On Continual Classes of Evolution Equations

Nonlinear Phenomena in Complex Systems, vol. 28, no. 3 (2025), pp. 308 - 311 On Continual Classes of Evolution Equations Sergei Sakovich∗ Institute of Physics, National Academy of Sciences of Belarus, 68-2 Nezavisimosti Ave., 220072 Minsk, BELARUS (Received 10 June, 2025) The original Miura transformation, considered as a nonlinear potential transformation, is applicable to a continual class of evolution equations, not only to discrete integrable equations and their hierarchies. The same continual class of evolution equations appears from a different problem, namely, from the gauge-invariant description of a zero-curvature representation with a definite x-part containing no essential parameter. PACS numbers: 02.30.Ik, 02.30.Jr Keywords: evolution equations, Miura transformation, zero-curvature representations DOI: https://doi.org/10.5281/zenodo.17243012 1. Introduction The original Miura transformation was introduced in [1] as a nonlinear potential transformation relating the Korteweg–de Vries (KdV) equation to the modified KdV equation. The Miura transformation is applicable not only to the KdV equation and its integrable hierarchy. For example, the Sawada–Kotera equation and the Kaup–Kupershmidt equation also admit the Miura transformation [2]. Moreover, as we have shown in [3], the Miura transformation is applicable to a very wide (continual) class of evolution equations, and most of those equations, of course, should be non-integrable in any reasonable sense. Let us note that this result of [3] was inspired by the Lie–B¨acklund algebra structure of the exactly solvable Liouville equation. In the present paper, in Section 2, we recall the direct derivation of the complete class of evolution equations admitting the Miura transformation, without any reference to Lie– B¨acklund algebras. In Section 3, we show that exactly the same continual class of evolution equations appears from a different problem, namely, from the gauge-invariant description of a zero-curvature representation with a definite ∗E-mail: [email protected] x-part containing no essential parameter. In Section 4, we comment on the obtained results. 2. The Miura transformation Let us find all the local evolution equations ut=f(x, t, u, u1, . . . , un)(1) which admit the Miura transformation u=v1−1 2v2,(2) where uiand vi(i= 1,2, . . . ) denote ∂i xuand ∂i xv, respectively. The Miura transformation (2) relates a local evolution equation (1) to a local evolution equation vt=g(x, t, v, v1, . . . , vn)(3) if the function u(x, t)determined by (2) satisfies (1) whenever the function v(x, t)satisfies (3). This can be equivalently expressed by the condition f(x, t, u, u1, . . . , un) =Dx−vg(x, t, v, v1, . . . , vn),(4) where Dxdenotes the total derivative with respect to x. Note that the condition (4) must be a differential consequence of the relation (2), 308 On Continual Classes of Evolution Equations 309 otherwise (4) and (2) would produce an ordinary differential equation restricting solutions vof the evolution equation (3). Now, using the relation (2) and its differential consequences, namely, v1=u+1 2v2, v2=u1+uv+1 2v3,v3=u2+u2+u1v+2uv2+3 4v4, etc., we can eliminate v1, v2, . . . , vn+1 from the condition (4), and in this way we rewrite (4) in the following equivalent form: f(x, t, u, u1, . . . , un) =e Dx−vh(x, t, v, u, u1, . . . , un−1),(5) where e Dx=∂x+u+1 2v2∂v+u1∂u +u2∂u1+u3∂u2+· · · (6) and h(x, t, v, u, u1, . . . , un−1) =gx, t, v, e Dxv, e D2 xv, . . . , e Dn xv.(7) The crucial point is that the condition (5) must be an identity, because it cannot be a differential consequence of the relation (2). Therefore, the right-hand side of (5) must be independent of v, this determines admissible functions h, and then (5) is simply a definition of admissible functions f. Next, repeatedly applying ∂vto (5) three times and using the evident identity ∂ve Dx=e Dx+v∂v,(8) we obtain the auxiliary condition e Dx+ 2v∂3 vh= 0.(9) Since hmust be a local expression, it follows from (9) that ∂3 vh= 0, that is, the function h is necessarily of the form h=1 2v2p+vq +r, (10) where p,qand rare functions of x, t, u, u1, . . . , un−1. Finally, substituting the expression (10) into the condition (5) and taking into account that p, q,rand fdo not depend on v, we obtain the following expressions: q=Dxp, r =D2 x+up, f=D3 x+ 2uDx+u1p, (11) where pis an arbitrary function of x, t, u, u1, . . . , un−3. The relations (7), (10) and (11) solve our problem. We have found that the local evolution equations (1) admitting the Miura transformation (2) constitute the continual class ut=D3 x+ 2uDx+u1 ×p(x, t, u, u1, . . . , un−3),(12) the corresponding local evolution equation (3) being vt=D2 x+vDx+v1 ×px, t, v1−1 2v2, Dx(v1−1 2v2), . . . , Dn−3 x(v1−1 2v2),(13) where the function pand the order nare arbitrary. 