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Erratum to “On Soliton structures in optical fiber communications with Kundu–Mukherjee–Naskar model (Open Physics 2021;19:679–682)”

  • Abhik Mukherjee EMAIL logo and Anurag
Published/Copyright: November 13, 2025
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Open Physics
From the journal Open Physics Volume 23 Issue 1

Abstract

In a recent paper (Open Physics 2021;19:679–682), Khalil Salim Al-Ghafri investigated the soliton structures of the integrable Kundu–Mukherjee–Naskar (KMN) equation in optical fiber communication. Some exact solutions such as the W-shaped soliton solution, the bright soliton solutions (type-I and type-II) and dark soliton solution are derived using the ansatz approach. In this erratum article, we show that all these four solutions presented in Open Physics 2021;19:679–682 do not satisfy the original KMN equation; hence they are not the exact solutions of the integrable KMN equation.

1 Introduction

Recently, Khalil Salim Al-Ghafri investigated the integrable Kundu–Mukherjee–Naskar (KMN) equation [1] in optical fiber communication and derived few exact solutions [2]. Different research groups have carried out various innovation activities to evaluate different properties of the KMN equation such as exact and approximate solutions, integrable structures, symmetries, conserved quantities, etc. The KMN equation is given as

(1) i Ψ t + a Ψ x y + i b Ψ ( Ψ Ψ x * − Ψ * Ψ x ) = 0 ,

where Ψ ( x , y , t ) represents the optical soliton profile. To investigate the various optical soliton solutions of KMN Eq. (1), Al-Ghafri [2] used the traveling wave transformation as:

(2) Ψ ( x , y , t ) = ψ [ ξ ( x , y , t ) ] e i ϕ ( x , y , t ) ,

where ψ [ ξ ( x , y , t ) ] and ϕ ( x , y , t ) denote the amplitude and phase of the soliton, respectively. The variable ξ ( x , y , t ) and phase component ϕ ( x , y , t ) are defined as:

(3) ξ ( x , y , t ) = κ x + λ y + v t ,

(4) ϕ ( x , y , t ) = α x + β y + ω t + θ [ ξ ( x , y , t ) ] ,

where, κ , λ , v , α , β , ω represent constant parameters and θ [ ξ ( x , y , t ) ] is another real function. Applying the transformation given by Eq. (2) to Eq. (1) and collecting the real and imaginary parts lead to the following pair of ordinary differential equations:

(5) ( a α λ + a β κ + v ) ψ ′ + 2 a κ λ ψ ′ θ ′ + a κ λ ψ θ ″ = 0 ,

(6) a κ λ ψ ″ − ( a α β + ω ) + 2 b α ψ 3 − ( a β κ + a α λ + v ) ψ θ ′ − a κ λ ψ θ ′ 2 + 2 b κ ψ 3 θ ′ = 0 ,

where prime in the superscript of the dependent variables ψ and θ denotes the derivative with respect to the variable ξ . On solving Eq. (5) for θ ′ , we arrive at:

(7) θ ′ = − a α λ + a β κ + v 2 a κ λ + C a κ λ ψ 2 ,

where C is the integration constant. Using Eq. (7) in Eq. (6), Al-Ghafri [2] obtained an ordinary nonlinear differential equation as:

(8) ψ ″ + A 1 + A 2 ψ 3 − A 3 ψ − 3 = 0 ,

where

(9) A 1 = ( a α λ + a β κ + v ) 2 − 4 a κ λ ( ω + a α β ) + 8 b κ C 4 a 2 κ 2 λ 2 ,

(10) A 2 = − b κ ( a α λ + a β κ + v ) + 2 a b α κ λ a 2 κ 2 λ 2 ,

(11) A 3 = C 2 a 2 κ 2 λ 2 .

On multiplying Eq. (8) by ψ ′ and further integrating with respect to variable ξ , Al-Ghafri [2] obtained:

(12) ψ ′ 2 + A 1 ψ 2 + A 2 2 ψ 4 + A 3 ψ − 2 + 2 A 0 = 0 ,

where A 0 is an arbitrary integration constant. Al-Ghafri [2] used the transformation ψ 2 = V in Eq. (12) and arrived at:

(13) V ′ 2 + 4 A 1 V 2 + 2 A 2 V 3 + 4 A 3 + 8 A 0 V = 0 .

Al-Ghafri [2] further differentiated the above equation [Eq. (13)] to obtain:

(14) V ″ + 4 A 0 + 4 A 1 V + 3 A 2 V 2 = 0 .

