3B{k}^2{ϵ}_k}}.

As shown in Fig.5, the ratio V_p^{\mathrm{K}-\mathrm{M}} /V_g^{\mathrm{K}-\mathrm{M}} is consistently greater than 1, indicating that the group velocity is smaller than the phase velocity, a characteristic of normal dispersion in the plasma. The ratio increases with k, {ϵ}_n, {ϵ}_k, or B_0, but decreases with increasing electron temperature T_e. Notably, Fig.5(a) demonstrates that the influence of {ϵ}_n on the ratio is negligible, while Fig.5(b) highlights the significant impact of {ϵ}_k. This distinction suggests that {ϵ}_k plays a more critical role in modulating the dispersive properties of the plasma.

The amplitude of the K−M breather solution is expressed as |A^{\mathrm{K}-\mathrm{M}}|={ϵ}_n{ϵ}_k \sqrt{\frac{2P}{Q}}(a+2 b). Rewriting it in terms of plasma parameters, we have

(22)| A^{\mathrm{K}-\mathrm{M}} | ={ϵ}_k{ϵ}_{n}6k\lambda {\displaystyle \frac{{\lambda }^2{m}_i{m}_e-k_B{m}_e\left(T_e+T_i\right)}{e^2\left(B_0^2+2m_i{\mu }_0{n}_0{\lambda }^2\right)}}\left(a+2b\right),

or

(23)| A^{\mathrm{K}-\mathrm{M}}|=A^{\mathrm{K}-\mathrm{M}}\left(B_0,T_e,T_i,{ϵ}_k,{ϵ}_n,a,b,k\right).

Eq. (23) describes the dependence of the amplitude |A^{\mathrm{K}-\mathrm{M}}| on the plasma parameters. Fig.6 illustrates this dependence for various parameters of the plasma system, including {ϵ}_n, {ϵ}_k, B_0, k, T_e, T_i, a and b. As shown in Fig.6, the amplitude |A^{\mathrm{K}-\mathrm{M}} | increases with the wave number k, external magnetic field B_0, and the parameters {ϵ}_n, {ϵ}_k, a and b. Conversely, it decreases with increasing electron temperature T_e. Interestingly, the ion temperature T_i has a negligible effect on | A^{\mathrm{K}-\mathrm{M}} |, as shown in Fig.6(d). These results highlight the intricate dependence of the amplitude on both the plasma and wave parameters, providing valuable insights into the nonlinear dynamics of magnetosonic waves in plasmas.

The width of the K−M breather solution can be derived from Eq. (16) is

(24)W^{\mathrm{K}-\mathrm{M}}={\displaystyle \frac{1}{K}}\ln \left({\displaystyle \frac{M+\sqrt{M^2-4}}{M-\sqrt{M^2-4}}}\right),

or

(25)W^{\mathrm{K}-\mathrm{M}}=W^{\mathrm{K}-\mathrm{M}}\left({ϵ}_k,{ϵ}_n,a,b\right),

where K=2{ϵ}_n{ϵ}_k^{1/2}\sqrt{b^2-a^2} and M=2\left[\frac{4\left(b^2-a^2\right)}{b\left(3a+2b\right)}+\frac{a}{b}\right].

The dependence of the wave width W^{\mathrm{K}-\mathrm{M}} on the parameters {ϵ}_k, {ϵ}_n, a and b is shown in Fig.7. The results indicate that W^{\mathrm{K}-\mathrm{M}} decreases as a and b increases, reflecting the effects of nonlinearity and localization on the solution’s spatial extent. Furthermore, W^{\mathrm{K}-\mathrm{M}} also decreases with increasing {ϵ}_k and {ϵ}_n, highlighting the role of these parameters in controlling the dispersive and nonlinear properties of the wave.

6.2 Bright soliton (BS) solution

When a=0 and b\ne 0, the K−M breather solution of Eq. (16) is a generalized BS solution

(26)v{}^{\prime }(X^{\prime },T^{\prime }) =2b\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}(2b{X}^{\prime }) {\mathrm{e}}^{\mathrm{i}2b^2{T}^{\prime }} .

