Pith. sign in

REVIEW 4 major objections 4 minor 66 references

Lie Symmetries for the Shallow Water Magnetohydrodynamics Equations in a Rotating Reference Frame

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read The paper classifies the Lie point symmetries of the rotating-frame SWMHD equations and, in the fully general case, derives the one-dimensional optimal system that exhausts the inequivalent similarity reductions.

desk verdict The three simpler cases may be right, but the general-case algebra and optimal system are built on a dilation that is not admitted when gravity is present. read the letter →

arxiv 2412.14578 v1 pith:XRP73QBJ submitted 2024-12-19 math-ph math.APmath.MPphysics.plasm-ph

Andronikos Paliathanasis , Amlan Halder This is my paper · ORCID
classification math-phmath.APmath.MPphysics.plasm-ph MSC 35A3076W0576B15
keywords Liesymmetriesshallowwatermagnetohydrodynamicsone-dimensionaloptimalsystemsimilaritytransformationsadjointrepresentationrotatingreferenceframeCoriolisforcehyperbolic
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

The paper classifies the Lie point symmetries of the one-dimensional shallow-water magnetohydrodynamics (SWMHD) equations in a rotating frame, working with the relaxed magnetic-divergence condition $\nabla(h\mathbf{B})\neq 0$ that keeps one-dimensional solutions physical and restores Galilean invariance in the non-rotating case. The symmetry algebra is shown to depend on whether the constant gravitational potential $g$ and the Coriolis parameter $f_0$ vanish: ten symmetries when both are absent, eight with gravity alone, seven with rotation alone, and six when both are present. For the general rotating case the paper computes the invariants of the adjoint action and lists the twenty-one one-dimensional subalgebras of the optimal system, then turns those generators into similarity transformations that reduce the five-equation hyperbolic system to ordinary differential equations. Several reductions yield closed-form solutions, including shock waves and solitons. A correct classification matters because it makes the search for analytic solutions systematic rather than ad hoc.

What carries the argument

The load-bearing object is the infinitesimal point symmetry generator $X=\xi^t\partial_t+\xi^x\partial_x+\eta^h\partial_h+\eta^u\partial_u+\eta^v\partial_v+\eta^a\partial_a+\eta^b\partial_b$ and its first prolongation; requiring the prolonged generator to annihilate the five equations modulo the system produces the determining equations whose solutions are the admitted symmetries. The classification is carried by the commutator table and the adjoint representation $\mathrm{Ad}(\exp(\varepsilon X_i))X_j$ of each algebra; the invariant functions of the adjoint action are found by solving linear first-order PDEs, and those invariants select the representatives of the one-dimensional optimal system. The relaxed divergence condition $\nabla(h\mathbf{B})\neq 0$ is itself part of the machinery, because it alters the symmetry algebra relative to the $\nabla(h\mathbf{B})=0$ formulation and restores Galilean invariance in the free system.

What would settle it

Recompute the adjoint action of $X_1$ on $X_5$: because $[X_1,X_5]=X_2$, formula (41) gives $\mathrm{Ad}(\exp(\varepsilon X_1))X_5=X_5-\varepsilon X_2$, while Table 2 lists $X_5-2X_2$; checking which expression is correct, and rerunning the invariant equations (62)--(67) with the corrected table, settles whether the printed optimal system is established.

Watch

Extended reading notes

Core claim

The central claim is that the Lie point symmetry algebras of the system (20)--(24) are, in the four parameter regimes, $L^{10}=\{A_{3,3}\rtimes A_{2,1}\}\otimes_s A^a_{5,34}$ (free), $L^{8}=A_{2,1}\rtimes A_{6,22}$ (gravity only), $L^{7}=A_{3,5}\rtimes\{A_{2,1}\rtimes A_{2,1}\}$ (rotation only), and $L^{6}=A_{3,5}\rtimes A_{3,3}$ (rotation plus gravity), where the names follow the Morozov--Mubarakzyanov--Patera classification of low-dimensional Lie algebras. The same tables show that Galilean symmetries are present in the non-rotating cases and are lost when the Coriolis term is switched on. For the general case $g f_0\neq 0$, the adjoint invariants reduce a generic generator to $a_1X_1+z_1Z_1$, with three special families, and the resulting one-dimensional optimal system contains twenty-one inequivalent subalgebras. Applying those subalgebras as similarity transformations reduces the five-field hyperbolic system to ODE systems, and for generators such as $Z_2$, $Z_3$, $X_{10}+z_2Z_2$, $X_{10}+z_3Z_3$, and $X_2+a_{10}X_{10}+z_2Z_2$, explicit closed-form solutions are given, describing shock waves and solitons.

