Nonlinear Self-Adjointness in PDEs

• Nonlinear self-adjointness is a property of differential equations that enables the derivation of conservation laws by embedding the original PDE into a formal Lagrangian and applying a strategic substitution.
• The framework unifies strict, quasi, and weak self-adjointness, allowing symmetry-based techniques to generate conservation laws even in non-variational settings.
• Applications include the KP equation, Burgers and KdV-type models, where the method yields infinite families of conservation laws and aids in the structural classification of evolution equations.

Searching arXiv for relevant papers on nonlinear self-adjointness and related developments. Nonlinear self-adjointness is a generalized adjointness property for differential equations, introduced to extend symmetry-based conservation-law constructions beyond variational and classically self-adjoint settings. In Ibragimov’s framework, a PDE system $F^\alpha(x,u,u_{(1)},\dots)=0$ is embedded into a formal Lagrangian $\mathcal{L}=v^\alpha F^\alpha$, where $v$ is an auxiliary adjoint variable, and the adjoint system is defined by Euler–Lagrange variation with respect to $u$. The system is nonlinearly self-adjoint if there exists a nontrivial substitution $v=\Phi(x,u,\partial u,\dots)$ such that the adjoint equations become equivalent to the original equations, typically up to a multiplier matrix or differential consequences. This concept includes linear self-adjointness, strict self-adjointness, and quasi self-adjointness as special cases, and it underpins a generalized Noether-type machinery for deriving conservation laws for non-variational PDEs (Ibragimov, 2011).

1. Definition and formal framework

For a general system

$F_a(x,u,u_{(1)},\dots,u_{(s)})=0,$

Ibragimov introduces the formal Lagrangian

$\mathcal{L}=v^aF_a(x,u,u_{(1)},\dots,u_{(s)}),$

where $v=(v^1,\dots,v^m)$ are new dependent variables. The adjoint system is then defined by

$F_a^*=\frac{\delta}{\delta u^a}(v^bF_b)=0,$

with the Euler–Lagrange operator written in the usual alternating-total-derivative form (Ibragimov, 2011).

In the linear case, classical self-adjointness means $L^*=L$, equivalently that the adjoint equation reduces to the original one under $\mathcal{L}=v^\alpha F^\alpha$0. For nonlinear equations this criterion is too restrictive. Ibragimov’s generalized definition requires the existence of a nontrivial substitution

$\mathcal{L}=v^\alpha F^\alpha$1

or, in a more general formulation,

$\mathcal{L}=v^\alpha F^\alpha$2

such that

$\mathcal{L}=v^\alpha F^\alpha$3

This is the defining relation of nonlinear self-adjointness in the point-substitution form (Ibragimov, 2011).

The framework subsumes earlier notions. In the terminology used across the cited literature, one distinguishes strict self-adjointness, quasi self-adjointness, weak self-adjointness, and nonlinear self-adjointness. Strict self-adjointness corresponds to $\mathcal{L}=v^\alpha F^\alpha$4; quasi self-adjointness to $\mathcal{L}=v^\alpha F^\alpha$5; weak self-adjointness to $\mathcal{L}=v^\alpha F^\alpha$6; and the fully nonlinear notion permits dependence on derivatives as well (Freire, 2012). In the KP setting, the system is characterized as quasi self-adjoint because the adjoint variables can be identified directly with the physical fields $\mathcal{L}=v^\alpha F^\alpha$7, $\mathcal{L}=v^\alpha F^\alpha$8, while still being treated as a case of nonlinear self-adjointness in the broader sense (Ibragimov, 2011).

A further refinement was given by Zhang, who formulated nonlinear self-adjointness with differential substitution: $\mathcal{L}=v^\alpha F^\alpha$9 and showed that this formulation is equivalent to the existence of an adjoint symmetry. The determining system for such substitutions is precisely the adjoint-symmetry system on the solution manifold (Zhang, 2014). This establishes a structural equivalence between differential substitutions and adjoint symmetries, and places multipliers inside the same hierarchy as a subclass.

2. Relation to classical self-adjointness and equivalent reformulations

The principal conceptual point is that nonlinear self-adjointness is not an operator identity $v$0, but a compatibility relation between the adjoint system and the original system after a prescribed substitution. In linear theory, the adjoint is symmetric in the sense $v$1, but for nonlinear equations the “double adjoint” is related instead to the linearization of the original equation, so the classical operator picture no longer suffices (Ibragimov, 2011).