2 Erratum to the study of Al-Ghafri [2]

Al-Ghafri [2] obtained different exact solutions of Eq. (14). But, it must be noted that the solutions obtained from Eq. (14) are the square of amplitude of exact solutions of KMN equation [Eq. (1)] only if it satisfies Eq. (13) at the same time. Hence, the solutions obtained from Eq. (14) must also satisfy Eq. (13) to be the exact solutions of Eq. (1).

If we carefully look at Eq. (13), it has an extra parameter A 3 which is not present in Eq. (14). When we use the ansatz of different kinds of solutions that were used in the study of Al-Ghafri [2] in Eq. (13), we would obtain an extra constraint condition for parameter A 3 for each solution. These constraint conditions give the relationship of A 3 with other parameters present in the system. Hence, one must not arbitrarily choose all the parameters present in the system.

But, all the four solutions presented in the study of Al-Ghafri [2] i.e., W shaped soliton, Bright soliton (type-I), Bright soliton (type-II) and the Dark soliton do not satisfy Eq. (14) and Eq. (13) simultaneously. Hence, it can be clearly stated that these four solutions (Eqs (19)–(22) presented in the study of Al-Ghafri [2]) are not the exact solutions of the KMN Eq. (1). These facts are explained in detail in the following subsections.

2.1 Erratum to the W-shaped soliton solution given in Eq. (19) of the study of Al-Ghafri [2]

Al-Ghafri [2] used the following ansatz for W-shaped soliton solution as:

(15) V ( ξ ) = l + m sech 2 p ξ 4 − [ 1 − tanh p ξ ] 2 + n sech 4 p ξ ( 4 − [ 1 − tanh p ξ ] 2 ) 2 ,

where the parameters l , m , n are related to A 1 , A 2 as l = − 4 A 1 3 A 2 , m = 16 A 1 A 2 , n = − 32 A 1 A 2 with p = A 1 . The constants A 1 , A 2 follow the conditions A 1 > 0 , A 2 < 0 and A 0 is taken to be zero as it is an arbitrary constant.

As we have discussed before, the solution (15) must satisfy Eq. (14) and Eq. (13) simultaneously to be an exact solution of the KMN equation. On substituting Eq. (15) in Eq. (13), we obtain a constraint:

(16) A 3 = − 16 A 1 3 27 A 2 2 .

As A 1 > 0 and A 2 < 0 , A 3 is going to be negative according to Eq. (16). However, Eq. (11) suggests that A 3 is always positive. As a result, we state that the W-shaped soliton Eq. (15) presented in the study of Al-Ghafri [2] is not an exact solution of KMN Eq. (1).

2.2 Erratum to the bright soliton solution of type-I given in Eq. (20) of the study of Al-Ghafri [2]

Al-Ghafri [2] used the following ansatz for bright solitons of type I:

(17) V ( ξ ) = l + m sech 2 p ξ 4 − [ 1 − tanh p ξ ] 2 + n sech 4 p ξ ( 4 − [ 1 − tanh p ξ ] 2 ) 2 ,

where the parameters l , m , n are related to A 1 , A 2 as l = 0 , m = − 16 A 1 A 2 , n = 32 A 1 A 2 , with p = − A 1 . The constants A 1 , A 2 follow the conditions A 1 < 0 and A 2 > 0 and A 0 is taken to be zero as it is an arbitrary constant.

As we have discussed before, the solution (17) must satisfy Eq. (14) and Eq. (13) simultaneously to be an exact solution of KMN equation. On substituting Eq. (17) in Eq. (13), we obtain:

(18) A 3 = 0 .

On comparing Eqs (18) and (11), we find that C = 0 . However, Fig. 1 in the study of Al-Ghafri [2] is plotted for C = 1 . Hence, the solution (20) presented in the study of Al-Ghafri [2] is not correct.

2.3 Erratum to the bright soliton solution of type-II given in Eq. (21) of the study of Al-Ghafri [2]

Al-Ghafri [2] used the following ansatz for bright solitons of type II:

(19) V ( ξ ) = l + m sech 2 p ξ 4 − [ 1 − tanh p ξ ] 2 + n sech 4 p ξ ( 4 − [ 1 − tanh p ξ ] 2 ) 2 ,

where the parameters l , m , n are related to A 1 , A 2 as l = − 6 A 1 11 A 2 , m = − 32 A 1 11 A 2 , n = 64 A 1 11 A 2 , with p = − 2 A 1 11 . The constants A 1 , A 2 follow the condition A 1 < 0 , A 2 > 0 along with 39 A 1 2 = 121 A 0 A 2 .

On substituting Eq. (19) in Eq. (13), we obtain an extra condition given by:

(20) A 3 = 180 A 1 3 1331 A 2 2 ,

which is missing in the original paper [2]. The value of C chosen for bright soliton of type II should be such that A 3 satisfies the above equation [Eq. (20)].