In this case, the amplitude of the background wave of BS solution is zero. The BS solution in the experimental coordinates can be expressed as

(27)\begin{array}{rl}{v}_{ix}^{\mathrm{B}\mathrm{S}}=& 2{ϵ}_k{ϵ}_n\sqrt{{\displaystyle \frac{2P}{Q}}}b\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\left\{2b{ϵ}_n[{ϵ}_k^{1/2}x+(3B{k}^2{ϵ}_k^{3/2}-{ϵ}_k^{1/2}\lambda )t]\right\}\\ & \times {\mathrm{e}}^{\mathrm{i}[k{ϵ}_k^{1/2}x-(\omega {ϵ}_k^{3/2}+k\lambda {ϵ}_k^{1/2}-4P{b}^2{ϵ}_n^2{ϵ}_k^{3/2})t]}.\end{array}

Fig.8 illustrates the BS solution v_{ix}^{\mathrm{B}\mathrm{S}}, which is a solitary wave with a single stable peak. Governed by Eq. (10), the BS solution demonstrates how nonlinearity and dispersion balance to form a self-maintaining wave. Its stability highlights its role in nonlinear plasma energy transport.

The phase velocity of the BS solution is given by

(28)V_p^{\mathrm{B}\mathrm{S}}=\lambda +{\displaystyle \frac{\omega }{k}}{ϵ}_k-{\displaystyle \frac{4P{b}^2{ϵ}_n^2{ϵ}_k}{k}}.

The amplitude of the BS solution is

(29)| A^{\mathrm{B}\mathrm{S}}|=2 b{ϵ}_k{ϵ}_n\sqrt{{\displaystyle \frac{2P}{Q}}} .

The group velocity is

(30)V_g^{\mathrm{B}\mathrm{S}}=\lambda -3B{k}^2{ϵ}_k .

The width of BS solution is expressed as

(31)W^{\mathrm{B}\mathrm{S}}={\displaystyle \frac{1}{K}}\ln \left({\displaystyle \frac{M+\sqrt{M^2-4}}{M-\sqrt{M^2-4}}}\right),

where K=2b{ϵ}_n{ϵ}_k^{1/2} and M=4.

Fig.9 illustrates the dependence of the phase velocity V_p^{\mathrm{B}\mathrm{S}} and amplitude |A^{\mathrm{B}\mathrm{S}} | of the BS solution on the wave number k and the parameter b. The parameters used are {ϵ}_k=0.10, {ϵ}_n=0.08, n_0=8.0 \times {10}^{14} m⁻³, T_e = 4.0 eV, T_i = 0.025 eV, B_0=5.00\times {10}^{-5} T. The results show that the phase velocity V_p^{\mathrm{B}\mathrm{S}} decreases as the wave number k increases, but increases with the parameter b. Additionally, the amplitude | A^{\mathrm{B}\mathrm{S}}| increases with both k and b, reflecting the nonlinear nature of the solution and the strong dependence on plasma and wave parameters.

6.3 Akhmediev breather (AB) solution

When , the K−M breather solution Eq. (16) transitions into the AB solution, expressed as

(32)v{}^{\prime }(X^{\prime },T^{\prime })=[a-2{\displaystyle \frac{(b^2-a^2)\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\zeta )-ib\sqrt{a^2-b^2}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\zeta )}{b\mathrm{c}\mathrm{o}\mathrm{s}(2X^{\prime }\sqrt{a^2-b^2})-a\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\zeta )}}]{\mathrm{e}}^{\mathrm{i}{a}^2{T}^{\prime }},

where \zeta =4bP{ϵ}_n^2{ϵ}_k^{3/2}\sqrt{a^2-b^2}t. In experimental coordinates, the AB solution takes the form

(33)\begin{array}{rl}{v}_{ix}^{\mathrm{A}\mathrm{B}}=& {ϵ}_k{ϵ}_n\sqrt{{\displaystyle \frac{2P}{Q}}}\left[a-2{\displaystyle \frac{(b^2-a^2)\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\zeta )-\mathrm{i}b\sqrt{a^2-b^2}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\zeta )}{b\mathrm{c}\mathrm{o}\mathrm{s}(2X^{\prime }\sqrt{a^2-b^2})-a\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\zeta )}}\right]\\ & \times {\mathrm{e}}^{\mathrm{i}[k{ϵ}_k^{1/2}x-(k\lambda {ϵ}_k^{1/2}+\omega {ϵ}_k^{3/2}-2P{a}^2{ϵ}_n^2{ϵ}_k^{3/2})t]},\end{array}

where v={ϵ}_n[{ϵ}_k^{1/2}x +(3B k^2{ϵ}_k^{3/2}-{ϵ}_k^{1/2}\lambda )t].

