Novikov Equation: Integrability & Peakon Dynamics

• Novikov Equation is an integrable one-dimensional PDE with cubic nonlinearity that models shallow-water dynamics and supports multiple coherent structures.
• It features a bi-Hamiltonian formulation and admits peakon, multipeakon, and smooth solitary wave solutions recoverable via inverse-scattering and spectral methods.
• The equation exhibits wave breaking and singularity formation, which necessitate conservative weak solution frameworks and detailed stability analyses.

The Novikov equation is an integrable one-dimensional evolution equation for a scalar field $u=u(t,x)$ with cubic nonlinearity, usually written in momentum form

$$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$

or, equivalently,

$$u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx}, \qquad x\in\mathbb R,\ t>0.$$

It belongs to the class of Camassa–Holm-type equations, but differs from the classical Camassa–Holm equation by replacing linear or quadratic transport with cubic nonlinear transport. The modern theory combines integrable-systems methods, conservative weak-solution theory, multipeakon dynamics, inverse-scattering and Riemann–Hilbert formulations, and stability analysis of both peaked and smooth coherent structures (Chen et al., 2015).

1. Definition and Camassa–Holm-type structure

The Novikov equation is commonly presented through the momentum variable $m=u-u_{xx}$, giving the compact transport law

$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0,$

which is equivalent to

$m_t + (u^2m)_x + 2u\,m\,u_x = 0$

after expanding $(u^2m)_x$ (Chen et al., 2015). In local $u$-form it is written as

$u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx},$

and in nonlocal form, with Green kernel $p(x)=\tfrac12 e^{-|x|}$,

$$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$0

where

$$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$1

These forms are used interchangeably in the PDE, weak-solution, and spectral literature (Chen et al., 2015).

Within the Camassa–Holm-type family, the defining feature is the transport of $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$2 by a nonlinear velocity. In the summary of Chen, Chen, and Liu, one recognizes the general structure

$$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$3

with cubic nonlinearity $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$4. This contrasts with the classical Camassa–Holm equation, where the transport is linear in $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$5 (Chen et al., 2015). The same comparison appears repeatedly in later work, where the Novikov equation is grouped with Camassa–Holm and Degasperis–Procesi as a peakon equation, but with cubic rather than quadratic nonlinearity (Lafortune, 2023).

The equation was introduced by Vladimir Novikov in 2009 in a symmetry classification of nonlocal evolution equations and was subsequently studied by Hone and Wang as an integrable Camassa–Holm-type model with cubic nonlinearity (Chang et al., 2017). Chen–Hu–Liu further showed that it can be derived as a higher-order approximation in shallow-water asymptotics and fits into the class of tri-Hamiltonian integrable systems (He et al., 2023). This places the equation simultaneously in the theory of shallow-water asymptotics, nonlocal quasilinear dispersive PDE, and integrable hierarchies.

A recurrent source of confusion is nomenclature. The Novikov equation is a one-dimensional Camassa–Holm-type PDE with cubic nonlinearity, whereas the Novikov–Veselov equation is a two-dimensional zero-energy integrable system for a real potential $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$6 or $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$7; the latter is treated through Manakov triples, Moutard transformations, and $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$8-D inverse scattering, and is a distinct equation despite the similar name (Perry, 2012).

2. Integrability and algebraic structure

Integrability is a central feature of the Novikov equation. Novikov showed that the equation admits a $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$9 Lax representation; in scalar form one may write

$$u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx}, \qquad x\in\mathbb R,\ t>0.$$0

together with an evolution equation for $$u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx}, \qquad x\in\mathbb R,\ t>0.$$1 involving $$u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx}, \qquad x\in\mathbb R,\ t>0.$$2, $$u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx}, \qquad x\in\mathbb R,\ t>0.$$3, and the spectral parameter $$u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx}, \qquad x\in\mathbb R,\ t>0.$$4 (Chen et al., 2015). Other papers use equivalent $$u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx}, \qquad x\in\mathbb R,\ t>0.$$5 matrix Lax systems or a $$u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx}, \qquad x\in\mathbb R,\ t>0.$$6 matrix representation, both encoding the same zero-curvature condition and confirming complete integrability (Monvel et al., 2016).