These facts are completely missing in the study of Al-Ghafri [2] and the solution (21) presented in the study of Al-Ghafri [2] is not correct.

2.4 Erratum to the dark soliton solution given in Eq. (22) of the study of Al-Ghafri [2]

Al-Ghafri [2] used the following ansatz for dark soliton:

(21) V ( ξ ) = l + m sech 2 p ξ 4 − [ 1 − tanh p ξ ] 2 + n sech 4 p ξ ( 4 − [ 1 − tanh p ξ ] 2 ) 2 ,

where the parameters l , m , n are related to A 1 , A 2 as l = − 26 A 1 33 A 2 , m = 32 A 1 11 A 2 , n = − 64 A 1 11 A 2 with p = 2 A 1 11 . The constants A 1 , A 2 follow the condition as A 1 > 0 , A 2 < 0 along with 39 A 1 2 = 121 A 0 A 2 . On substituting Eq. (21) in Eq. (13), we obtain an extra condition given by:

(22) A 3 = 4732 A 1 3 35937 A 2 2 ,

which is missing in the original paper [2]. The value of C for dark soliton should be chosen such that A 3 satisfies the above equation [Eq. (22)].

These facts are completely missing in the study of Al-Ghafri [2] and the solution (22) presented in the study of Al-Ghafri [2] is not correct.

Specifically, if such incorrect soliton profiles are assumed to be realizable in optical fiber systems, they could misrepresent pulse stability, lead to incorrect predictions of optical signal propagation [3]. Bright solitons are localized pulses of light that maintain their shape during propagation due to a balance between dispersion and nonlinearity. They are widely used in long-distance optical communication to minimize pulse broadening and signal distortion caused by group velocity dispersion [3]. Dark solitons are intensity dips (or voids) in a continuous wave background. They are less affected by perturbations such as amplifier noise compared to bright solitons, making them suitable for noise-resilient communication channels [3]. W-shaped solitons are localized waveforms with a double-dip or oscillatory profile, often arising in higher-order nonlinear systems. They are explored for error correction and regeneration of distorted signals, where the W-profile can reshape or stabilize degraded optical pulses. They may provide robustness in systems prone to higher-order dispersion and nonlinear instabilities. Including this perspective underscores the importance of verifying mathematical solutions before their application in physical models.

3 Conclusions

In the present article, we have shown that the exact soliton solutions of the integrable KMN equation that was presented in the study of Al-Ghafri [2] are not correct. All the four solutions, i.e., the W-shaped soliton, the bright soliton (type-I and type-II) and dark soliton solutions, which are presented in the study of Al-Ghafri [2] must also satisfy Eqs (14) and (13) simultaneously to be exact solutions of the KMN equation. But, we have shown that those solutions (Eqs (19)–(22) presented in the study of Al-Ghafri [2]) do not satisfy Eq. (14) and Eq. (13) simultaneously; hence, they are not the exact solutions of the KMN Eq. (1). This discrepancy highlights the necessity for rigorous verification of solutions in mathematical physics, particularly in the context of optical fiber communication. Future research should focus on identifying valid solutions that truly adhere to the integrable nature of the KMN equation, ensuring that they can be reliably applied in practical scenarios.

,

Acknowledgments

Abhik Mukherjee acknowledges Enhanced Seed Grant through Endowment Fund 2023-24, Manipal University Jaipur, Rajasthan, India for the financial support during the progress of the work.

  1. Author contribution: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  2. Conflict of interest: The authors state no conflict of interest.

  3. Data availability statement: All data generated or analysed during this study are included in this published article.

References

[1] Kundu A, Mukherjee A, Naskar T. Modelling rogue waves through exact dynamical lump soliton controlled by ocean currents. Proc Soc A. 2014;470:20130576. 10.1098/rspa.2013.0576Search in Google Scholar PubMed PubMed Central

[2] Al-Ghafri KS. Soliton structures in optical fiber communications with Kundu–Mukherjee–Naskar model. Open Phys. 2021;19:679–82. 10.1515/phys-2021-0074Search in Google Scholar

[3] Agarwal GP. Nonlinear fiber optics. USA: Academic Press; 2007. Search in Google Scholar
Received: 2025-07-07
Revised: 2025-10-05
Accepted: 2025-10-28
Published Online: 2025-11-13

