Novikov Equations: Integrability & Peakons

• Novikov equations are integrable cubic Camassa–Holm type PDEs characterized by cubic nonlinearities and the presence of peakon solutions.
• They feature rich symmetry structures and self-adjoint properties that yield nontrivial conservation laws through Lie-point symmetry analysis.
• Extensions include weakly dissipative, multicomponent, and algebraic-geometric variants, broadening their applications in inverse scattering and spectral theory.

“Novikov equations” denotes a small but non-uniform family of mathematical objects centered on the scalar Novikov equation

$u_t-u_{txx}+4u^2u_x-3uu_xu_{xx}-u^2u_{xxx}=0,$

equivalently

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$

an integrable Camassa–Holm type equation with cubic nonlinearities, Lax pair representations, and peakon solutions. In current usage, the term also covers weakly dissipative, multicomponent, $b$-family, and $\lambda$-family variants, the associated Novikov equation, and several distinct Novikov-named equations that are not equivalent to the $1+1$-dimensional cubic shallow-water model, notably the Novikov–Veselov equation and the algebraic-geometric “Novikov equations” arising from commuting differential operators (Tiglay, 2010, Lenells et al., 2012, Lassas et al., 2011, Shabat et al., 20 Jul 2025).

1. Terminological scope and canonical scalar form

In the PDE literature, the canonical Novikov equation is the cubic analogue of the Camassa–Holm and Degasperis–Procesi equations. Besides the momentum form

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$

it appears in periodic, nonperiodic, weakly dissipative, and nonzero-background formulations. In the periodic Cauchy problem it is also written in nonlocal form

$$u_t+u^2u_x=-A^{-2}\bigl(3u^2u_x+2u_x^3+3uu_xu_{xx}\bigr),\qquad A^2=1-\partial_x^2,$$

while the nonzero-background inverse-scattering treatment uses $\hat m=m+1=u-u_{xx}+1$ after reduction to zero background (Tiglay, 2010, Monvel et al., 2016).

Usage	Defining equation	Relation
Scalar Novikov equation	$u_t-u_{txx}+4u^2u_x-3uu_xu_{xx}-u^2u_{xxx}=0$	Central $1+1$-dimensional cubic CH-type equation (Tiglay, 2010)
Associated Novikov equation	$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$0	Related to Novikov by a chain of transformations (Rasin et al., 2019)
Novikov–Veselov equation	$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$1	Unrelated $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$2-dimensional KdV analogue (Lassas et al., 2011)
Novikov equations for commuting operators	Conditions equivalent to $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$3 for $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$4	Stationary algebraic-geometric usage (Shabat et al., 20 Jul 2025)

This multiplicity of usage is substantive rather than merely terminological. The scalar Novikov equation belongs to the peakon and shallow-water literature; the associated Novikov equation is a transform-related scalar representative with a third-order scalar spectral problem; the Novikov–Veselov equation belongs to $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$5-dimensional inverse scattering; and the commuting-operator Novikov equations arise as stationary commutativity conditions in the theory of rank-one commutative rings of differential operators (Rasin et al., 2019, Lassas et al., 2011, Shabat et al., 20 Jul 2025).

2. Symmetry structure, self-adjointness, and local conservation laws

The Lie-point symmetry analysis of the scalar Novikov equation yields a five-dimensional algebra with basis

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$6

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$7

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$8

together with the discrete symmetry $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$9, which interchanges $b$0 and $b$1. Using Ibragimov’s formal Lagrangian

$b$2

the adjoint equation satisfies

$b$3

so the equation is strictly self-adjoint. In this framework, among all Lie point symmetries only the dilation symmetry $b$4 yields a nontrivial local conserved vector, namely

$b$5

on solutions, whereas $b$6 give only trivial conservation laws. The same analysis produces explicit invariant solutions, including the infinite families

$b$7

dilation-invariant ansätze $b$8, and travelling waves $b$9, with $\lambda$0 as an explicit example (Bozhkov et al., 2012).