The equation is bi-Hamiltonian in the formulation of Chen, Chen, and Liu, with compatible Hamiltonian operators

$$u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx}, \qquad x\in\mathbb R,\ t>0.$$7

$$u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx}, \qquad x\in\mathbb R,\ t>0.$$8

and Hamiltonians

$$u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx}, \qquad x\in\mathbb R,\ t>0.$$9

so that

$m=u-u_{xx}$0

Other summaries describe a bi-/tri-Hamiltonian structure and infinitely many conservation laws, including

$m=u-u_{xx}$1

and higher-order conserved quantities generated by the integrable hierarchy (He et al., 2023).

The conservation laws most frequently used in analysis are the $m=u-u_{xx}$2-energy

$m=u-u_{xx}$3

and the fourth-order energy

$m=u-u_{xx}$4

which are conserved for smooth solutions and remain central in the conservative weak theory (Cai et al., 2016). These two functionals also underlie the stability theory of peakons and smooth solitary waves (Chen et al., 2019).

Integrability persists in finite-dimensional reductions. Under the multipeakon ansatz

$m=u-u_{xx}$5

the momentum becomes a discrete measure and the PDE reduces to an integrable ODE system for the peak positions and amplitudes (Chang et al., 2017). Chang, Li, and Szmigielski showed that the peakon ODEs admit a Pfaffian formulation and can be interpreted as an isospectral flow on a manifold cut out by Pfaffian identities; they also established a link with the finite Toda lattice of BKP type (Chang et al., 2017). This algebraic structure is specific to the BKP/Sawada–Kotera side of the integrable hierarchy and is one of the distinctive features separating the Novikov equation from the better-known Camassa–Holm peakon problem.

3. Conservative weak solutions and global well-posedness

Because smooth solutions can develop gradient blow-up in finite time while $m=u-u_{xx}$6 remains bounded, the Novikov equation is studied in a conservative weak-solution class adapted to wave breaking. In the formulation of Chen, Chen, and Liu, an energy-conservative weak solution satisfies $m=u-u_{xx}$7 for each $m=u-u_{xx}$8, the map $m=u-u_{xx}$9 is Lipschitz in the $m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0,$0-metric, the equation holds in weak nonlocal form, and one supplements a measure-valued balance law for a higher-order energy measure whose absolutely continuous part has density $m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0,$1 (Chen et al., 2015).

Their main theorem states that if

$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0,$2

then there exists a unique global energy-conservative solution. Moreover, $m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0,$3 is Hölder continuous in $m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0,$4 with exponent $m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0,$5; the first energy $m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0,$6 is exactly conserved for all $m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0,$7; the second energy is nonincreasing and becomes constant once the singular part of the higher-order energy measure is included; and solutions depend continuously on the initial data in the sense of uniform convergence on compact subsets of $m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0,$8 (Chen et al., 2015).

The proof is built around a semilinear reformulation in characteristic-Lagrangian variables. One introduces an energy-stretched coordinate $m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0,$9, together with

$m_t + (u^2m)_x + 2u\,m\,u_x = 0$0

and obtains a closed semilinear ODE system for $m_t + (u^2m)_x + 2u\,m\,u_x = 0$1 in the $m_t + (u^2m)_x + 2u\,m\,u_x = 0$2 variables. Local well-posedness follows by Picard iteration, while global existence follows from the conserved energies $m_t + (u^2m)_x + 2u\,m\,u_x = 0$3 and $m_t + (u^2m)_x + 2u\,m\,u_x = 0$4, which control $m_t + (u^2m)_x + 2u\,m\,u_x = 0$5 and $m_t + (u^2m)_x + 2u\,m\,u_x = 0$6 (Chen et al., 2015). The inverse transform recovers $m_t + (u^2m)_x + 2u\,m\,u_x = 0$7, and uniqueness is proved by re-parameterizing solutions with a generalized characteristic variable

$m_t + (u^2m)_x + 2u\,m\,u_x = 0$8

so that along good characteristics one returns to the same semilinear ODE system (Chen et al., 2015).

A complementary well-posedness result was obtained by constructing a Finsler-type optimal-transport metric that renders the solution map Lipschitz continuous on bounded subsets of $m_t + (u^2m)_x + 2u\,m\,u_x = 0$9 (Cai et al., 2016). The motivation is that the standard Sobolev metric fails at wave breaking: even when $(u^2m)_x$0 remains finite, the $(u^2m)_x$1-distance between nearby solutions may jump at a peakon collision because the higher-order energy measure develops a Dirac mass (Cai et al., 2016).

