Pith. sign in

REVIEW 3 major objections 6 minor 22 references

Dynamics and modulation of cosmic ray modified magnetosonic waves in a galactic gaseous rotating plasma

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Cosmic-ray pressure materially modifies magnetosonic waves in the interstellar medium of spiral galaxies, accelerating linear damping, producing KdVB solitons and shocks, and reducing modulational-instability growth.

desk verdict Honest, self-contained derivation of CR-modified KdVB/NLS for magnetosonic waves, but the rotation parameters violate the model's stated slow-rotation assumption and the MI section has coefficient inconsistencies, so the quantitative claims need reworking before they can be trusted. read the letter →

arxiv 2411.16286 v3 pith:EDBLTSZP submitted 2024-11-25 physics.plasm-ph

Jyoti Turi , Gadadhar Banerjee This is my paper · ORCID
classification physics.plasm-ph PACS 52.35.Bj52.35.Mw95.30.Qd
keywords cosmicraysmagnetosonicwavesinterstellarmediumspiralgalaxiesrotatingmagnetizedplasmaKorteweg-deVries-Burgersequationmodulationalinstabilityrogue
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper sets out to show that the cosmic-ray component of the interstellar medium of spiral galaxies is an active fluid that changes how magnetosonic waves behave, both linearly and nonlinearly. Adding a cosmic-ray pressure term to the dissipative, rotating magnetohydrodynamic equations produces a modified dispersion relation, and the authors show that cosmic-ray diffusivity together with magnetic resistivity gives a damping rate of order $|\gamma| \approx (C_c^2 \kappa + \eta) k^4/(2\omega_1^2)$. In the nonlinear regime they derive a Korteweg–de Vries–Burgers (KdVB) equation whose coefficients carry the cosmic-ray and rotation parameters, yielding one- and two-soliton solutions when dissipation is weak and shock solutions when it is strong. From that equation they then derive a nonlinear Schrödinger equation and show that the modulational-instability growth rate is $\Gamma = |M| K^2 \sqrt{K_c^2/K^2 - 1}$, with cosmic-ray pressure reducing the growth. If these claims hold, cosmic rays measurably alter wave damping, magnetic-field structure, and energy transport in spiral galaxies.

What carries the argument

The load-bearing object is the cosmic-ray-modified pressure law, $P = P_g + P_c$, with the cosmic-ray pressure obeying the diffusion–convection equation $\partial P_c/\partial t + v\cdot\nabla P_c + \gamma_c \nabla\cdot v - \kappa \nabla^2 P_c = 0$. This single modification threads through every stage of the argument: it adds the $C_c^2$ terms to the dispersion relation and to the phase velocity $\lambda^2 = (1 + C_g^2 + C_c^2)/(1 + \cot^2\theta)$; it supplies the $C_c^2 \kappa$ contribution to the damping rate $|\gamma| \approx (C_c^2 \kappa + \eta) k^4/(2\omega_1^2)$; and it fixes the KdVB coefficients $Q$, $R$, $S$, hence the NLS coefficients $M = 6Rk$ and $N = Q^2/(6Rk)$. The KdVB equation $\partial_\tau B + Q B \partial_\xi B + R \partial_\xi^3 B = S \partial_\xi^2 B$ is the nonlinear engine, and the NLS equation $i\partial_T B + (M/2)\partial_X^2 B + N |B|^2 B = 0$ provides the modulational-instability growth rate and the rogue-wave solutions. The factors $\omega/(\omega + i\kappa k^2)$ and $\omega/(\omega + i\eta k^2)$ in the dispersion relation are the explicit carriers of dissipation.