This selective symmetry–conservation correspondence is characteristic of the scalar Novikov equation. A broader polynomial family

$\lambda$1

contains Camassa–Holm at $\lambda$2 and Novikov at $\lambda$3. Within that family, the joint requirements of $\lambda$4-peakons and the low-order conservation law

$\lambda$5

single out the generalized Camassa–Holm–Novikov equation

$\lambda$6

which reduces to Novikov at $\lambda$7 (Anco et al., 2014).

3. Cauchy theory, global continuation, inverse scattering, and weak dissipation

For the periodic Cauchy problem on $\lambda$8, the Novikov equation is locally well posed in Sobolev spaces $\lambda$9 for

$1+1$0

with solutions

$1+1$1

A continuation criterion of Beale–Kato–Majda type is obtained from

$1+1$2

If $1+1$3 and the initial momentum satisfies

$1+1$4

then the solution exists globally and uniquely. In the analytic category, a Cauchy–Kowalevski type theorem yields short-time existence and uniqueness of solutions analytic in both $1+1$5 and $1+1$6 (Tiglay, 2010).

A different global theory is available in the $1+1$7-family

$1+1$8

whose case $1+1$9 is the Novikov equation. For $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$0, absolutely continuous initial data

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$1

generate global energy-conservative weak solutions that are Hölder continuous with exponent

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$2

These solutions preserve

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$3

for every $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$4 and continue past wave breaking in a conservative weak-solution framework (Chen et al., 2021).

On the line with nonzero constant background, the inverse scattering transform has been formulated through a $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$5 matrix Riemann–Hilbert problem. After reduction to zero background and introduction of

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$6

the Cauchy problem is encoded in a singular $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$7 RH problem on a six-ray contour in the $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$8-plane. The solution is reconstructed parametrically from the RH solution $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$9 at the distinguished point $$u_t+u^2u_x=-A^{-2}\bigl(3u^2u_x+2u_x^3+3uu_xu_{xx}\bigr),\qquad A^2=1-\partial_x^2,$$0 through formulas for $$u_t+u^2u_x=-A^{-2}\bigl(3u^2u_x+2u_x^3+3uu_xu_{xx}\bigr),\qquad A^2=1-\partial_x^2,$$1 and $$u_t+u^2u_x=-A^{-2}\bigl(3u^2u_x+2u_x^3+3uu_xu_{xx}\bigr),\qquad A^2=1-\partial_x^2,$$2. This framework exhibits the Novikov equation as a “modified DP equation” in the sense that its RH geometry is closely parallel to that of Degasperis–Procesi, but with a different nonlinear reconstruction (Monvel et al., 2016).

The weakly dissipative Novikov equation,

$$u_t+u^2u_x=-A^{-2}\bigl(3u^2u_x+2u_x^3+3uu_xu_{xx}\bigr),\qquad A^2=1-\partial_x^2,$$3

is exactly reducible to the non-dissipative equation by

$$u_t+u^2u_x=-A^{-2}\bigl(3u^2u_x+2u_x^3+3uu_xu_{xx}\bigr),\qquad A^2=1-\partial_x^2,$$4

Hence the dissipative flow is the non-dissipative flow viewed through exponential amplitude damping and the compressed time variable

$$u_t+u^2u_x=-A^{-2}\bigl(3u^2u_x+2u_x^3+3uu_xu_{xx}\bigr),\qquad A^2=1-\partial_x^2,$$5

This makes the weakly dissipative model structurally equivalent to the ordinary Novikov equation rather than an independent cubic evolution law (Lenells et al., 2012).

4. Peakons, characteristics, wave breaking, and stability

The multipeakon ansatz

$$u_t+u^2u_x=-A^{-2}\bigl(3u^2u_x+2u_x^3+3uu_xu_{xx}\bigr),\qquad A^2=1-\partial_x^2,$$6

plays the same role for Novikov that it does for Camassa–Holm and Degasperis–Procesi, but the cubic nonlinearity changes both characteristic transport and finite-dimensional dynamics. In the Novikov case,

$$u_t+u^2u_x=-A^{-2}\bigl(3u^2u_x+2u_x^3+3uu_xu_{xx}\bigr),\qquad A^2=1-\partial_x^2,$$7

so the particle speed is the square of the wave elevation, and for peakons

$$u_t+u^2u_x=-A^{-2}\bigl(3u^2u_x+2u_x^3+3uu_xu_{xx}\bigr),\qquad A^2=1-\partial_x^2,$$8