Load-bearing premise

The classification and the optimal system assume that the determining equations were solved exhaustively and that the commutator and adjoint tables are free of algebraic slips; if any table entry is wrong, the named algebras and the Section 5 optimal system would need to be recomputed.

Editorial extensions

If this is right

  • In the non-rotating free system the ten symmetries include the Galilean boosts $X_5=t\partial_x+\partial_u$ and the scaling $X_3$, so the relaxed divergence condition restores Galilean invariance for one-dimensional SWMHD.
  • Switch on a constant gravitational field and the algebra drops from ten to eight dimensions, with the combined scaling $Y=X_9-2X_4$ replacing two separate scaling symmetries.
  • Introducing the Coriolis term removes the Galilean boost; the rotating algebras are seven-dimensional without gravity and six-dimensional with gravity, so the two parameters together are not equivalent to either alone.
  • For the general rotating case, the twenty-one elements of the one-dimensional optimal system exhaust the inequivalent similarity reductions, so no one-dimensional symmetry reduction is missed.
  • Closed-form reductions include shock-wave solutions with walls at $f_0 t=n\pi$ or $f_0 t=n\pi/2$ and soliton-type solutions for the $X_2+a_{10}X_{10}+z_2Z_2$ reduction when $|z_2|\le 1$.

Reading between the lines

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

  • The same adjoint-invariant procedure could be applied to the seven- and eight-dimensional algebras of the other parameter regimes; the paper computes the optimal system only for the general rotating case.
  • The closed-form $Z_2$ and $Z_3$ solutions are natural manufactured solutions for numerical Riemann solvers built on the relaxed condition; a scheme that fails to reproduce the shock wall at $\sin(f_0 t)=0$ would be suspect.
  • Because the completeness of the determining-equation solution is asserted rather than shown, a direct computer-algebra rerun of Sections 4 and 5 would settle whether the printed classification and optimal system are exhaustive.
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

4 major / 4 minor

Summary. The manuscript performs a Lie point symmetry classification for the one-dimensional shallow water magnetohydrodynamics (SWMHD) system with a rotating reference frame and a constant gravitational field, considering the physically motivated relaxation ∇(hB) ≠ 0. For the four parameter cases (g=0,f0=0), (g≠0,f0=0), (g=0,f0≠0), (g≠0,f0≠0), the authors claim ten-, eight-, seven-, and six-dimensional Lie algebras, respectively, identify them in the Morozov–Mubarakzyanov–Patera scheme, and for the general case construct a one-dimensional optimal system and use it to generate similarity reductions and analytic solutions. The paper includes the determining equations, explicit vector fields, commutator and adjoint tables, and several reduced ordinary differential systems.

Significance. If correct, the paper would provide a complete symmetry classification and an optimal-system catalog for a physically relevant hyperbolic system, which is a standard and useful contribution to the symmetry-analysis literature. The explicit listing of the determining equations and the attempt at a full optimal system are commendable features. However, the correctness of the main results is not established: several listed vector fields do not satisfy the authors' own symmetry conditions, the optimal system contains elements that are not in the admitted algebra, and at least one similarity reduction is internally inconsistent. These are load-bearing issues that affect the classification and the catalog of solutions, so the paper's central claims are not supported in its current form.