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  39. Fractal formation and chaotic soliton phenomena in nonlinear conformable Heisenberg ferromagnetic spin chain equation
  40. Single-step fabrication of Mn(iv) oxide-Mn(ii) sulfide/poly-2-mercaptoaniline porous network nanocomposite for pseudo-supercapacitors and charge storage
  41. Novel constructed dynamical analytical solutions and conserved quantities of the new (2+1)-dimensional KdV model describing acoustic wave propagation
  42. Tavis–Cummings model in the presence of a deformed field and time-dependent coupling
  43. Spinning dynamics of stress-dependent viscosity of generalized Cross-nonlinear materials affected by gravitationally swirling disk
  44. Design and prediction of high optical density photovoltaic polymers using machine learning-DFT studies
  45. Robust control and preservation of quantum steering, nonlocality, and coherence in open atomic systems
  46. Coating thickness and process efficiency of reverse roll coating using a magnetized hybrid nanomaterial flow
  47. Dynamic analysis, circuit realization, and its synchronization of a new chaotic hyperjerk system
  48. Decoherence of steerability and coherence dynamics induced by nonlinear qubit–cavity interactions
  49. Finite element analysis of turbulent thermal enhancement in grooved channels with flat- and plus-shaped fins
  50. Modulational instability and associated ion-acoustic modulated envelope solitons in a quantum plasma having ion beams
  51. Statistical inference of constant-stress partially accelerated life tests under type II generalized hybrid censored data from Burr III distribution
  52. On solutions of the Dirac equation for 1D hydrogenic atoms or ions
  53. Entropy optimization for chemically reactive magnetized unsteady thin film hybrid nanofluid flow on inclined surface subject to nonlinear mixed convection and variable temperature
  54. Stability analysis, circuit simulation, and color image encryption of a novel four-dimensional hyperchaotic model with hidden and self-excited attractors
  55. A high-accuracy exponential time integration scheme for the Darcy–Forchheimer Williamson fluid flow with temperature-dependent conductivity
  56. Novel analysis of fractional regularized long-wave equation in plasma dynamics
  57. Development of a photoelectrode based on a bismuth(iii) oxyiodide/intercalated iodide-poly(1H-pyrrole) rough spherical nanocomposite for green hydrogen generation
  58. Investigation of solar radiation effects on the energy performance of the (Al2O3–CuO–Cu)/H2O ternary nanofluidic system through a convectively heated cylinder
  59. Quantum resources for a system of two atoms interacting with a deformed field in the presence of intensity-dependent coupling
  60. Studying bifurcations and chaotic dynamics in the generalized hyperelastic-rod wave equation through Hamiltonian mechanics
  61. A new numerical technique for the solution of time-fractional nonlinear Klein–Gordon equation involving Atangana–Baleanu derivative using cubic B-spline functions
  62. Interaction solutions of high-order breathers and lumps for a (3+1)-dimensional conformable fractional potential-YTSF-like model
  63. Hydraulic fracturing radioactive source tracing technology based on hydraulic fracturing tracing mechanics model
  64. Numerical solution and stability analysis of non-Newtonian hybrid nanofluid flow subject to exponential heat source/sink over a Riga sheet
  65. Numerical investigation of mixed convection and viscous dissipation in couple stress nanofluid flow: A merged Adomian decomposition method and Mohand transform
  66. Effectual quintic B-spline functions for solving the time fractional coupled Boussinesq–Burgers equation arising in shallow water waves
  67. Analysis of MHD hybrid nanofluid flow over cone and wedge with exponential and thermal heat source and activation energy
  68. Solitons and travelling waves structure for M-fractional Kairat-II equation using three explicit methods
  69. Impact of nanoparticle shapes on the heat transfer properties of Cu and CuO nanofluids flowing over a stretching surface with slip effects: A computational study
  70. Computational simulation of heat transfer and nanofluid flow for two-sided lid-driven square cavity under the influence of magnetic field
  71. Irreversibility analysis of a bioconvective two-phase nanofluid in a Maxwell (non-Newtonian) flow induced by a rotating disk with thermal radiation
  72. Hydrodynamic and sensitivity analysis of a polymeric calendering process for non-Newtonian fluids with temperature-dependent viscosity
  73. Exploring the peakon solitons molecules and solitary wave structure to the nonlinear damped Kortewege–de Vries equation through efficient technique
  74. Modeling and heat transfer analysis of magnetized hybrid micropolar blood-based nanofluid flow in Darcy–Forchheimer porous stenosis narrow arteries
  75. Activation energy and cross-diffusion effects on 3D rotating nanofluid flow in a Darcy–Forchheimer porous medium with radiation and convective heating
  76. Insights into chemical reactions occurring in generalized nanomaterials due to spinning surface with melting constraints