What would settle it

A controlled measurement of the decay of a magnetosonic wave packet in a rotating, resistive plasma with controlled cosmic-ray-like diffusivity would falsify the claim if the damping does not scale as $|\gamma| \approx (C_c^2 \kappa + \eta) k^4/(2\omega_1^2)$ with the predicted coefficient; equivalently, a particle-in-cell simulation that resolves the two-fluid pressure dynamics could check the predicted $k^4$ dependence.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that a two-fluid pressure description—thermal gas plus a diffusing cosmic-ray gas with negligible density—is enough to capture how cosmic rays alter magnetosonic waves in the interstellar medium of spiral galaxies. Linear analysis of the modified dispersion relation yields a damped mode with damping rate $|\gamma| \approx (C_c^2 \kappa + \eta) k^4/(2\omega_1^2)$, meaning cosmic-ray diffusivity $\kappa$ and magnetic resistivity $\eta$ jointly accelerate wave decay. Reductive perturbation around the cosmic-ray-modified phase velocity $\lambda^2 = (1 + C_g^2 + C_c^2)/(1 + \cot^2\theta)$ produces the KdVB equation whose nonlinear, dispersive, and dissipative coefficients depend on cosmic-ray pressure, thermal pressure, rotation, and dissipation. In the weak-dissipation limit the same framework yields a nonlinear Schrödinger equation with dispersion coefficient $M = 6Rk$ and nonlinearity $N = Q^2/(6Rk)$, giving modulational-instability growth rate $\Gamma = |M| K^2 \sqrt{K_c^2/K^2 - 1}$ whose critical wavenumber is $K_c^2 = 2N|B_0|^2/M$. The central conclusion is that cosmic rays, by modifying the pressure, both accelerate linear damping and suppress the growth of modulation instability, while rotation acts in the opposite direction on the instability.

Load-bearing premise

The derivation requires slow rotation, $\Omega_0/\omega_{ci} \ll 1$, so that higher-order Coriolis and centrifugal terms can be dropped; the numerical work nevertheless uses $\Omega_0/\omega_{ci}$ near 0.8, placing the quantitative predictions outside the stated validity regime.

Editorial extensions

If this is right

  • In the interstellar medium of spiral galaxies, magnetosonic wave energy is expected to decay faster when cosmic-ray diffusivity or magnetic resistivity is high, with the damping rate growing as the fourth power of the wavenumber.
  • In regions where cosmic-ray diffusion and resistivity are weak, the nonlinear outcome is a KdV soliton; where they are strong, the model predicts monotonic or oscillatory magnetosonic shocks, with the transition controlled by the dissipation coefficient $S$.
  • Cosmic-ray pressure reduces the modulational-instability growth rate, so cosmic-ray-loaded gas should be less prone to envelope collapse into rogue waves, while faster rotation (larger $\Omega_0$) increases the growth rate.
  • The explicit coefficient formulas and the sensitivity indices for $C_g$, $C_c$, $\Omega_0$, $\theta$, $\kappa$, $\eta$ provide quantitative anchors for comparing soliton, shock, and rogue-wave amplitudes with observed magnetic-field fluctuations in galactic ISM.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the qualitative result survives the slow-rotation inconsistency, then galaxy rotation curves imply that magnetosonic waves in the inner, faster-rotating regions should show shorter envelope modulations and smaller rogue-wave amplitudes than in the outer disk; this is testable with high-resolution Faraday-rotation or synchrotron-polarization maps.
  • The same pressure-modification mechanism should apply to other MHD wave families, such as Alfvén and slow magnetosonic modes, and to molecular-cloud cores where cosmic-ray ionization dominates; the specific coefficients would change, but the acceleration of linear damping by $\kappa$ and the suppression of MI by cosmic-ray pressure are natural analogues.
  • Because the paper's numerical regime violates its own slow-rotation assumption, the safest next step is to re-derive the KdVB and NLS coefficients without dropping higher-order Coriolis terms; the qualitative conclusions may hold, but the fitted numbers would shift.
  • The rogue-wave solutions are envelope rational functions of the NLS equation; linking them to observable 'magnetic spikes' in the ISM requires translating the normalized coefficients $M$ and $N$ into physical modulation timescales, a step the paper does not take.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The manuscript studies linear and nonlinear magnetosonic waves in a rotating, resistive, cosmic-ray-modified magnetized fluid intended to model the interstellar medium of spiral galaxies. It derives a modified linear dispersion relation (Eq. 15), an estimate for the linear damping rate, a Korteweg-de Vries-Burgers (KdVB) equation (Eq. 21) with coefficients (22)-(24), soliton and shock solutions, and a nonlinear Schrödinger (NLS) equation (Eq. 46) used to study modulational instability (MI) and first- and second-order rogue waves. The central claims are that cosmic-ray pressure accelerates the linear damping, that cosmic-ray diffusivity and magnetic resistivity generate shock structures, and that cosmic-ray pressure reduces the MI growth rate.