3. The zero-curvature representation The continual class of evolution equations (12) appears from a completely different problem as well. Let us find all the local evolution equations (1) which admit zero-curvature representations (ZCRs) DtX−DxT+ [X, T] = 0 (14) with the fixed matrix Xgiven by X=0u −1 20(15) and any 2×2traceless matrices T(x, t, u, u1, . . . , un−1), where Dtand Dxstand for the total derivatives, the square brackets denote the matrix commutator, and ui=∂i xu Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 310 Sergei Sakovich (i= 1,2, . . . ). We solve this problem by the cyclic basis method [4–9]. In the present case of the matrix Xgiven by (15), the characteristic form of the ZCR (14) of an evolution equation (1) is fC =∇T, (16) where fis the right-hand side of (1), C=∂X/∂u, and the operator ∇is defined as ∇M=DxM− [X, M]for any 2×2matrix M. Computing C, ∇C,∇2Cand ∇3C, we find that the cyclic basis is {C, ∇C, ∇2C}, with the closure equation ∇3C=−u1C−2u∇C. (17) Decomposing Tover the cyclic basis as T=a0C+a1∇C+a2∇2C, (18) where a0,a1and a2are functions of x, t, u, u1, . . . , un−1, we find from (16) and (17) that a1=−Dxa2, a0=−Dxa1+ 2ua2, f=Dxa0−u1a2.(19) Note that the function a2= p(x, t, u, u1, . . . , un−3)and the order nremain undetermined. The explicit expressions T= 1 2Dxp D2 xp+up −1 2p−1 2Dxp!(20) and f=D3 xp+ 2uDxp+u1p, (21) which follow from (18) and (19), solve our problem. We have found that the local evolution equations (1) admitting ZCRs (14) with the matrix Xgiven by (15) constitute the continual class (12), the corresponding matrices Tbeing determined by (20). 4. Conclusion We have shown that the original Miura transformation (2), considered as a nonlinear potential transformation, is applicable to a continual class of evolution equations (12), not only to discrete integrable equations and their hierarchies. We have also shown that exactly the same continual class of evolution equations (1) with (21) appears from a different problem, namely, from the gauge-invariant description of a zero-curvature representation (14) with the given x-part (15) containing no essential parameter. Not every expression of the form u= s(x, t, v, v1, . . . , vm)can serve as a Miura-type transformation between two local evolution equations, as was shown by a simple polynomial generalization of the original (quadratic) Miura transformation [10]. In many interesting special cases, the general Miura-type transformations can be analyzed and represented as chains of simpler transformations [11], and this definitely deserves further investigation. We believe that the relation between the differential substitutions (another name of Miura-type transformations) and the socalled pseudosymmetries [12] may be very useful, for the following reason. Pseudosymmetries correspond to ZCRs with some fixed x-parts [12], whereas ZCRs with fixed x-parts always represent some continual classes of evolution equations [4]. References [1] R.M. Miura. Korteweg–de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation. J. Math. Phys. 9, 1202 (1968). Нелинейные явления в сложных системах Т. 28, № 3, 2025 On Continual Classes of Evolution Equations 311 [2] A.P. Fordy, J. Gibbons. Some remarkable nonlinear transformations. Phys. Lett. A 75, 325 (1980). [3] S.Yu. Sakovich. The Miura transformation and Lie–B¨acklund algebras of exactly solvable equations. Phys. Lett. A 132, 9 (1988). [4] S.Yu. Sakovich. On zero-curvature representations of evolution equations. J. Phys. A: Math. Gen. 28, 2861 (1995). [5] S.Yu. Sakovich. Cyclic bases of zero-curvature representations: five illustrations to one concept. Acta Appl. Math. 83, 69 (2004). [6] A. Karasu-Kalkanlı, A. Karasu, S.Yu. Sakovich. A strange recursion operator for a new integrable system of coupled Korteweg–de Vries equations. Acta Appl. Math. 83, 85 (2004). [7] S. Sakovich. A note on Lax pairs of the Sawada– Kotera equation. J. Math. 2014, 906165 (2014). [8] S. Sakovich. True and fake Lax pairs: how to distinguish them. Int. J. Nonlinear Phenom. Complex Syst. 23, 338 (2020). [9] S. Sakovich. An integrable hierarchy without a recursion operator. Int.J. Nonlinear Phenom. Complex Syst. 26, 131 (2023). [10] S.Yu. Sakovich. On the polynomial Miura transformation. Phys. Lett. A 146, 32 (1990). [11] S.Yu. Sakovich. On Miura transformations of evolution equations. J. Phys. A: Math. Gen. 26, L369–L373 (1993). [12] V.V. Sokolov. Pseudosymmetries and differential substitutions. Funct. Anal. Appl. 22, 121 (1988). Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025