major comments (4)
  1. [§4.3, Eq. (58) and §4.3.1] The symmetry condition (58) for the rotating case requires ηh = 2h(ξx,x − ξt,t) (with H evidently a typographical form of h). For the vector field Z1 = X3 + X8 − X9 = x∂x + u∂u + v∂v + a∂a + b∂b listed in §4.3.1, one has ξx,x = 1 and ξt,t = 0, so the condition forces ηh = 2h, whereas the written vector field has ηh = 0. Therefore Z1 as defined is not a Lie symmetry of system (52)–(56), which invalidates the seven-dimensional algebra L7 and the commutator Table 6.
  2. [§4.4.1 and §5] For the general case f0g ≠ 0, the admitted dilation must include the term 2h∂h because the condition in §4.4 gives ηh = 2hξx,x. The paper does not redefine Z1 in §4.4, tacitly reusing the Z1 of §4.3, which lacks the h-term. Consequently, the bracket [X10, Z1] is not 2X10 as listed in Table 6; for X10 = (ah)^{−1}∂b the commutation with the corrected Z1 gives 3X10. Since the invariant system (62)–(67) and the optimal system of Section 5 are built from the incorrect 2X10 bracket, the optimal system is not established for the actually admitted algebra. Moreover, the optimal system lists {X3} even though X3 = t∂t + x∂x is not among the six admitted symmetries of L6, and §6.3 explicitly reduces with X3, which is not a symmetry for f0g ≠ 0.
  3. [§6.1, Eqs. (73)–(81)] The reduction with X1 begins with the static ansatz h = h(x), u = u(x), v = v(x), a = a(x), b = b(x), but the result contains explicit time dependence: equations (79)–(80) give v and b as linear functions of t, and equation (81) is declared to be an ordinary differential equation for h(t). This contradicts the original ansatz and means the presented solution is not a similarity reduction with X1 in the sense defined in Section 2.2. The inconsistency indicates that the reduction was not derived correctly from the invariants of X1.
  4. [§4.1, Eqs. (35)–(40)] The solution of the determining equations (35)–(40) (and the similar systems in §§4.2–4.4) is asserted without a derivation or an explanation of the splitting procedure used to separate the conditions in the dependent variables. Because the claimed algebra classifications and all subsequent optimal-system and reduction results depend on these solutions, the reader cannot verify that the listed vector fields are complete. In particular, the absence of a verifiable derivation makes the asserted identifications of L10, L8, L7, and L6 unsupported, especially given the concrete errors identified in the listed vector fields above.
minor comments (4)
  1. [Table 2] The adjoint entry Ad(exp(εX1))X5 is listed as X5 − 2X2, but Table 1 gives [X1, X5] = X2, so the standard adjoint formula would give X5 − εX2. Unless a different convention is intended, this appears to be an error in the adjoint representation table.
  2. [Eq. (58)] The symbol H in the expression ηh = 2H(ξx,x − ξt,t) is undefined; it should presumably be the dependent variable h, and this typographical issue should be corrected.
  3. [Throughout] There are numerous typographical errors, including 'Galileon symmetries' (should be 'Galilean'), 'shoch waves' in §6.10, 'fist two cases' in the introductory paragraph of Section 4, and 'W remark' in §4.1.1. These should be corrected in a revision.
  4. [References] The reference list and in-text citations contain apparent duplicate numbering, such as [30] appearing twice and [31] being cited twice in the sentence at the end of Section 1. The references should be renumbered consistently.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Lie symmetries are derived by solving the invariance conditions (35)–(40) and reductions are obtained by substituting the resulting invariants; self-citations are background.

full rationale

The paper's central derivation chain is self-contained: the four symmetry algebras are obtained by solving the determining equations (35)–(40), (47)–(51), (57)–(61) and the Section 4.4 conditions, i.e. by imposing the Lie invariance condition (8) on the given PDE system (20)–(24). The constants g and f0 are model parameters, not fitted values, and no quantity called a prediction is recovered from a fit. The optimal system in Section 5 is constructed from adjoint invariants (62)–(67) computed from the commutator table, and the similarity reductions in Section 6 are direct substitutions of invariant forms such as h=x^2H(t), u=xU(t). The Morozov–Mubarakzyanov–Patera labels are external classification results, not self-citations; the author self-citations (e.g. [32], [33], [40]) provide context and do not carry the load of the classification. There are internal consistency concerns (e.g. Table 2's Ad(exp(εX1))X5 = X5−2X2 appears inconsistent with [X1,X5]=X2, and the reuse of Tables 6–7 for L6 in Section 4.4 assumes the same Z1 with 2X10 bracket), but those are correctness risks, not circularity: they do not make the output equal to the input by construction. No step was found in which a result is defined in terms of the quantity it is supposed to predict, or in which a fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; g and f0 are symbolic model constants. The analysis assumes the physical validity of the relaxed-divergence SWMHD model and the completeness of the standard symmetry computation. No new entities are introduced.

