Published on: October 2026
EXACT SOLUTIONS OF THE KDV EQUATION — COMMENT ON GHOSH’S PAPER
Weiguang Huang
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Abstract
By applying the travelling wave method within a symbolic computation framework (MathHandbook), new exact analytical solutions of the nonlinear KdV equation were derived. Both the defocusing and focusing regimes were analyzed, revealing a duality in their solution structures. Rigorous validation through symbolic substitution confirmed the correctness of the solutions, providing exact benchmarks for nonlinear dynamics. A relationship between the defocusing and focusing solutions was found, which reveals the observed symmetry directly from the mathematical structure of the PDE. The analytical structure and stability of the obtained solution branches were discussed, and the Appendix corrects errors in previously published results.
Keywords: symbolic computation; nonlinear evolution equations; KdV equation; exact solutions; focusing–defocusing duality; solitons.
How to Cite this Paper
Huang, W. (2026). Exact Solutions of the KdV Equation — Comment on Ghosh’s paper. International Journal of Creative and Open Research in Engineering and Management, <i>02</i>(10), 1-9. https://doi.org/10.55041/ijcope.v2i10.122
Huang, Weiguang. "Exact Solutions of the KdV Equation — Comment on Ghosh’s paper." International Journal of Creative and Open Research in Engineering and Management, vol. 02, no. 10, 2026, pp. 1-9. doi:https://doi.org/10.55041/ijcope.v2i10.122.
Huang, Weiguang. "Exact Solutions of the KdV Equation — Comment on Ghosh’s paper." International Journal of Creative and Open Research in Engineering and Management 02, no. 10 (2026): 1-9. https://doi.org/https://doi.org/10.55041/ijcope.v2i10.122.
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References
[1] W. Huang, (2026). Exact analytic solution for the time-fractional Hunter-Saxton equation with Caputo derivative. Nonlinear Science. 8:100193. https://doi.org/10.1016/j.nls.2026.100193[2] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, (1967). “Method for solving the Korteweg–deVries equation”, Phys. Rev. Lett., 19(19), pp. 1095–1097. doi: 10.1103/PhysRevLett.19.1095.
[3] P. D. Lax, (1968). “Integrals of nonlinear equations of evolution and solitary waves”, Commun. Pure Appl. Math., 21(5), pp. 467–490. doi: 10.1002/cpa.3160210503.
[4] R. M. Miura, (1968). “Korteweg–de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation”, J. Math. Phys., 9(8), pp. 1202–1204. doi: 10.1063/1.1664700.
[5] V. E. Zakharov and L. D. Faddeev, (1971). “Korteweg–de Vries equation: A completely integrable Hamiltonian system”, Funct. Anal. Appl., 5(4), pp. 280–287. doi: 10.1007/BF01086739.
[6] B. Dey, (1986). “Domain wall solutions of KdV like equations with higher order nonlinearity”, J. Phys. A: Math. Gen., 19(1), pp. L9–L12. doi: 10.1088/0305-4470/19/1/003.
[7] K. Konno and Y. H. Ichikawa, (1974). “A modified Korteweg–de Vries equation for ion acoustic waves”, J. Phys. Soc. Jpn., 37(6), pp. 1631–1636. doi: 10.1143/JPSJ.37.1631.
[8] M. Wadati, (1975). “Wave propagation in nonlinear lattice. I”, J. Phys. Soc. Jpn., 38(3), pp. 673–680. doi: 10.1143/JPSJ.38.673.
[9] W. Malfliet, (1992), “Solitary wave solutions of nonlinear wave equations”, Am. J. Phys., 60(7), pp. 650–654. doi: 10.1119/1.17120.
[10] M. W. Coffey, (1990). “On series expansions giving closed-form solutions of Korteweg–de Vries-like equations”, SIAM J. Appl. Math., 50(6), pp. 1580–1592. doi: 10.1137/0150093.
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