Significance. The derivations are largely self-contained: the linear dispersion relation, KdVB coefficients, NLS coefficients, and MI growth-rate formula are obtained algebraically from the stated fluid model without fitting free parameters. If the validity-regime and coefficient inconsistencies are repaired, the paper would be a useful systematic account of cosmic-ray and rotation effects on magnetosonic solitons, shocks, and rogue waves in the ISM, and the sensitivity analysis in Section VI provides a concrete quantitative summary. At present, however, the numerical demonstrations lie outside the declared slow-rotation validity regime, and the MI equations contain internal algebraic inconsistencies. The physical conclusions should therefore be treated as provisional until these load-bearing issues are resolved.

major comments (3)
  1. [Section II, Table I, and Figs. 1-10] The stated validity condition is violated by every numerical demonstration. Section II explicitly assumes a slowly rotating plasma, Ω0/ωci ≪ 1, and says that centrifugal and second- and higher-order Coriolis terms are neglected. However, Table I gives B0 = 1×10⁻⁹ T and mi = 1.67×10⁻²⁷ kg, so ωci = eB0/mi ≈ 0.096 s⁻¹ while Ω0 = 0.076 s⁻¹; the ratio is about 0.8, not ≪ 1. The figures use normalized Ω0 = 0.6-0.9 (e.g., Figs. 1, 2, 6-9), i.e., Ω0 ≈ ωci. Because the KdVB dispersion coefficient R in Eq. (23) and the NLS coefficients M and N in Eqs. (47)-(48) all depend on Ω0, the soliton, shock, MI, and rogue-wave results in Sections IV-VI are computed in a regime where the neglected Ω0² terms are comparable to the retained leading-order terms. The quantitative conclusions therefore do not follow from the model as derived.
  2. [Section V, Eqs. (46)-(52)] The MI analysis does not follow from the displayed NLS equation. Eq. (46) is written as iB_T + (1/2)M B_XX + N|B|²B = 0, but the perturbation dispersion relation in Eq. (51) and the growth rate in Eq. (52) are those for an NLS equation with coefficient M, not (1/2)M. Linearizing Eq. (46) around a constant-amplitude carrier gives Ω̃² = M N |B0|² K² - M²K⁴/4, whereas Eq. (51) gives Ω̃² = M²K²(K² - 2N|B0|²/M); these differ in both the factor 1/2 and the sign structure of the K⁴ term. In addition, Eq. (49) uses the KdV nonlinear coefficient Q in the nonlinear frequency shift, where the NLS coefficient N is required. Because Figs. 6-10 are generated from these expressions, the reported MI and rogue-wave parameter dependence is not supported as written.
  3. [Section V, Eq. (45)] The group velocity is inconsistent with the linear dispersion relation. From Eq. (44), ω = -Rk³, so the group velocity is vg = ∂ω/∂k = -3Rk², not 3Rk² as stated in Eq. (45). This sign error enters the definition of the slow coordinate X = ε(ξ - vgτ) and propagates into the reductive perturbation derivation of the NLS equation. The authors should correct Eq. (45) and re-derive or re-verify the NLS coefficients M and N and all quantities that depend on them.
minor comments (6)
  1. [Eq. (6)] The continuity equation is written as ∂ρ/∂t + ∂/∂t (ρvx) = 0, but the second term should be a spatial derivative, ∂/∂x (ρvx); this is presumably a typographical error.
  2. [Eq. (9)] Eq. (9) contains sin λ, but λ is not defined; from the surrounding equations and the rotation geometry, this should be sin θ.
  3. [Eq. (54)] The second-order rogue wave solution introduces the symbol P without a definition; if P is a scaling parameter or is meant to be M, this should be stated explicitly.
  4. [Fig. 4 caption] The figure caption says "cosmic ray diffusivity (η0), magnetic resistivity (κ0)", but the text and legends use κ0 for cosmic-ray diffusivity and η0 for magnetic resistivity; the parenthetical labels should be swapped.
  5. [Throughout] Several typographical and wording issues should be corrected: "evaluation" appears where "evolution" is meant (e.g., Section IV and Fig. 4 discussions), "rouge" appears for "rogue" in Section VI and the Conclusions, and the Conclusions contain the incomplete phrase "significant refinement in the wave propon frequency (Ω0)".
  6. [Abstract and Section V] The phrase "carrier wave frequency is considerably lower than the wave frequency" is obscure; the conditions under which the NLS reduction from the KdVB equation is valid should be stated more precisely.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the KdVB, NLS, MI, and rogue-wave results are computed in-paper from the stated fluid equations; minor self-citations are not load-bearing.