assumptions (4)
  • domain assumption The one-dimensional SWMHD system (20)-(24) with the relaxed condition ∇(hB)≠0 is the correct physical model for the intended applications.
    Section 3 adopts this model as physically acceptable based on cited numerical studies [44-47,52,54-56]; if the relaxation is not accepted, the symmetry results describe a different system.
  • standard math The Lie symmetry condition (8), with first prolongation, captures all point symmetries of the system.
    Sections 2.1 and 4 use the standard invariance criterion; completeness of the determining-equation solution is assumed.
  • standard math The identification of the symmetry algebras with A3,3⋊A2,1⋊A^a_{5,34}, A2,1⋊A6,22, A3,5⋊{A2,1⋊A2,1}, and A3,5⋊A3,3 follows the Morozov-Mubarakzyanov-Patera classification.
    Sections 4.1.1, 4.2.1, 4.3.1, and 4.4.1 rely on the external classification scheme [57-61].
  • ad hoc to paper The adjoint representation tables used for the optimal system are correct.
    Section 5 depends on Tables 2, 3, 5, 7, and 8; at least one entry (Table 2, X1 row, X5 column) appears inconsistent with Table 1, so this assumption is currently doubtful.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Lie Symmetries for the Shallow Water Magnetohydrodynamics Equations in a Rotating Reference Frame." pith.science (2026). https://pith.science/paper/XRP73QBJ

@misc{pith2026241214578,
  author       = {Pith},
  title        = {Pith review of: Lie Symmetries for the Shallow Water Magnetohydrodynamics Equations in a Rotating Reference Frame},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XRP73QBJ}},
  note         = {Machine review of arXiv:2412.14578}
}
abstract

We perform a detailed Lie symmetry analysis for the hyperbolic system of partial differential equations that describe the one-dimensional Shallow Water magnetohydrodynamics equations within a rotating reference frame. We consider a relaxing condition $\mathbf{\mathbf{\nabla }}\left( h\mathbf{B} \right) \neq 0$ for the one-dimensional problem, which has been used to overcome unphysical behaviors. The hyperbolic system of partial differential equations depends on two parameters: the constant gravitational potential $g$ and the Coriolis term $f_{0}$, related to the constant rotation of the reference frame. For four different cases, namely $g=0,~f_{0}=0$; $g\neq 0\,,~f_{0}=0$; $g=0$, $f_{0}\neq 0$; and $g\neq 0$, $f_{0}\neq 0$ the admitted Lie symmetries for the hyperbolic system form different Lie algebras. Specifically the admitted Lie algebras are the $L^{10}=\left\{ A_{3,3}\rtimes A_{2,1}\right\} \otimes _{s}A_{5,34}^{a}$; $% L^{8}=A_{2,1}\rtimes A_{6,22}$; $L^{7}=A_{3,5}\rtimes\left\{ A_{2,1}\rtimes A_{2,1}\right\} $; and $L^{6}=A_{3,5}\rtimes A_{3,3}~$respectively, where we use the Morozov-Mubarakzyanov-Patera classification scheme. For the general case where $f_{0}g\neq 0$, we derive all the invariants for the Adjoint action of the Lie algebra $L^{6}$ and its subalgebras, and we calculate all the elements of the one-dimensional optimal system. These elements are then considered to define similarity transformations and construct analytic solutions for the hyperbolic system.

Figures

Figures reproduced from arXiv: 2412.14578 by the authors.

Figure 1
Figure 1. Qualitative evolution of the functions h (ξ) and v (ξ) as they are given by numerical simulation of the nonlinear system (111), (112). 6.8 Reduction with X2 + z2Z2 From the vector field X2 + z2Z2 we define the similarity transformation ζ = x + z2 f0 cos (f0t), h = h (ζ), a = a (ζ), b = b (ζ), (113) u = z2 sin (f0t) + U (ζ), v = −z2 cos (f0t) + V (ζ). (114) Therefore the hyperbolic system (20), (21), (22), (23) and (… view at source ↗
Figure 2
Figure 2. Qualitative evolution for the function h (ζ) as it is given by numerical simulation of the nonlinear differential equation (117). 6.9 Reduction with X10 + z2Z2 From the Lie symmetry vector X10 + z2Z2 we find the similarity transformation h = h (t), u = cot (f0t) f0x + U (t), v = f0x + V (t), (118) a = a (t), b = x z2 sin (f0t) h (t) a (t) + B (t). (119) Thus, from the SWMHD equations it follows h (t) = h0 sin (f0t) … view at source ↗
Figure 3
Figure 3. Qualitative evolution for the similarity solution of [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

66 extracted references · 66 canonical work pages

  1. [1]

    A.J.C. Barre de Saint-Venant, Th´ eorie du mouvement non permanent des eaux, avec appli- cation aux crues des rivi` eres et a l’introduction de mar´ ees dans leurs lits, Comptes Rendus de l’Acad´ emie des Sciences 73, 147 (1871)

    work page

  2. [2]