The tangent norm in that construction weights both vertical and horizontal deformations against the measure

$(u^2m)_x$2

and the induced geodesic distance generates the same topology as $(u^2m)_x$3 while giving a Grönwall estimate of the form

$(u^2m)_x$4

for conservative solutions (Cai et al., 2016). The same paper also proves, via Thom’s transversality theorem, that solutions are piecewise smooth for an open dense set of initial data in $(u^2m)_x$5, which allows the metric theory to be extended from generic piecewise regular solutions to general weak solutions (Cai et al., 2016).

These results single out the conservative continuation as the distinguished global dynamics beyond wave breaking. A plausible implication is that, for the Novikov equation, uniqueness at low regularity is inseparable from the measure-valued encoding of concentrated higher-order energy.

4. Singularities, wave breaking, and generic cusp structure

Wave breaking for the Novikov equation is characterized by bounded $(u^2m)_x$6 together with blow-up of $(u^2m)_x$7. In the conservative framework, singularities arise where the Jacobian of the transformation from characteristic coordinates $(u^2m)_x$8 to Eulerian coordinates $(u^2m)_x$9 vanishes, equivalently where

$u$0

(He et al., 2023). The singularity theory developed by He, Luo, and Yin gives a local classification of generic singular points for conservative solutions.

For generic smooth initial data they prove that only two local patterns occur. In Type I, one has

$u$1

and near the corresponding Eulerian point $u$2,

$u$3

In Type II, one has

$u$4

and

$u$5

These asymptotics identify two generic cuspidal singularities of orders $u$6 and $u$7 (He et al., 2023).

The geometric interpretation is given in characteristic variables. Type I corresponds to a tangential collision of order $u$8, while Type II corresponds to a transverse first-order collision (He et al., 2023). This connects the fine singular structure directly to the semilinear characteristic formulation used in the existence theory.

Chen, Chen, and Liu emphasize a qualitative difference between the Novikov and Camassa–Holm cases: because the transport speed $u$9 vanishes at $u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx},$0, one can maintain a “stuck” characteristic carrying concentrated energy as long as $u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx},$1, whereas in the Camassa–Holm case the slope “instantaneously recovers” after peak breaking (Chen et al., 2015). This suggests that persistent concentration of higher-order energy is a structural feature of the cubic model rather than an artifact of a particular solution class.

Regularity theory is therefore naturally formulated below $u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx},$2. The $u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx},$3-Hölder continuity obtained in the global conservative theory (Chen et al., 2015) is consistent with the Type I cusp exponent of the generic singularity analysis (He et al., 2023). A plausible implication is that the Hölder threshold is not merely technical, but reflects the geometry of characteristic focusing built into conservative weak solutions.

5. Peakons, multipeakons, and stability theory

The Novikov equation supports peaked traveling waves. On the line, the single peakon takes the form

$u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx},$4

which is a weak solution with a cusp at the crest (Chen et al., 2019). In the scaled $u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx},$5 form one often writes simply

$u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx},$6

The identity $u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx},$7 makes clear why the momentum of a peakon is a point mass and why the peakon sector leads to a finite-dimensional integrable system (Lafortune, 2023).

For multipeakons,

$u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx},$8

substitution into the PDE yields

$u_t-u_{xxt}+4u^2u_x=3u\,u_x\,u_{xx}+u^2u_{xxx},$9

or, more explicitly,

$p(x)=\tfrac12 e^{-|x|}$0

$p(x)=\tfrac12 e^{-|x|}$1

Hone–Lundmark–Szmigielski solved this system by inverse-spectral methods, and Chang–Li–Szmigielski later recast it in a Pfaffian framework tied to BKP-type tau-functions (Chang et al., 2017).

The stability theory of peakons is norm-dependent. Orbital stability in $p(x)=\tfrac12 e^{-|x|}$2 had been established previously and is summarized in the instability paper of Liu, Pelinovsky, and Shimabukuro: if $p(x)=\tfrac12 e^{-|x|}$3, then the global weak solution remains uniformly close in $p(x)=\tfrac12 e^{-|x|}$4 to the translation orbit of $p(x)=\tfrac12 e^{-|x|}$5, with the proof based on the conserved quantities

$p(x)=\tfrac12 e^{-|x|}$6

and a Lyapunov functional built from them (Chen et al., 2019).

By contrast, the same paper proves $p(x)=\tfrac12 e^{-|x|}$7-instability. Linearization about a peakon shows that $p(x)=\tfrac12 e^{-|x|}$8 grows like $p(x)=\tfrac12 e^{-|x|}$9, even though the $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$00-norm of the linearized perturbation is exactly conserved on each half-line (Chen et al., 2019). For the full nonlinear perturbation problem, characteristic ODEs and a Riccati-type inequality at the peak imply that arbitrarily small $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$01-perturbations can produce $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$02, and with further refinement can force finite-time blow-up of the slope while $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$03 remains bounded (Chen et al., 2019).