full rationale

The derivation chain is self-contained: the linear dispersion relation (Eqs. 13-15), the KdVB equation (21) with coefficients Q, R, S (Eqs. 22-24), the NLS equation (46) with M and N (Eqs. 47-48), and the MI growth rate (52) are all obtained by the paper's own normal-mode and reductive-perturbation algebra from the stated MHD equations (6)-(12). No parameter is fitted to the claimed damping, shock, MI, or rogue-wave behavior, and the solutions (26), (29), (38), (53), and (54) are standard mathematical solutions applied to in-paper coefficients. The self-citations to the authors' own Refs. 32, 37, and 18 for the model equations and typical parameters are real but not load-bearing, because the equations are written out in full and the new KdVB/NLS/MI results do not assume those papers' conclusions. The main caveat is a correctness concern, not circularity: Sec. II explicitly assumes slow rotation and states 'Denormalization of Ω0 implies Ω0/ωci ≪ 1', yet Table I lists Ω0 = 0.076 s^-1 with ωci ≈ 0.096 s^-1 (ratio ≈ 0.79) and the figures use normalized Ω0 = 0.7-0.9, placing the quantitative plots outside the stated validity regime. Similarly, Eq. (46) has the dispersion coefficient M/2, while Eq. (51) and Eq. (52) as written correspond to a different NLS convention, so the plotted MI curves may not follow from the displayed equation; this is an internal consistency issue, not a reduction to inputs. These concerns do not make the derivation circular, so the circularity score remains low.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a standard two-fluid cosmic-ray MHD model with parameter values from earlier literature, not on fitted constants. No new particles, forces, or dimensions are introduced. The main unstated burden is the small-rotation assumption, which the figures violate, and the ϵ-scaling choices that make the dissipation enter the KdVB equation.