    Castro-Orgaz and W.H

    O. Castro-Orgaz and W.H. Hager, Shallow Water Hydraulics, Springer Cham, New York (2019)

    work page 2019

  3. [3]

    Paldor, Shallow Water Waves on the Rotating Earth, Springer Cham, New York (2015) 27

    N. Paldor, Shallow Water Waves on the Rotating Earth, Springer Cham, New York (2015) 27

    work page 2015

  4. [4]

    Valery and P

    B.K. Valery and P. James Lynch, Fundamentals of Shallow Water Acoustics, Springer New York, New York (2012)

    work page 2012

  5. [5]

    Garc ´ ıa-Navarro, J

    P. Garc ´ ıa-Navarro, J. Murillo, J. Fern´ andez-Pato, I. Echeverribar and M. Morales- Hernandez, The shallow water equations and their application to realistic cases, Envi- romental Fluid Mechanics 19, 1235 (2019)

    work page 2019

  6. [6]

    Kurganov, Finite-volume schemes for shallow-water equations, Acta Numerica 27, 289 (2018)

    A. Kurganov, Finite-volume schemes for shallow-water equations, Acta Numerica 27, 289 (2018)

    work page 2018

  7. [7]

    Kinnmark, The Shallow Water Wave Equations: Formulation, Analysis and Application, Lecture Notes in Engineering, Springer Berlin, Heidelberg (1986)

    I. Kinnmark, The Shallow Water Wave Equations: Formulation, Analysis and Application, Lecture Notes in Engineering, Springer Berlin, Heidelberg (1986)

    work page 1986

  8. [8]

    Taitel and D

    Y. Taitel and D. Barena, Encyclopedia of Two-Phase Heat Transfer and Flow I, Funda- mentals and Methods Volume 1 :Modeling of Gas Liquid Flow in Pipes, edited by J.R. Thome, World Scientiffic Publishing Co. Pte. Ltd., Singapure, (2015)

    work page 2015

Show all 66 references
  1. [9]

    Hewitt, Two-Phase Flow and Its Applications: Past, Present, and Future, Heat Transfer Engineering, 4, 67 (1983)

    G.F. Hewitt, Two-Phase Flow and Its Applications: Past, Present, and Future, Heat Transfer Engineering, 4, 67 (1983)

    1983

  2. [10]

    shallow water

    P.A. Gilman, Magnetohydrodynamic “shallow water” equations for the solar tachocline, Astroph. J. 544, L79 (2000)

    2000

  3. [11]

    Zahn, The solar tachocline, A&A 265, 106 (1992)

    E.A Spiegel and J.-P. Zahn, The solar tachocline, A&A 265, 106 (1992)

    1992

  4. [12]

    Hyghes, R

    D.W. Hyghes, R. Rosner and N.O. Weiss, The Solar Tachocline, Cambridge University Press, Cambridger (2009)

    2009

  5. [13]

    Schecter, J.F Boyd and P.A

    D.A. Schecter, J.F Boyd and P.A. Gilman, ”Shallow-Water” Magnetohydrodynamic Waves in the Solar Tachocline, Ap. J. 551, L185 (2001)

    2001

  6. [14]

    shallow water

    T. V. Zaqarashvili, R. Oliver, J.L. Ballester and B.M. Shergelashvili, Rossby waves in “shallow water” magnetohydrodynamics, A&A 470, 815 (2007)

    2007

  7. [15]

    Hindle, P.J

    A.W. Hindle, P.J. Bunsby and T.M. Rogers, Observational Consequences of Shallow-water Magnetohydrodynamics on Hot Jupiters, Ap.J. Lett. 916, L8 (2021)

    2021

  8. [16]

    Alonso-Oran, Asymptotic Shallow Models Arising in Magnetohydrodynamics, Water Waves 3, 371 (2021) 28

    D. Alonso-Oran, Asymptotic Shallow Models Arising in Magnetohydrodynamics, Water Waves 3, 371 (2021) 28

    2021

  9. [17]

    Petrosyan, D

    A. Petrosyan, D. Klimachkov, M. Fedotova and T. Zinyakov, Shallow Water Magnetohy- drodynamics in Plasma Astrophysics. Waves, Turbulence, and Zonal Flows, Atmosphere 11, 314 (2020)

    2020

  10. [18]

    Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, Volume I: Symmetries, Exact Solutions, and Conservation Laws, CRS Press LLC, Florida (2000)