A key theorem in Ibragimov’s 2011 paper states that, for a scalar equation, nonlinear self-adjointness is equivalent to the existence of a multiplier $v$2 such that the multiplied equation

$v$3

is strictly self-adjoint, with the relation

$v$4

This theorem converts nonlinear self-adjointness into strict self-adjointness after a suitable nonvanishing factor is introduced (Ibragimov, 2011). The heat equation is the canonical example: although $v$5 is not strictly self-adjoint, it becomes strictly self-adjoint when rewritten as

$v$6

(Ibragimov, 2011).

Another notable consequence is that all linear equations are nonlinearly self-adjoint. The reason is that the adjoint equation of a linear PDE is independent of the original field $v$7, so any nonzero solution of the adjoint equation defines a substitution $v$8, which satisfies the nonlinear self-adjointness definition automatically (Ibragimov, 2011). This result broadens the reach of the method far beyond the classically self-adjoint linear operators.

Zhang’s 2014 analysis adds a further reformulation: the set of differential substitutions for nonlinear self-adjointness coincides with the set of adjoint symmetries, while the set of multipliers is a subset of these substitutions. This identifies nonlinear self-adjointness with the adjoint-symmetry method rather than treating it as a purely separate construction (Zhang, 2014). A plausible implication is that many computational techniques developed for adjoint symmetries can be used directly to test nonlinear self-adjointness.

3. Conservation laws and Ibragimov’s theorem

The operational significance of nonlinear self-adjointness lies in conservation-law construction. Given a symmetry

$v$9

with characteristic

$u$0

Ibragimov’s conservation theorem produces a conserved current from the formal Lagrangian $u$1 (Ibragimov, 2011). In the third- and fifth-order evolution setting, the corresponding conserved vector is written as

$u$2

with terms continuing to the order of the PDE (Freire, 2012). For the KP system, an explicit three-component version is used to derive

$u$3

on solutions (Ibragimov, 2011).

Without self-adjointness, the resulting current depends on both $u$4 and the auxiliary variables $u$5, so it is a conservation law only for the combined original-plus-adjoint system. Nonlinear self-adjointness is exactly the condition that permits elimination of $u$6: once $u$7 is known, the conserved vector becomes a local conservation law of the original PDE alone (Freire, 2011).

This mechanism appears repeatedly in the cited literature. For the KP system

$u$8

the formal Lagrangian is

$u$9

and the adjoint system has the same form as the original one under

$v=\Phi(x,u,\partial u,\dots)$0

Therefore every Lie point symmetry of the KP system yields a conservation law, and the infinite-dimensional Lie algebra generates an infinite family of conservation laws (Ibragimov, 2011).

A related methodological observation appears in the potential KP equation. The potential form admits a genuine Lagrangian,

$v=\Phi(x,u,\partial u,\dots)$1

so Noether’s theorem could be used there. Nonlinear self-adjointness, however, allows one to work directly with the original $v=\Phi(x,u,\partial u,\dots)$2-system rather than passing to potentials (Ibragimov, 2011). This suggests that the method is best regarded not as a replacement for Noether theory, but as a formal-Lagrangian extension of it.

4. Classification results for evolution equations

A substantial part of the early literature is devoted to classifying families of evolution equations that satisfy nonlinear self-adjointness.

Freire’s study of fourth-order evolution equations with time-dependent coefficients considers

$v=\Phi(x,u,\partial u,\dots)$3

and determines the self-adjoint subclass for the substitution $v=\Phi(x,u,\partial u,\dots)$4 (Freire, 2011). The defining relations are expressed as

$v=\Phi(x,u,\partial u,\dots)$5

together with corresponding restrictions on $v=\Phi(x,u,\partial u,\dots)$6 and $v=\Phi(x,u,\partial u,\dots)$7 involving arbitrary functions of $v=\Phi(x,u,\partial u,\dots)$8 (Freire, 2011). In the derivation, the proportionality factor is $v=\Phi(x,u,\partial u,\dots)$9, so

$F_a(x,u,u_{(1)},\dots,u_{(s)})=0,$0

This shows that time dependence is compatible with nonlinear self-adjointness, but only under differential constraints on the coefficient functions.

The same paper also notes a first-order corollary for

$F_a(x,u,u_{(1)},\dots,u_{(s)})=0,$1

which is self-adjoint if and only if

$F_a(x,u,u_{(1)},\dots,u_{(s)})=0,$2

for some smooth function $F_a(x,u,u_{(1)},\dots,u_{(s)})=0,$3 (Freire, 2011).