assumptions (5)
  • domain assumption The interstellar medium is modeled as a homogeneous, fully ionized, polytropic thermal gas plus a massless cosmic-ray pressure fluid governed by a diffusion-convection equation.
    Equations (1)-(5) and (12) define this two-fluid pressure coupling; it neglects cosmic-ray density, momentum, and any anisotropic pressure effects.
  • domain assumption The plasma rotates slowly enough that Ω0/ωci ≪ 1, allowing centrifugal force and higher-order Coriolis terms to be neglected.
    Stated in Section II, but the numerical values used in Table I and the figures have Ω0/ωci near 0.8, violating this assumption.
  • standard math Cosmic-ray diffusion and magnetic resistivity are weak and are scaled as η, κ ∼ ϵ^{1/2}η0, κ0 in the reductive perturbation expansion.
    This is the standard asymptotic ordering that produces a KdVB balance in Section IV A; it is internally used to keep dissipation at the same order as nonlinearity and dispersion.
  • domain assumption For the MI and rogue wave analysis, cosmic-ray diffusion and magnetic resistivity are neglected, justified by high magnetic Reynolds number and frozen-in-field conditions.
    Section V assumes this to reduce the KdVB equation to the KdV equation and then to the NLS equation, discarding the very dissipation terms that are central to the shock section.
  • domain assumption The normalized parameter values in Table I are representative of the spiral galaxy interstellar medium.
    The values are taken from Refs. 32 and 39, with no error bars or sensitivity to the chosen normalization.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Dynamics and modulation of cosmic ray modified magnetosonic waves in a galactic gaseous rotating plasma." pith.science (2026). https://pith.science/paper/EDBLTSZP

@misc{pith2026241116286,
  author       = {Pith},
  title        = {Pith review of: Dynamics and modulation of cosmic ray modified magnetosonic waves in a galactic gaseous rotating plasma},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EDBLTSZP}},
  note         = {Machine review of arXiv:2411.16286}
}
read the original abstract

The influence of the presence of cosmic fluid on the magnetosonic waves and modulation instabilities in the interstellar medium of spiral galaxies is investigated. The fluid model is developed by modifying the pressure equation in such dissipative rotating magnetoplasmas incorporating thermal ionized gas and cosmic rays. Applying the normal mode analysis, a modified dispersion relation is derived to study linear magnetosonic wave modes and their instabilities. The cosmic rays influence the wave damping by accelerating the damping rate. The standard reductive perturbation method is employed in the fluid model leading to a Korteweg de Vries Burgers (KdVB) equation in the small-amplitude limit. Several nonlinear wave shapes are assessed by solving the KdVB equation, analytically and numerically. The cosmic ray diffusivity and magnetic resistivity are responsible for the generation of shock waves. The modulational instability (MI) and the rogue wave solutions of the magnetosonic waves are studied by deriving a nonlinear Schrodinger (NLS) equation from the obtained KdVB equation under the assumption that the cosmic ray diffusion and magnetic resistivity are weak and the carrier wave frequency is considerably lower than the wave frequency. The influence of various plasma parameters on the growth rate of MI is examined. The modification of the pressure term due to cosmic fluid reduces the MI growth in the interstellar medium. In addition, a quantitative analysis of the characteristics of rogue wave solutions is presented. Our investigation's applicability to the interstellar medium of spiral galaxies is traced out.

Figures

Figures reproduced from arXiv: 2411.16286 by the authors.

Figure 1
Figure 1. FIG. 1: The real part [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The typical one-soliton profiles [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Two-soliton profiles [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The magnetosonic shock profiles [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The magnetosonic oscillatory shock profiles [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The development of the instability growth rate Γ [see Eq. (52 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The three-dimensional first-order rogue wave profiles [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: The first-order rogue wave pulses [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: The second-order rogue wave pulses [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: A comparison between first- and second-order rogue wa [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

  1. [2]

    The coefficients M and N are modified due to the inclusion of cosmic ray pressure, which affects the growth of the instability

    The growth rate of instability depends on the coefficients M and N (which are further influenced by various plasma parameters), any changes in the plasma parameters certainly change the growth rate of instabil- ity. The coefficients M and N are modified due to the inclusion of cosmic ray pressure, which affects the growth of the instability. The graphs of the gr...

    work page

  2. [3]

    displayed in Fig. 1. The plots of the real part ωr = ωr(k) are shown in the subplot (a) for different values of the parameters Ω 0, Cg and Cc associated with the rotational frequency, thermal pressure and cosmic ray pressure, respectively. It is found that initially, for k ≪ 1, the profile of ωr remains parallel to the axis of wave num- ber, i.e., the phase...