  77. Influence of a magnetic field on double-porosity photo-thermoelastic materials under Lord–Shulman theory
  78. Soliton-like solutions for a nonlinear doubly dispersive equation in an elastic Murnaghan's rod via Hirota's bilinear method
  79. Analytical and numerical investigation of exact wave patterns and chaotic dynamics in the extended improved Boussinesq equation
  80. Nonclassical correlation dynamics of Heisenberg XYZ states with (x, y)-spin--orbit interaction, x-magnetic field, and intrinsic decoherence effects
  81. Exact traveling wave and soliton solutions for chemotaxis model and (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation
  82. Unveiling the transformative role of samarium in ZnO: Exploring structural and optical modifications for advanced functional applications
  83. On the derivation of solitary wave solutions for the time-fractional Rosenau equation through two analytical techniques
  84. Analyzing the role of length and radius of MWCNTs in a nanofluid flow influenced by variable thermal conductivity and viscosity considering Marangoni convection
  85. Advanced mathematical analysis of heat and mass transfer in oscillatory micropolar bio-nanofluid flows via peristaltic waves and electroosmotic effects
  86. Exact bound state solutions of the radial Schrödinger equation for the Coulomb potential by conformable Nikiforov–Uvarov approach
  87. Some anisotropic and perfect fluid plane symmetric solutions of Einstein's field equations using killing symmetries
  88. Nonlinear dynamics of the dissipative ion-acoustic solitary waves in anisotropic rotating magnetoplasmas
  89. Curves in multiplicative equiaffine plane
  90. Exact solution of the three-dimensional (3D) Z2 lattice gauge theory
  91. Propagation properties of Airyprime pulses in relaxing nonlinear media
  92. Symbolic computation: Analytical solutions and dynamics of a shallow water wave equation in coastal engineering
  93. Wave propagation in nonlocal piezo-photo-hygrothermoelastic semiconductors subjected to heat and moisture flux
  94. Comparative reaction dynamics in rotating nanofluid systems: Quartic and cubic kinetics under MHD influence
  95. Laplace transform technique and probabilistic analysis-based hypothesis testing in medical and engineering applications
  96. Physical properties of ternary chloro-perovskites KTCl3 (T = Ge, Al) for optoelectronic applications
  97. Gravitational length stretching: Curvature-induced modulation of quantum probability densities
  98. The search for the cosmological cold dark matter axion – A new refined narrow mass window and detection scheme
  99. A comparative study of quantum resources in bipartite Lipkin–Meshkov–Glick model under DM interaction and Zeeman splitting
  100. PbO-doped K2O–BaO–Al2O3–B2O3–TeO2-glasses: Mechanical and shielding efficacy
  101. Nanospherical arsenic(iii) oxoiodide/iodide-intercalated poly(N-methylpyrrole) composite synthesis for broad-spectrum optical detection
  102. Sine power Burr X distribution with estimation and applications in physics and other fields
  103. Numerical modeling of enhanced reactive oxygen plasma in pulsed laser deposition of metal oxide thin films
  104. Dynamical analyses and dispersive soliton solutions to the nonlinear fractional model in stratified fluids
  105. Computation of exact analytical soliton solutions and their dynamics in advanced optical system
  106. An innovative approximation concerning the diffusion and electrical conductivity tensor at critical altitudes within the F-region of ionospheric plasma at low latitudes
  107. An analytical investigation to the (3+1)-dimensional Yu–Toda–Sassa–Fukuyama equation with dynamical analysis: Bifurcation
  108. Swirling-annular-flow-induced instability of a micro shell considering Knudsen number and viscosity effects
  109. Review Article
  110. Examination of the gamma radiation shielding properties of different clay and sand materials in the Adrar region
  111. Erratum
  112. Erratum to “On Soliton structures in optical fiber communications with Kundu–Mukherjee–Naskar model (Open Physics 2021;19:679–682)”
  113. Special Issue on Fundamental Physics from Atoms to Cosmos - Part II
  114. Possible explanation for the neutron lifetime puzzle
  115. Special Issue on Nanomaterial utilization and structural optimization - Part III
  116. Numerical investigation on fluid-thermal-electric performance of a thermoelectric-integrated helically coiled tube heat exchanger for coal mine air cooling
  117. Special Issue on Nonlinear Dynamics and Chaos in Physical Systems
  118. Analysis of the fractional relativistic isothermal gas sphere with application to neutron stars
  119. Abundant wave symmetries in the (3+1)-dimensional Chafee–Infante equation through the Hirota bilinear transformation technique
  120. Successive midpoint method for fractional differential equations with nonlocal kernels: Error analysis, stability, and applications
  121. Novel exact solitons to the fractional modified mixed-Korteweg--de Vries model with a stability analysis
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