This implies $$u_t+u^2u_x=-A^{-2}\bigl(3u^2u_x+2u_x^3+3uu_xu_{xx}\bigr),\qquad A^2=1-\partial_x^2,$$9, so both peakons and antipeakons move to the right. The explicit inverse-spectral formulas for ordinary $\hat m=m+1=u-u_{xx}+1$0-peakons can be extended to all characteristic curves by introducing “ghostpeakons,” namely zero-amplitude peakons. In each interval between adjacent peakons, the characteristics are

$\hat m=m+1=u-u_{xx}+1$1

and the signed field along such curves is recovered by a separate determinant formula because the characteristic speed only gives $\hat m=m+1=u-u_{xx}+1$2, not the sign of $\hat m=m+1=u-u_{xx}+1$3 (Lundmark et al., 2018).

The Novikov equation also admits explicit unbounded solutions with finite-time peakon creation. A Lie symmetry analysis produces nonlinear transformations that send the one-peakon $\hat m=m+1=u-u_{xx}+1$4 to

$\hat m=m+1=u-u_{xx}+1$5

which is smooth up to

$\hat m=m+1=u-u_{xx}+1$6

when a peak is created at $\hat m=m+1=u-u_{xx}+1$7. For $\hat m=m+1=u-u_{xx}+1$8, the solution becomes piecewise smooth,

$\hat m=m+1=u-u_{xx}+1$9

and remains a weak solution. This provides an explicit smooth-to-peaked transition inside the Novikov dynamics (Kardell, 2013).

Stability theory bifurcates according to the class of solitary waves considered. For smooth solitary waves on nonzero background, there is a one-parameter family $u_t-u_{txx}+4u^2u_x-3uu_xu_{xx}-u^2u_{xxx}=0$0 with asymptotic endstate

$u_t-u_{txx}+4u^2u_x-3uu_xu_{xx}-u^2u_{xxx}=0$1

and these waves are critical points of a renormalized action functional

$u_t-u_{txx}+4u^2u_x-3uu_xu_{xx}-u^2u_{xxx}=0$2

The associated Hessian is a nonlocal integro-differential operator on $u_t-u_{txx}+4u^2u_x-3uu_xu_{xx}-u^2u_{xxx}=0$3, and the analysis, together with numerical Evans-function evidence and a Vakhitov–Kolokolov condition, indicates that all smooth solitary wave solutions are nonlinearly orbitally stable (Ehrman et al., 2024).

For peakons in the cubic $u_t-u_{txx}+4u^2u_x-3uu_xu_{xx}-u^2u_{xxx}=0$4-family

$u_t-u_{txx}+4u^2u_x-3uu_xu_{xx}-u^2u_{xxx}=0$5

the integrable Novikov equation is the case $u_t-u_{txx}+4u^2u_x-3uu_xu_{xx}-u^2u_{xxx}=0$6. The linearized $u_t-u_{txx}+4u^2u_x-3uu_xu_{xx}-u^2u_{xxx}=0$7-based operator has full spectrum

$u_t-u_{txx}+4u^2u_x-3uu_xu_{xx}-u^2u_{xxx}=0$8

which yields spectral and linear instability on $u_t-u_{txx}+4u^2u_x-3uu_xu_{xx}-u^2u_{xxx}=0$9 for every $1+1$0. In contrast, on $1+1$1 the peakons are spectrally or linearly stable only in the case $1+1$2. Thus the actual Novikov peakon is $1+1$3-unstable at the linearized level but uniquely distinguished within the $1+1$4-family by $1+1$5-stability (Deng et al., 2024).

5. Generalizations, associated equations, and multicomponent systems

A natural higher-degree continuation of Camassa–Holm and Novikov is the generalized Camassa–Holm–Novikov hierarchy

$1+1$6

equivalently

$1+1$7

This family reduces to Camassa–Holm at $1+1$8 and Novikov at $1+1$9, preserves the $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$00 norm for all $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$01, supports $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$02-peakons for all $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$03, and exhibits wave breaking in mixed-sign $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$04-peakon collisions. Novikov is nevertheless distinguished inside this family by an additional point symmetry

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$05

that is not generic for higher $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$06 (Anco et al., 2014).