    work page

  3. [4]

    Due to the effect of cosmic ray modified pressure, cosmic ray diffusivity, and magnetic resistivity, the dissipation term in Eq

    It has been no- ticed that when Cc is fixed, the amplitude is shifted to lower values and the width becomes wider for increasing values of Cg; however, the amplitude increases and the width becomes narrower for increasing values of Cc. Due to the effect of cosmic ray modified pressure, cosmic ray diffusivity, and magnetic resistivity, the dissipation term in Eq. (

    work page

  4. [5]

    frozen-in-field

    Subplots (a) and (b) show the effects of Cg and Cc on the oscillatory shock structures. We ob- serve that as Cg increases, the strength and width of the oscillatory shock grow, whereas they become smaller and tend to damp faster as Cc increases. Subplots (c) and (d) illustrate how the rotational frequency Ω 0 with the angle of rotation θ affects the oscilla...

    work page

  5. [6]

    Now, we move on to find various types of soliton and shock wave solutions from Eq

    reduces to the Burgers equation having a stationary shock profile. Now, we move on to find various types of soliton and shock wave solutions from Eq. (

    work page

  6. [7]

    frozen-in-field

    for different cases. B. Soliton Solutions In the turbulence zones of ISM of spiral galaxies, such as galaxy centre, star-forming regions, and supernova remnants, generally the magnetic Reynold number is high, and the magnetic fields follow the “frozen-in-field” condition on large scales. In such regions, magnetic fields can sustain over a long period of time ...

    work page

  7. [8]

    and Cg, Cc (subplots (b) of Fig. 8). The figures illustrate that as the values of the parameters Ω 0 and θ increase both the amplitude and width decrease, i.e., an increase in Ω 0 and/or θ leads to the contraction of the rogue wave pulse. On the other hand, we notice that the inclusion of cosmic ray pressure with the ther- mal pressure significantly enlarge...

    work page

  8. [10]

    Similar assumptions have been made previously in many experi- mental situations 43,44

    and ( 12), respectively, are al- most linearly proportional to the number density. Similar assumptions have been made previously in many experi- mental situations 43,44. For large values of η0 and κ0, one can choose a higher order of ǫ (i.e., η ∼ ǫη0, κ ∼ ǫκ0) and in those cases sharp rising shock waves are noticed compared to oscillatory and monotonic sh...

    work page

Show all 22 references
  1. [11]

    Therefore, Fig

    is dominant. Therefore, Fig. 4 indicates that the energy dissipation caused by the cosmic ray effects leads to the temporal evaluation of shock wave profiles in the ISM of spiral galaxies. In other words, the ISM’s shock wave structures are key mechanisms for accelerat- ing cosm...

  2. [12]

    to look into further potential structures and their characteristics. To 9 0 10 20 30 40 50 0.999995 0.999996 0.999997 0.999998 0.999999 1 1.000001 1.000002 1.000003B Cg=0.5 Cg=0.7 Cg=0.9 (a) 0 10 20 30 40 50 0.999994 0.999995 0.999996 0.999997 0.999998 0.999999 1 1.000001 1.00...

  3. [13]

    (15) The dispersion ( 15) includes the effect of the Coriolis force due to the rotation of the plasma

    reduces to ω2 = ( C2 g + C2 c ω ω + iκk2 + ω ω + iηk 2 ) k2 + 4Ω 2 0. (15) The dispersion ( 15) includes the effect of the Coriolis force due to the rotation of the plasma. Assuming the plasma to be a highly conducting fluid, if the effect of magnetic resistivity is ignored, the ...

