TY  - JOUR
AU  - Pippal, Sarita
PY  - 2026
DA  - 2026/04/28
TI  - Unified Fractional Balancing Principle for Nonlinear Wave Equations: Exponential versus Algebraic Traveling Waves
JO  - ICCK Journal of Applied Mathematics
T2  - ICCK Journal of Applied Mathematics
JF  - ICCK Journal of Applied Mathematics
VL  - 2
IS  - 2
SP  - 166
EP  - 172
DO  - 10.62762/JAM.2026.568364
UR  - https://www.icck.org/article/abs/JAM.2026.568364
KW  - fractional simplest equation method
KW  - nonlinear wave equations
KW  - power-law nonlinearity
KW  - travelling waves
KW  - algebraic decay
KW  - integrable systems
KW  - dominant balance
KW  - phase geometry
KW  - stability analysis
KW  - bifurcation
AB  - A unified fractional simplest equation framework is developed for nonlinear power-law wave equations whose travelling-wave reductions lead to $U'' + \lambda U^p = 0$, $p>1$. In contrast to classical simplest equation methods restricted to integer polynomial expansions, we introduce a fractional power ansatz $U(\xi)=A\phi^\alpha(\xi)$ derived from a dominant balance principle. A general scaling theorem is established, proving that the admissible travelling-wave exponents are uniquely determined by $\alpha(p-1)=2$, yielding the explicit fractional law $\alpha=\frac{2}{p-1}$. Consequently, the equation admits algebraic travelling waves of the universal form $U(\xi)=A(\mu\xi+C_0)^{-2/(p-1)}$, with the amplitude constraint $A^{p-1}=\frac{4\mu^2}{(p-1)^2\lambda}$. This result provides closed-form algebraic representations that complement classical energy integration, which produces only implicit elliptic-type solutions of the form $\xi=\int \frac{dU}{\sqrt{2C-\frac{\lambda}{p+1}U^{p+1}}}$. The theory establishes a scaling hierarchy in which the decay exponent $\alpha=\frac{2}{p-1}$ decreases monotonically as $p$ increases, revealing a systematic transition from exponential localization in integrable quadratic and cubic systems ($p=2,3$) to algebraic decay for higher-power nonlinearities ($p>3$). Two benchmark cases ($p=5$ and $p=7$) are analyzed explicitly, confirming the universality of the fractional exponent formula. Stability analysis leads to a singular perturbation equation of the form $w''+\frac{C}{\xi^2}w=0$, demonstrating algebraic stability away from the singular point. Numerical phase portraits and bifurcation analysis show that the fractional solutions correspond to singular invariant branches in the conservative phase geometry, distinct from homoclinic or heteroclinic soliton manifolds in completely integrable systems. The framework unifies integer and non-integer expansion approaches, provides a classification mechanism based on nonlinear scaling, and offers a systematic tool for constructing explicit algebraic travelling waves in nonlinear wave dynamics.
SN  - 3068-5656
PB  - Institute of Central Computation and Knowledge
LA  - English
ER  - 

@article{Pippal2026Unified,
  author = {Sarita Pippal},
  title = {Unified Fractional Balancing Principle for Nonlinear Wave Equations: Exponential versus Algebraic Traveling Waves},
  journal = {ICCK Journal of Applied Mathematics},
  year = {2026},
  volume = {2},
  number = {2},
  pages = {166-172},
  doi = {10.62762/JAM.2026.568364},
  url = {https://www.icck.org/article/abs/JAM.2026.568364},
  abstract = {A unified fractional simplest equation framework is developed for nonlinear power-law wave equations whose travelling-wave reductions lead to \$U'' + \lambda U^p = 0\$, \$p>1\$. In contrast to classical simplest equation methods restricted to integer polynomial expansions, we introduce a fractional power ansatz \$U(\xi)=A\phi^\alpha(\xi)\$ derived from a dominant balance principle. A general scaling theorem is established, proving that the admissible travelling-wave exponents are uniquely determined by \$\alpha(p-1)=2\$, yielding the explicit fractional law \$\alpha=\frac{2}{p-1}\$. Consequently, the equation admits algebraic travelling waves of the universal form \$U(\xi)=A(\mu\xi+C\_0)^{-2/(p-1)}\$, with the amplitude constraint \$A^{p-1}=\frac{4\mu^2}{(p-1)^2\lambda}\$. This result provides closed-form algebraic representations that complement classical energy integration, which produces only implicit elliptic-type solutions of the form \$\xi=\int \frac{dU}{\sqrt{2C-\frac{\lambda}{p+1}U^{p+1}}}\$. The theory establishes a scaling hierarchy in which the decay exponent \$\alpha=\frac{2}{p-1}\$ decreases monotonically as \$p\$ increases, revealing a systematic transition from exponential localization in integrable quadratic and cubic systems (\$p=2,3\$) to algebraic decay for higher-power nonlinearities (\$p>3\$). Two benchmark cases (\$p=5\$ and \$p=7\$) are analyzed explicitly, confirming the universality of the fractional exponent formula. Stability analysis leads to a singular perturbation equation of the form \$w''+\frac{C}{\xi^2}w=0\$, demonstrating algebraic stability away from the singular point. Numerical phase portraits and bifurcation analysis show that the fractional solutions correspond to singular invariant branches in the conservative phase geometry, distinct from homoclinic or heteroclinic soliton manifolds in completely integrable systems. The framework unifies integer and non-integer expansion approaches, provides a classification mechanism based on nonlinear scaling, and offers a systematic tool for constructing explicit algebraic travelling waves in nonlinear wave dynamics.},
  keywords = {fractional simplest equation method, nonlinear wave equations, power-law nonlinearity, travelling waves, algebraic decay, integrable systems, dominant balance, phase geometry, stability analysis, bifurcation},
  issn = {3068-5656},
  publisher = {Institute of Central Computation and Knowledge}
}