The associated Novikov equation

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$07

is connected to the Novikov equation by the reciprocal/potential transformation framework used in Matsuno’s treatment of Novikov. It carries a third-order scalar Lax pair,

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$08

a Bäcklund transformation obtained from the Riccati substitution $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$09, two infinite hierarchies of conservation laws, an infinite hierarchy of continuous symmetries, Hirota bilinearization, and special regular, singular, merging, and splitting soliton sectors. In this sense it serves as a simpler scalar relative of the Novikov, Hirota–Satsuma, and Sawada–Kotera equations rather than as a direct reformulation of the scalar Novikov PDE (Rasin et al., 2019).

Multicomponent extensions preserve the special role of the Novikov exponent. The two-component $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$10-dependent system

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$11

reduces to the scalar Novikov equation under $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$12 when $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$13. That value is exceptional: the point symmetry algebra jumps from dimension $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$14 to dimension $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$15, higher symmetries appear up to the computed order $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$16, a bilinear $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$17-type conserved quantity

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$18

exists only for $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$19, and the system becomes an instance of a broader $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$20-valued zero-curvature hierarchy. Its peakon sector also allows $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$21-peakons with non-constant amplitudes, a phenomenon absent in the scalar one-peakon picture (Ferraioli et al., 2016).

A more recent hyperbolic two-component Novikov equation,

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$22

has a pure multipeakon sector with a non-self-adjoint $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$23 Lax operator. In that setting, the forward spectral problem is converted into a finite matrix eigenvalue problem, global existence of the peakon flow is proved by ideas modeled on Moser’s deformation method, and the inverse problem is solved through three Weyl functions and simultaneous rational approximation involving tensor products and a symmetry constraint. Under positivity assumptions, the spectrum is positive and simple, the eigenvalues are isospectral, and the peakons scatter with ordered asymptotic velocities (Chang et al., 2023).

6. Distinct Novikov-named equations and algebraic-geometric usage

The Novikov–Veselov equation is not a variant of the scalar Novikov equation. It is the classical Veselov–Novikov equation introduced by Novikov and Veselov, a $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$24-dimensional generalization of KdV. At zero energy it takes the form

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$25

and is solved by a nonlinear Fourier transform built from the zero-energy Schrödinger equation

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$26

For smooth compactly supported rotationally symmetric conductivity-type data, the inverse scattering evolution is real-valued, preserves conductivity type, remains free of exceptional points, and satisfies

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$27

with no smallness assumption on the initial data. The same literature also reviews multisolitons, ring solitons, breathers, and the role of exceptional points in the zero-energy scattering theory (Lassas et al., 2011, Croke et al., 2013).

A further, entirely different usage appears in the theory of commuting differential operators. For a rank-one commutative ring generated by

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$28

together with commuting operators $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$29 and $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$30 of orders $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$31 and $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$32, the “Novikov equations” are the total-derivative conditions on the coefficients $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$33 and $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$34 equivalent to

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$35

After reduction to passive form, the nontrivial branch becomes a genus-$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$36 integrable Hamiltonian system with first integrals $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$37, a compatible pair of Poisson brackets, separation variables on a hyperelliptic curve

$m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$38

and Abel–Jacobi quadratures. In this algebraic-geometric setting, “Novikov equations” means stationary commutativity conditions rather than a cubic shallow-water evolution equation (Shabat et al., 20 Jul 2025).

Taken together, these usages justify the plural title. In the narrow PDE sense, the Novikov equation is the integrable cubic Camassa–Holm type equation with momentum form $m_t+u^2m_x+3uu_xm=0,\qquad m=u-u_{xx},$39. In a broader research sense, Novikov equations comprise a network of dissipative, multicomponent, associated, and algebraic-geometric systems, as well as unrelated Novikov-named equations such as Novikov–Veselov. The common thread is not a single canonical hierarchy, but a recurring combination of integrability, nonlocality, and Novikov-associated nomenclature across several adjacent areas of analysis, spectral theory, and algebraic geometry.