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Volume 14
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Article

On the Invariant Subspace Method for Nonlinear Second-Order Evolution Equations

by
Manal Badgaish
1,
Lhoucine Hmidouch
2 and
Nacir Hmidouch
3,*
1
Mathematics Department, Faculty of Science, Umm Al-Qura University, Makkah 24382, Saudi Arabia
2
Department of Mathematics, Faculty of Sciences, Chouaib Doukkali University, BP. 20, El Jadida 24000, Morocco
3
Department of Mathematical Sciences, Talladega College, Talladega, AL 35160, USA
*
Author to whom correspondence should be addressed.
Mathematics 2026, 14(12), 2155; https://doi.org/10.3390/math14122155
Submission received: 15 April 2026 / Revised: 2 June 2026 / Accepted: 6 June 2026 / Published: 16 June 2026
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Abstract

We investigate invariant finite-dimensional linear spaces for a class of nonlinear evolution equations with second-order time dependence of the type ∂ttu=−∂x(A(x,u)u)+∂xx(B(x,u)u)+f(u), where the coefficients may depend nonlinearly on the solution. Using direct algebraic verification and invariant subspace reductions, we derive several exact solutions and reduce the governing equations to finite-dimensional systems of ordinary differential equations. Various examples are presented to illustrate the applicability of the method to nonlinear transport, diffusion, and reaction models.
Keywords: nonlinear evolution equations; invariant subspace method; exact solutions nonlinear evolution equations; invariant subspace method; exact solutions

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MDPI and ACS Style

Badgaish, M.; Hmidouch, L.; Hmidouch, N. On the Invariant Subspace Method for Nonlinear Second-Order Evolution Equations. Mathematics 2026, 14, 2155. https://doi.org/10.3390/math14122155

AMA Style

Badgaish M, Hmidouch L, Hmidouch N. On the Invariant Subspace Method for Nonlinear Second-Order Evolution Equations. Mathematics. 2026; 14(12):2155. https://doi.org/10.3390/math14122155

Chicago/Turabian Style

Badgaish, Manal, Lhoucine Hmidouch, and Nacir Hmidouch. 2026. "On the Invariant Subspace Method for Nonlinear Second-Order Evolution Equations" Mathematics 14, no. 12: 2155. https://doi.org/10.3390/math14122155

APA Style

Badgaish, M., Hmidouch, L., & Hmidouch, N. (2026). On the Invariant Subspace Method for Nonlinear Second-Order Evolution Equations. Mathematics, 14(12), 2155. https://doi.org/10.3390/math14122155

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