    N.H. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, Volume I: Symmetries, Exact Solutions, and Conservation Laws, CRS Press LLC, Florida (2000)

    2000

  11. [19]

    Bluman and S

    G.W. Bluman and S. Kumei, Symmetries of Differential Equations, Springer-Verlag, New York, (1989)

    1989

  12. [20]

    Stephani, Differential Equations: Their Solutions Using Symmetry, Cambridge Univer- sity Press, New York, (1989)

    H. Stephani, Differential Equations: Their Solutions Using Symmetry, Cambridge Univer- sity Press, New York, (1989)

    1989

  13. [21]

    Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, (1993)

    P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, (1993)

    1993

  14. [22]

    Gwaxa, S

    B. Gwaxa, S. Jamal and A.G. Johnpillai, On the Optimal System and Series Solutions of Fifth-Order Fujimoto-Watanabe Equations 17, 557 (2023)

    2023

  15. [23]

    Moyo and P.G.L

    S. Moyo and P.G.L. Leach, Symmetry Properties of Autonomous Integrating Factors, Sigma 1, 024 (2005)

    2005

  16. [24]

    Christodoulakis, A

    T. Christodoulakis, A. Karagiorgos and A. Zampeli, Symmetries in Classical and Quantum Treatment of Einstein’s Cosmological Equations and Mini-Superspace Actions, Symmetry 10, 70 (2018)

    2018

  17. [25]

    Christodoulakis, N

    T. Christodoulakis, N. Dimakis and P.A. Terzis, Lie point and variational symmetries in minisuperspace Einstein gravity, J. Math. Phys.: Math. Theor. 47, 095202 (2014)

    2014

  18. [26]

    Paliathanasis, Geometric Linearization for Constraint Hamiltonian Systems, Symmetry 16, 988 (2024)

    A. Paliathanasis, Geometric Linearization for Constraint Hamiltonian Systems, Symmetry 16, 988 (2024)

    2024

  19. [27]

    Aliy, J.S

    A.I. Aliy, J.S. Yusuf, M.M. Nauman, D.U. Ozsahin, B.G. Agaie, J.H. Zaini and H. Umar, Lie Symmetry Analysis and Explicit Solutions to the Estevez–Mansfield–Clarkson Equa- tion, Symmetry 16, 1194 (2024)

    2024

  20. [28]

    Chesnokov, Symmetries and exact solutions of the shallow water equations for a two-dimensional shear flow, J

    A.A. Chesnokov, Symmetries and exact solutions of the shallow water equations for a two-dimensional shear flow, J. Appl. Mech. Techn. Phys. 49, 737 (2008)

    2008

  21. [29]

    Bira, T.R

    B. Bira, T.R. Sekhar and D. Zeidan, Application of Lie groups to compressible model of two-phase flows, Comput. Math. Appl. 71, 46 (2016) 29

    2016

  22. [30]

    Chesnokov, Symmetries and exact solutions of the rotating shallow-water equations, Eur

    A.A. Chesnokov, Symmetries and exact solutions of the rotating shallow-water equations, Eur. J. Appl. Math. 20, 461 (2009)

    2009

  23. [31]

    Aksenov and K.P

    A.V. Aksenov and K.P. Druzhkov, Conservation laws and symmetries of the shallow water system above rough bottom, J. Phys.: Conf. Ser. 722, 012001 (2016)

    2016

  24. [32]

    Paliathanasis, One-dimensional optimal system for 2D Rotating Ideal Gas, Symmetry 11, 1115 (2019)

    A. Paliathanasis, One-dimensional optimal system for 2D Rotating Ideal Gas, Symmetry 11, 1115 (2019)

    2019

  25. [33]

    Paliathanasis, Similarity transformations for modified shallow water equations with density dependence on the average temperature, Int

    A. Paliathanasis, Similarity transformations for modified shallow water equations with density dependence on the average temperature, Int. J. Nonl. Sci. Num. Sim. 24, 1095 (2023)

    2023

  26. [34]

    Pavlenko, Symmetries and solutions to equations of two-dimensional motions of poly- tropic gas, Sib

    A.S. Pavlenko, Symmetries and solutions to equations of two-dimensional motions of poly- tropic gas, Sib. `Elektron. Mat. Izv., 2, 291 (2005)

    2005

  27. [35]

    Huo and L

    C. Huo and L. Li, Lie Symmetry Analysis, Particular Solutions and Conservation Laws of a New Extended (3+1)-Dimensional Shallow Water Wave Equation, Symmetry 14, 1855 (2022)