A sharper spectral picture was obtained by El Dika and Molinet. In $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$04, the linearized operator about the peakon has spectrum

$$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$05

a closed vertical strip, so the peakon is spectrally unstable in $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$06. In $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$07, the spectrum satisfies

$$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$08

again giving instability, whereas in the energy space $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$09,

$$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$10

and the linearized $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$11-norm is exactly conserved (Lafortune, 2023). These results align $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$12 and $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$13 instability with $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$14 orbital stability, rather than contradicting it.

The periodic setting has an analogous peakon theory. On the circle $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$15, Wang and Tian showed that the periodic peakon profile

$$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$16

defines a global periodic weak solution, and they proved orbital stability in $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$17 by controlling the maximum and minimum of the solution through the conserved functionals

$$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$18

(Wang et al., 2018).

6. Smooth solitary waves, periodic traveling waves, and analytical formulations

The Novikov equation also admits smooth solitary waves on nonzero background. For a traveling-wave ansatz $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$19, $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$20, one derives, under the nondegeneracy assumptions

$$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$21

the profile equation

$$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$22

together with a first integral

$$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$23

A phase-plane analysis shows that for every fixed $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$24 and every

$$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$25

there is a unique even homoclinic orbit $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$26 with $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$27, $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$28, and $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$29. Equivalently, the waves can be parameterized by the asymptotic end-state

$$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$30

with

$$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$31

(Ehrman et al., 2024).

The orbital stability theory for these smooth solitary waves is formulated in the momentum variable $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$32. Dunlap, Kozlov, and Marangell introduce an action functional

$$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$33

whose critical points coincide with solitary-wave profiles, and analyze the Hessian $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$34, a nonlocal integro-differential operator on $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$35 (Ehrman et al., 2024). Under a spectral hypothesis consisting of one simple negative eigenvalue, a simple zero eigenvalue spanned by $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$36, and positive essential spectrum, together with a Vakhitov–Kolokolov condition, they obtain orbital stability in $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$37. Their Evans-function computations and numerical study of the constrained functional indicate that all smooth solitary waves satisfy the required hypotheses (Ehrman et al., 2024).

Periodic smooth traveling waves have also been studied in the $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$38-family of Novikov equations,

$$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$39

for which $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$40 corresponds to the classical Novikov equation (Zhao et al., 28 Feb 2026). For small-amplitude $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$41-periodic waves

$$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$42

spectral perturbation theory and Floquet–Bloch analysis reduce modulational stability near $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$43 to the sign of the index

$$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$44

The condition $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$45 is exactly the Benjamin–Feir instability criterion; when it holds, sufficiently small periodic traveling waves are spectrally unstable to long-wave modulations, while if $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$46 and $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$47, the waves are spectrally stable (Zhao et al., 28 Feb 2026).

Analytical formulations beyond inverse scattering also exist. Bozhkov, Freire, and Ibragimov identified a five-dimensional Lie point symmetry algebra generated by

$$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$48

proved strict self-adjointness through a formal Lagrangian, and derived a nontrivial local conservation law associated with the dilation symmetry: $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$49 on the solution manifold (Bozhkov et al., 2012). They also obtained group-invariant solutions, including stationary, separated, and traveling-wave reductions (Bozhkov et al., 2012).

Inverse-scattering on nonzero background has been developed through a $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$50 Riemann–Hilbert problem. Boutet de Monvel, Shepelsky, and Zielinski transformed the Cauchy problem with $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$51 as $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$52 into a meromorphic RH problem on a six-ray contour in the complex $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$53-plane, with reconstruction of $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$54 in parametric form from the value of the RH solution at a special point $$m_t + u^2\,m_x + \tfrac32\,(u^2)_x\,m = 0, \qquad m=(1-\partial_x^2)u,$$55 (Monvel et al., 2016). They also derived reflectionless soliton solutions and described the Novikov equation in this setting as a “modified DP equation,” in analogy with the relation between KdV and mKdV (Monvel et al., 2016).

Taken together, these developments show that the Novikov equation supports several distinct coherent-structure sectors—peakons, smooth solitons, and periodic traveling waves—whose analysis requires different combinations of variational, spectral, geometric, and integrable-systems techniques.