Fig.10 presents the AB solution v_{ix}^{\mathrm{A}\mathrm{B}} at t=0, showing periodic modulation on a uniform background. This structure arises from MI, with localized growth and decay. The solution’s periodicity and sensitivity to plasma parameters reflect transient behaviors in nonlinear systems.

The phase velocity of the AB solution is identical to that of the K−M breather solution, i.e.,

(34)V_p^{\mathrm{A}\mathrm{B}}=\lambda +{\displaystyle \frac{\omega }{k}}{ϵ}_k-{\displaystyle \frac{2P{a}^2}{k}}{ϵ}_n^2{ϵ}_k.

Similarly, the amplitude of the AB solution is the same as that of the K−M breather solution:

(35)| A^{\mathrm{A}\mathrm{B}}|={ϵ}_n{ϵ}_k\sqrt{{\displaystyle \frac{2P}{Q}}}(a+2b).

The AB solution represents a periodic and localized structure that modulates in both space and time. These properties are directly influenced by the parameters a, b, {ϵ}_k, and {ϵ}_n, reflecting the interaction between nonlinearity and dispersion in plasma. The equivalence of phase velocity and amplitude between the AB and K−M breather solutions underscores their shared fundamental characteristics, while the specific spatial and temporal evolution of the AB solution highlights its distinct dynamics in modulated wave systems.

6.4 Peregrine rogue wave (RW) solution

When |b|=|a|, the K−M breather solution transitions into the RW solution, described as

(36)v{}^{\prime }( X^{\prime }, T^{\prime })=\left[1-{\displaystyle \frac{4(2\mathrm{i}{a}^2{T}^{\prime }+1)}{4a^4{T}^{\prime 2}+4a^2{X}^{\prime 2}+1}}\right]a{\mathrm{e}}^{\mathrm{i}{a}^2{T}^{\prime }}.

In experimental coordinates, the RW solution can be expressed as

(37)\begin{array}{rl}{v}_{ix}^{\mathrm{R}\mathrm{W}}=& {ϵ}_k{ϵ}_n\sqrt{{\displaystyle \frac{2P}{Q}}}a\left[1-{\displaystyle \frac{4(2\mathrm{i}{a}^2{T}^{\prime }+1)}{4a^4{T}^{\prime }{u}^2+4a^2{X}^{\prime 2}+1}}\right]\\ & \times {\mathrm{e}}^{\mathrm{i}[k{ϵ}_k^{1/2}x-(k\lambda {ϵ}_k^{1/2}+\omega {ϵ}_k^{3/2}-2P{a}^2{ϵ}_n^2{ϵ}_k^{3/2})t]}.\end{array}

Fig.11 depicts the RW solution v_{ix}^{\mathrm{R}\mathrm{W}}, showing an extreme isolated peak at t=0. The RW arises from modulational instability, concentrating energy into a single transient event. Its unique, localized nature highlights extreme plasma dynamics and energy bursts.

The phase velocity of the RW solution is identical to that of both the K−M breather and AB solutions

(38)V_p^{\mathrm{R}\mathrm{W}}=\lambda +{\displaystyle \frac{\omega }{k}}{ϵ}_k-{\displaystyle \frac{2P{a}^2}{k}}{ϵ}_n^2{ϵ}_k.

The amplitude of the RW solution is given by

(39)| A^{\mathrm{R}\mathrm{W}}|=3 a{ϵ}_n{ϵ}_k\sqrt{{\displaystyle \frac{2P}{Q}}} .

The RW solution describes an extreme, localized wave structure characterized by its rapid amplification and subsequent decay, which is distinct from the periodic or breather-like features of the K−M breather and AB solutions. It highlight the critical role of plasma parameters{ϵ}_k, {ϵ}_n, and a in shaping the formation and evolution of rogue waves in plasmas.