    2022

  28. [36]

    Dorodnitsyn, E.I

    V.A. Dorodnitsyn, E.I. Kaptsov, R.V. Kozlov and S. Meleshko, One-dimensional MHD flows with cylindrical symmetry: Lie symmetries and conservation laws, Int. J. Non-Linear Mechanics 148, 104290 (2023)

    2023

  29. [37]

    Webb and G.P

    G.M. Webb and G.P. Zank, Fluid relabelling symmetries, Lie point symmetries and the Lagrangian map in magnetohydrodynamics and gas dynamics, J. Phys. A.: Math. Theor. 40, 545 (2007)

    2007

  30. [38]

    Bogoyavlenskij, Restricted Lie point symmetries and reductions for ideal magnetohy- drodynamics equilibria, J

    O. Bogoyavlenskij, Restricted Lie point symmetries and reductions for ideal magnetohy- drodynamics equilibria, J. Engineering Mathematics 66, 141 (2010)

    2010

  31. [39]

    Liu and H

    M. Liu and H. Dong, On the existence of solution, Lie symmetry analysis and conservation law of magnetohydrodynamic equations, CNSNS 87, 105277 (2020)

    2020

  32. [40]

    Paliathanasis, Group properties and solutions for the 1D Hall MHD system in the cold plasma approximation, Eur

    A. Paliathanasis, Group properties and solutions for the 1D Hall MHD system in the cold plasma approximation, Eur. Phys. J. Plus 136, 538 (2021)

    2021

  33. [41]

    Webb and R.L

    G.M. Webb and R.L. Mace, Potential Vorticity in Magnetohydrodynamics, J. Plasma Phys. 81, 905810115 (2015) 30

    2015

  34. [42]

    Meleshko, E.I

    S.V. Meleshko, E.I. Kaptsov, S. Moyo and G.M. Webb, Group classification of the two- dimensional magnetogasdynamics equations in Lagrangian coordinates, Math.Meth. Appl. Sci. 46, 15367 (2023)

    2023

  35. [43]

    Kaptsov, S.V

    E.I. Kaptsov, S.V. Meleshko and V.A. Dorodnitsyn, Symmetries and conservation laws of the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coor- dinates, J. Phys. A.: Math. Theor. 55, 495202 (2022)

    2022

  36. [44]

    Bouchut and X

    F. Bouchut and X. Lhebrand, A 5-wave relaxation solver for the shallow water MHD system, J. Sci. Comput. 68, 92 (2015)

    2015

  37. [45]

    Bouchut and X

    F. Bouchut and X. Lhebrand, A multi well-balanced scheme for the shallow water MHD system with topography, Numerische Mathematik 136, 875 (2017)

    2017

  38. [46]

    J.L. Ye, H.B. Guo and Y.B. Hu, An initial-boundary value problem for the one-dimensional rotating shallow water magnetohydrodynamic equations, J. Math. Anal. Appl. 527, 127422 (2023)

    2023

  39. [47]

    Mak, S.D Griffiths and D.W

    J. Mak, S.D Griffiths and D.W. Hughes, Shear flow instabilities in shallow-water magne- tohydrodynamics, J. Fluid Mechanics 788, 767 (2016)

    2016

  40. [48]

    Dellar, Lattice Kinetic Schemes for Magnetohydrodynamics, J

    P.J. Dellar, Lattice Kinetic Schemes for Magnetohydrodynamics, J. Comput. Phys. 172, 392 (2001)

    2001

  41. [49]

    shallow water

    H. De Sterck, Hyperbolic theory of the “shallow water” magnetohydrodynamics equation, Physics of Plasmas 8, 3293 (2001)

    2001

  42. [50]

    Dellar, Dispersive shallow water magnetohydrodynamics, Physics of Plasmas 10, 581 (2003)

    P.J. Dellar, Dispersive shallow water magnetohydrodynamics, Physics of Plasmas 10, 581 (2003)

    2003

  43. [51]

    Powell, An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension), NASA CR-194902 and ICASE Report No

    K.G. Powell, An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension), NASA CR-194902 and ICASE Report No. 94-24, NASA Langley Research Center (1994)

    1994

  44. [52]

    Janhunen, A Positive Conservative Method for Magnetohydrodynamics Based on HLL and Roe Methods, J

    P. Janhunen, A Positive Conservative Method for Magnetohydrodynamics Based on HLL and Roe Methods, J. Comput. Phys. 160, 649 (2000)

    2000

  45. [53]

    Sokolov, K.G

    I.V. Sokolov, K.G. Powell, O. Cohen and T.I. Gombosi, Computational MagnetoHydro- dynamics, Based on Solution of the Well-Posed Riemann Problem, Numerical Modeling of Space Plasma Flows: Astronum 2007, Edited by N.V. Pogorelov, E. Audit, and G.P. Zank, ASP Conference Series, V...