  4. [14]

    to study the effects of different parameters in- volved in the model, and the results obtained are dis- played in Fig

  5. [15]

    reduces to the dispersion relation as obtained by Turi and Misra 32 upon denormalization, in the absence of self-gravitation. Further, disregarding the effect of Coriolis force due to the rotation of the fluid, cosmic ray pressure, and cosmic ray diffusion, one can re- cover the ...

  6. [17]

    (53) The first-order rational rogue solution ( 53) reveals that a significant amount of magnetosonic wave energy is con- centrated in a relatively small area in space

    admits rational solutions lo- calized in both space and time variables 20,49 as follows, B1(X, T ) = √ M/N [ 4(1 + 2iM T) 1 + 4X 2 + 4M 2T 2 − 1 ] exp(iM T). (53) The first-order rational rogue solution ( 53) reveals that a significant amount of magnetosonic wave energy is con- ...

  7. [18]

    is given by 51, B2(X, T ) = √ M/N ( 1 + M2 + iN2 O2 ) exp(iM T), (54) where M2 = 3 8 − 1 2 X 4 − 3 2 X 2 − 6(P XT )2 − 10(P T)4 − 9(P T)2, N2 = −P T [ − 15 4 + X 4 − 3X 2 + 4(P XT )2 + 4(P T)4 +2(P T)2 ] , O2 = 3 32 + 1 12 X 6 + 1 8 X 4 + 1 2 X 4(P T)2 + 9 16 X 2 + X 2(P T)4 −...

  8. [21]

    In other words, the parameters κ0 and η0 in the present model yield the formation of shock structures

    reduces to the well-known KdV equation, and the solution can be obtained in terms of a solitary wave pulse. In other words, the parameters κ0 and η0 in the present model yield the formation of shock structures. In the limiting case, θ = π/2, the dispersion coefficient vanishes, ...

  9. [22]

    It has been reported that the second-order rogue waves absorb more energy from the surrounding waves than the first-order rogue waves do, making them spikier than the first-order

    The figure illustrates that the second-order correction of the rogue wave solution leads to an enhancement of the amplitude and width of the wave pulses. It has been reported that the second-order rogue waves absorb more energy from the surrounding waves than the first-order rog...

    1966

  10. [23]

    The waves become unstable due to the increase in the values of the parameters η and κ

    remains the same as obtained in Turi and Misra 32 in the absence of both cosmic ray dif- fusivity (κ) and magnetic resistivity ( η). The waves become unstable due to the increase in the values of the parameters η and κ. However, interestingly, the damping caused by cosmic ray ...

    2020

  11. [38]

    The graphical representations of such profiles have been displayed in Fig

    rep- resents an admixture of sech 2 and tanh type waveform. The graphical representations of such profiles have been displayed in Fig. 4 for distinct parametric values, as stated in the caption of the corresponding figure. The figures reveal that, due to the increments in the val...

  12. [46]

    The study of such low-frequency waves and the modulation of sinusoidal waves, which describe various physical phenomena in plasma, is quite natural

    in the limit of low-frequency wave48. The study of such low-frequency waves and the modulation of sinusoidal waves, which describe various physical phenomena in plasma, is quite natural. To study the MI, we consider a small perturbation δB such that B = B0 + δB exp i∆ B. B0 is...

  13. [51]

    However, in our case where M N > 0, the region carries bright (unstable) envelope solitons in the presence of small external perturbation

    possesses real solutions in terms of ˜Ω for all real values of K, consequently, the magnetosonic waves become dark (stable) envelope soli- tons in this region. However, in our case where M N > 0, the region carries bright (unstable) envelope solitons in the presence of small e...

  14. [53]

    Initially, the typical first-order rogue wave profiles |B1| are plotted against X and T for two differ- ent values of Ω 0 in Fig

    and ( 54). Initially, the typical first-order rogue wave profiles |B1| are plotted against X and T for two differ- ent values of Ω 0 in Fig. 7, keeping all other parameters fixed as mentioned in the figure caption. It has been ob- served that the amplitude and width of the rogue wa...

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.