We investigate a local $M$-deformation of the $(2+1)$-dimensional Akbota–Myrzakulov–Tolkynay–Zhaidary (AMTZ) system and clarify its integrability and traveling-wave structure. For sufficiently smooth functions on the positive domain, we establish an exact two-parameter coordinate equivalence between the local $M$-AMTZ and classical AMTZ systems for $0<\alpha\leq1$ and $\eta>0$. The same transformation shows that the corresponding Lax representation, zero-curvature condition, and conservation structure are inherited from the classical AMTZ system under coordinate pullback.
A general traveling-wave reduction is derived while retaining the integration constant, leading to a generalized Duffing-type equation with an independent constant term. Its first integral is a quartic polynomial and provides a phase-plane description of the reduced dynamics. Generic root configurations lead to elliptic traveling-wave solutions, for which an explicit Weierstrass representation is obtained, whereas repeated-root degenerations yield elementary traveling-wave profiles.
The analysis further shows that the parameters $\alpha$ and $\eta$ do not modify the intrinsic reduced dynamics. Instead, they determine the coordinate representation and the traveling-wave phase, with $\alpha$ producing a nonlinear power-law reparametrization and $\eta$ contributing a constant scaling. Thus, the local $M$-parameters do not by themselves generate distinct intrinsic traveling-wave families. The results provide a unified coordinate-equivalence framework that distinguishes inherited integrability structures from their local $M$-representation.
Citation: Zh.R. Myrzakulova, A. Taishiyeva. Local $ M $-fractional deformation of the (2+1)-dimensional AMTZ system: Coordinate equivalence, Lax integrability, and exact solutions[J]. AIMS Mathematics, 2026, 11(10): 33350-33376. doi: 10.3934/math.20261305
We investigate a local $M$-deformation of the $(2+1)$-dimensional Akbota–Myrzakulov–Tolkynay–Zhaidary (AMTZ) system and clarify its integrability and traveling-wave structure. For sufficiently smooth functions on the positive domain, we establish an exact two-parameter coordinate equivalence between the local $M$-AMTZ and classical AMTZ systems for $0<\alpha\leq1$ and $\eta>0$. The same transformation shows that the corresponding Lax representation, zero-curvature condition, and conservation structure are inherited from the classical AMTZ system under coordinate pullback.
A general traveling-wave reduction is derived while retaining the integration constant, leading to a generalized Duffing-type equation with an independent constant term. Its first integral is a quartic polynomial and provides a phase-plane description of the reduced dynamics. Generic root configurations lead to elliptic traveling-wave solutions, for which an explicit Weierstrass representation is obtained, whereas repeated-root degenerations yield elementary traveling-wave profiles.
The analysis further shows that the parameters $\alpha$ and $\eta$ do not modify the intrinsic reduced dynamics. Instead, they determine the coordinate representation and the traveling-wave phase, with $\alpha$ producing a nonlinear power-law reparametrization and $\eta$ contributing a constant scaling. Thus, the local $M$-parameters do not by themselves generate distinct intrinsic traveling-wave families. The results provide a unified coordinate-equivalence framework that distinguishes inherited integrability structures from their local $M$-representation.
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Zh.R. Myrzakulova, A. Taishiyeva. Local $ M $-fractional deformation of the (2+1)-dimensional AMTZ system: Coordinate equivalence, Lax integrability, and exact solutions[J]. AIMS Mathematics, 2026, 11(10): 33350-33376. doi: 10.3934/math.20261305
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