    2008

  46. [54]

    P. J. Morrison, in Mathematical Methods in Hydrodynamics and Integrability of Dynami- cal Systems, edited by M. Tabor and Y. M. Treve, AIP Conference Proceedings: American Institute of Physics, New York, Vol. 88 of (1982)

    1982

  47. [55]

    Dellar, Hamiltonian and symmetric hyperbolic structures of shallow water magneto- hydrodynamics, Physics of Plasmas 9, 1130 (2002)

    P.J. Dellar, Hamiltonian and symmetric hyperbolic structures of shallow water magneto- hydrodynamics, Physics of Plasmas 9, 1130 (2002)

    2002

  48. [56]

    Bouchut and X

    F. Bouchut and X. Lh´ ebrard, A 5-Wave Relaxation Solver for the Shallow Water MHD System, J. Sci. Comput. 68, 98 (2016)

    2016

  49. [57]

    Morozov, Classification of six-dimensional nilpotent Lie algebras, Izvestia Vysshikh Uchebn Zavendenii Matematika 5, 161 (1958)

    V.V. Morozov, Classification of six-dimensional nilpotent Lie algebras, Izvestia Vysshikh Uchebn Zavendenii Matematika 5, 161 (1958)

    1958

  50. [58]

    Mubarakzyanov, On solvable Lie algebras, Izvestia Vysshikh Uchebn Zavendenii Matematika 32, 114 (1963)

    G.M. Mubarakzyanov, On solvable Lie algebras, Izvestia Vysshikh Uchebn Zavendenii Matematika 32, 114 (1963)

    1963

  51. [59]

    Mubarakzyanov, Classification of real structures of five-dimensional Lie algebras, Izvestia Vysshikh Uchebn Zavendenii Matematika 34, 99 (1963)

    G.M. Mubarakzyanov, Classification of real structures of five-dimensional Lie algebras, Izvestia Vysshikh Uchebn Zavendenii Matematika 34, 99 (1963)

    1963

  52. [60]

    Mubarakzyanov, Classification of solvable six-dimensional Lie algebras with one nilpotent base element, Izvestia Vysshikh Uchebn Zavendenii Matematika 35, 104 (1963)

    G.M. Mubarakzyanov, Classification of solvable six-dimensional Lie algebras with one nilpotent base element, Izvestia Vysshikh Uchebn Zavendenii Matematika 35, 104 (1963)

    1963

  53. [61]

    Patera, R

    J. Patera, R. T. Sharp, P. Winternitz and H. Zassenhaus, Invariants of real low-dimensional Lie algebras, J. Math. Phys. 17, 986 (1976)

    1976

  54. [62]

    Noether, Invariante Variationsprobleme, Nach

    E. Noether, Invariante Variationsprobleme, Nach. Konig. Gesell. Wissen. Gottingen, Math.-phys. Kl 235, (1918)

    1918

  55. [63]

    Ibragimov, A new conservation theorem, J

    N.H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl. 333, 311 (2007)

    2007

  56. [64]

    Gandarias, Weak self-adjoint differential equations, J

    M.L. Gandarias, Weak self-adjoint differential equations, J. Phys. A: Math. Theor. 44, 262001 (2011)

    2011

  57. [65]

    Yi, Symmetries and Two Types of Non-Noether Conservation Laws of Birkhoffian Sys- tem with Unilateral Constraints, Commun

    Z. Yi, Symmetries and Two Types of Non-Noether Conservation Laws of Birkhoffian Sys- tem with Unilateral Constraints, Commun. Theor. Phys. 45, 239 (2006)

    2006

  58. [66]

    Anco, On the incompleteness of Ibragimov’s conservation law theorem and its equiv- alence to a standard formula using symmetries and adjoint-symmetries,Symmetry 9, 33 (201) 32

    S.C. Anco, On the incompleteness of Ibragimov’s conservation law theorem and its equiv- alence to a standard formula using symmetries and adjoint-symmetries,Symmetry 9, 33 (201) 32

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.