7 Discussion

Present results show that the group velocity, phase velocity, amplitude, and width of the K−M breather solution depend on several plasma parameters, including the external magnetic field B_0, electron and ion temperatures, as well as the parameters of {ϵ}_k, {ϵ}_n, a, b and k. Specifically, these dependencies can be expressed as

\{\begin{array}{l}{V}_p^{\mathrm{K}-\mathrm{M}}=V_p^{\mathrm{K}-\mathrm{M}}(B_0,T_e,T_i,{ϵ}_k,{ϵ}_n,a,k),\\ \\ V_g^{\mathrm{K}-\mathrm{M}}=V_g^{\mathrm{K}-\mathrm{M}}(B_0,T_e,T_i,{ϵ}_k,k),\\ \\ A^{\mathrm{K}-\mathrm{M}}=A^{\mathrm{K}-\mathrm{M}}(B_0,T_e,T_i,{ϵ}_k,{ϵ}_n,a,b,k),\\ \\ W^{\mathrm{K}-\mathrm{M}}=W^{\mathrm{K}-\mathrm{M}}(B_0,T_e,T_i,{ϵ}_k,k).\end{array}

These equations provide a direct link between the measurable properties of the K−M breather solution and the underlying plasma parameters. By exciting a K−M wave in a plasma, experimentally accessible quantities such as the group velocity, phase velocity, amplitude, width, wave number (k), and background wave amplitude (a) can be measured. These measurements, in turn, allow for the inference of more challenging plasma parameters, such as T_e, T_i, {ϵ}_k, {ϵ}_n, and b. While there appear to be five independent variables, only four independent equations are available. This limitation can be resolved by exciting a second K−M breather solution with a different wavenumber k, accompanied by variations in {ϵ}_k, {ϵ}_n, and b. This approach generates eight independent equations, enabling the determination of eight parameters: T_e, T_i, {ϵ}_k^{(1)}, {ϵ}_n^{(1)}, b^{(1)}, {ϵ}_k^{(2)}, {ϵ}_n^{(2)}, b^{(2)}. Here, T_e and T_i remain constant for the same plasma, while {ϵ}_k, {ϵ}_n, and b differ for the two K−M breather solutions.

This method highlights the potential of nonlinear wave structures as diagnostic tools for characterizing plasma properties. By exciting multiple K−M breather solutions and measuring readily accessible parameters, it is possible to deduce key plasma characteristics that are otherwise difficult to measure directly. These findings not only hold promise for controlled laboratory experiments but also have significant implications for astrophysical plasmas. The ability to predict and characterize nonlinear structures in magnetized plasmas contributes to our understanding of energy transport, particle acceleration, and wave-particle interactions in both space and fusion environments. Future research should focus on experimentally validating these theoretical predictions and extending the framework to incorporate relativistic and quantum effects, thereby broadening its applicability to more extreme plasma conditions.

8 Conclusion

Present study derived a nonlinear NLSE to describe weakly nonlinear magnetosonic waves in electron-ion plasmas, accounting for both magnetic and thermal pressure effects. predicts and explains various nonlinear wave structures, including K−M breather, AB, BS, and RW, along with their existence conditions and parameter dependencies.

The NLSE derived in this work can offer more comprehensive and generalized model for the analysis of nonlinear magnetosonic wave dynamics. Through a detailed exploration of its solutions, we demonstrated how key plasma parameters such as the external magnetic field, electron and ion temperature, wave number govern wave velocities, amplitudes, and widths. This analysis highlights the intricate interplay between nonlinearity and dispersion, offering deeper insights into the underlying plasma physics.

We propose a potential method to predict the plasma parameters that are difficult to measure directly. By analyzing multiple nonlinear wave structures and their measurable properties, such as velocities, amplitudes, and widths, we may obtain related information of the critical plasma parameters, including those related to temperature, magnetic field. This work aims to bridge the gap between nonlinear wave theory and plasma diagnostics, providing a possible approach for probing complex plasma systems. The implications of these findings extend across a wide range of physical environments, including magnetized laboratory plasmas, space plasmas, and astrophysical systems.

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