New 5th-order Schwarzian evolution equations and their higher-order symmetries

Abstract: We report new quasilinear and fully-nonlinear 5th-order Schwarzian evolution equations. These are symmetry-integrable evolution equations in 1+1 dimensions, i.e. equations that admit Lie-Bäcklund symmetries, whereby it is required that the equations are kept invariant under the Möbius transformation for their dependent variable.

• The paper completes the classification of Möbius-invariant, symmetry-integrable 5th-order Schwarzian evolution equations by identifying six genuinely new cases with explicit 7th-order Lie–Bäcklund symmetries.
• The authors use an equivalent auxiliary Schwarzian equation to analyze both quasilinear and fully nonlinear forms, proving uniqueness within the considered classes and recovering three new families of solutions.
• All six equations lack a 9th-order symmetry, producing a distinctive sequence beginning at orders 7, 11, 13, 17, and 19, while recursion operators and higher hierarchy members remain open problems.

This paper by Marianna Euler and Norbert Euler completes the classification of 5th-order Schwarzian evolution equations that are symmetry-integrable and invariant under the Möbius (projective) transformation of the dependent variable. Building on the authors' earlier classification work, it establishes all quasilinear and fully-nonlinear 5th-order equations of the form $u_t = u_x\Phi(S,S_x,S_{xx})$ admitting a 7th-order Lie-Bäcklund symmetry, where $S$ is the Schwarzian derivative. The analysis yields six genuinely new equations, each supplied with its explicit 7th-order $\tau_1$-equation.

Framework: Schwarzian equations and their $\tau$-equations

A Schwarzian evolution equation of order $n$ has the form $u_t = u_x\Phi(S,S_x,\ldots,S_{(n-3)x})$, where

$$S := \frac{u_{3x}}{u_x} - \frac{3}{2}\left(\frac{u_{xx}}{u_x}\right)^2$$

is the Schwarzian derivative. Such an equation is automatically invariant under the Möbius transformation $$u \mapsto (\alpha_1 u + \beta_1)/(\alpha_2 u + \beta_2)$$ together with translations in $x$ and $t$. Symmetry-integrability is characterized by the existence of infinitely many local commuting Lie-Bäcklund symmetries whose characteristic functions take the form $S$0. Each such symmetry generator defines a $S$1-equation of order $S$2, namely $S$3, satisfying the compatibility condition $S$4; the base equation generates a hierarchy of commuting flows.

A central technical device is the auxiliary $S$5-equation,

$S$6

which is equivalent to the original equation: one admits a Lie-Bäcklund symmetry of order $S$7 if and only if the other does. Because this operator applied to any function is never nonlinear in its highest derivative, the auxiliary formulation is always quasilinear and therefore more tractable for symmetry computations. The paper exploits this throughout, solving the invariance condition $S$8 for a 7th-order characteristic $S$9.

Prior classifications

The paper situates itself against three earlier results. First, the 3rd-order semilinear case reduces to the Schwarzian Korteweg–de Vries equation $\tau_1$0, whose hierarchy is generated by a 2nd-order recursion operator and includes $\tau_1$1-equations at every odd order. Second, exactly two semilinear 5th-order equations exist beyond the SKdV hierarchy:

$\tau_1$2

each admitting a 6th-order recursion operator generating two hierarchies with symmetries at orders $\tau_1$3 and $\tau_1$4 — notably skipping order nine. Third, three fully-nonlinear 3rd-order equations were classified, with $\tau_1$5, $\tau_1$6, and $\tau_1$7; the first two possess 2nd-order recursion operators, while for the third only a recursion operator for the corresponding $\tau_1$8-equation is known. The 5th- and 7th-order quasilinear $\tau_1$9-equations of these base equations are themselves quasilinear 5th-order Schwarzian equations and must be excluded from the new classification.

New quasilinear 5th-order equations

The quasilinear ansatz $\tau$0 yields two families after excluding known $\tau$1-equations. The first, under the genericity condition $\tau$2, is parametrized by an arbitrary nonzero constant $\tau$3 with $\tau$4:

$\tau$5

The second family, with $\tau$6 depending on $\tau$7 alone, is parametrized by an arbitrary constant $\tau$8:

$\tau$9

Both propositions are stated as uniqueness results: no other quasilinear 5th-order Schwarzian equations outside the previously classified classes admit a 7th-order Lie-Bäcklund symmetry. The corresponding 7th-order $n$0 functions are given explicitly, and they are structurally consistent with the fractional-power structure ($n$1 exponents on the leading term) observed in the fully-nonlinear 3rd-order hierarchies.

New fully-nonlinear 5th-order equations

For the fully-nonlinear case, $n$2, the classification yields exactly three equations. Two share the structural form

$n$3

and the third is

$n$4

The authors note that the two-parameter family form was derived independently in prior work as part of a classification of equations invariant under a larger projective transformation group acting on both $n$5 and $n$6; the present result confirms those two members within the purely Möbius-invariant setting and adds $n$7 as new. Explicit 7th-order $n$8-equations are provided for all three cases.

Symmetry structure and open problems

A salient structural finding is that none of the six new equations admits a Lie-Bäcklund symmetry of order nine. Their symmetry orders proceed as $n$9, mirroring the gap pattern already seen for the semilinear 5th-order equations but distinct from the SKdV hierarchy, which has symmetries at every odd order. This absence of a 9th-order symmetry is asserted without qualification and constitutes the clearest distinguishing feature of the 5th-order class.

The paper concedes two limitations explicitly. First, recursion operators for the new 5th-order equations have not been constructed, so the higher members of their hierarchies beyond the computed 7th-order $u_t = u_x\Phi(S,S_x,\ldots,S_{(n-3)x})$0-equations are not yet available in closed form; establishing these operators is left open. Second, the same gap persists for the fully-nonlinear 3rd-order equation of Case 1.4.3, for which only the $u_t = u_x\Phi(S,S_x,\ldots,S_{(n-3)x})$1-equation recursion operator is known. A further open question noted by the authors is whether any 7th-order Schwarzian evolution equations exist that are not $u_t = u_x\Phi(S,S_x,\ldots,S_{(n-3)x})$2-equations of lower-order base equations; none are currently known.

Conclusion

The paper completes the 5th-order tier of the classification of Möbius-invariant, symmetry-integrable evolution equations initiated in the authors' earlier work: combined with the previously known semilinear cases, the quasilinear and fully-nonlinear 5th-order classes are now exhausted, with six new equations identified and verified through their 7th-order Lie-Bäcklund symmetries. The remaining programmatic tasks are concrete: constructing recursion operators for the new equations, extending the hierarchies accordingly, and determining whether the classification pattern extends to genuinely new 7th-order base equations.