Freire’s subsequent classification of third- and fifth-order evolution equations with time-dependent coefficients extends this program. For the fifth-order class

$F_a(x,u,u_{(1)},\dots,u_{(s)})=0,$4

the adjoint equation is

$F_a(x,u,u_{(1)},\dots,u_{(s)})=0,$5

and the determining system splits according to whether $F_a(x,u,u_{(1)},\dots,u_{(s)})=0,$6 or $F_a(x,u,u_{(1)},\dots,u_{(s)})=0,$7 (Freire, 2012). The result is four subclasses of third-order and five subclasses of fifth-order nonlinearly self-adjoint equations, with substitutions ranging from constants to quadratic polynomials in $F_a(x,u,u_{(1)},\dots,u_{(s)})=0,$8, time-dependent functions, and expressions such as

$F_a(x,u,u_{(1)},\dots,u_{(s)})=0,$9

The classification includes classes containing Lax, Ito, Kaup–Kupershmidt, Sawada–Kotera, and generalized fifth-order KdV-type equations by suitable parameter choices (Freire, 2012).

These classifications establish a recurrent structural theme: nonlinear self-adjointness imposes algebraic relations among coefficients and often selects substitutions of low differential complexity. This suggests that the property is neither generic nor exceptional, but instead organizes broad integrable and near-integrable families into identifiable subclasses.

5. Representative equations and model applications

Several canonical nonlinear PDEs serve as standard examples.

KP equation

The KP equation is treated in system form

$\mathcal{L}=v^aF_a(x,u,u_{(1)},\dots,u_{(s)}),$0

Its adjoint system is

$\mathcal{L}=v^aF_a(x,u,u_{(1)},\dots,u_{(s)}),$1

which coincides with the original system under $\mathcal{L}=v^aF_a(x,u,u_{(1)},\dots,u_{(s)}),$2 (Ibragimov, 2011). The infinite-dimensional symmetry algebra then yields infinite conservation-law families. Explicit currents are given for three symmetry families $\mathcal{L}=v^aF_a(x,u,u_{(1)},\dots,u_{(s)}),$3, producing compact formulae such as

$\mathcal{L}=v^aF_a(x,u,u_{(1)},\dots,u_{(s)}),$4

for the $\mathcal{L}=v^aF_a(x,u,u_{(1)},\dots,u_{(s)}),$5-family (Ibragimov, 2011).

Burgers- and KdV-type equations

In Freire’s fourth-order note, the lower-order example

$\mathcal{L}=v^aF_a(x,u,u_{(1)},\dots,u_{(s)}),$6

is used to illustrate the method. With the symmetry

$\mathcal{L}=v^aF_a(x,u,u_{(1)},\dots,u_{(s)}),$7

and the substitution $\mathcal{L}=v^aF_a(x,u,u_{(1)},\dots,u_{(s)}),$8, the conserved vector reduces to

$\mathcal{L}=v^aF_a(x,u,u_{(1)},\dots,u_{(s)}),$9

Special cases include the time-dependent inviscid Burgers equation and the KdV equation, with conserved vectors

$v=(v^1,\dots,v^m)$0

respectively (Freire, 2011).

Anisotropic nonlinear heat equation

For the anisotropic heat equation

$v=(v^1,\dots,v^m)$1

the adjoint equation becomes

$v=(v^1,\dots,v^m)$2

Under the assumptions that $v=(v^1,\dots,v^m)$3 are positive, linearly independent, and satisfy $v=(v^1,\dots,v^m)$4, the admissible substitution is independent of $v=(v^1,\dots,v^m)$5 and has the form

$v=(v^1,\dots,v^m)$6

(Ibragimov et al., 2012). This yields a family of conserved vectors, including the basic divergence-form conservation law

$v=(v^1,\dots,v^m)$7

which is simply the equation itself written as local energy balance (Ibragimov et al., 2012).

The same paper shows that source terms usually destroy self-adjointness, but special forms restore it. In two dimensions,

$v=(v^1,\dots,v^m)$8

is generically not nonlinearly self-adjoint, but becomes so when

$v=(v^1,\dots,v^m)$9

leading to trigonometric or exponential substitutions depending on the sign of $F_a^*=\frac{\delta}{\delta u^a}(v^bF_b)=0,$0 (Ibragimov et al., 2012).

Internal-wave equations

The rotating stratified internal-wave system

$F_a^*=\frac{\delta}{\delta u^a}(v^bF_b)=0,$1

is shown to be quasi-self-adjoint under the scaling

$F_a^*=\frac{\delta}{\delta u^a}(v^bF_b)=0,$2

for the adjoint variables $F_a^*=\frac{\delta}{\delta u^a}(v^bF_b)=0,$3 (Ibragimov et al., 2011). This makes the adjoint system identical to the original one and permits construction of local conservation laws, including the energy density

$F_a^*=\frac{\delta}{\delta u^a}(v^bF_b)=0,$4

(Ibragimov et al., 2011).

6. Extensions, refinements, and conceptual boundaries

One line of development concerns approximate nonlinear self-adjointness for perturbed PDEs. Zhang considers systems

$F_a^*=\frac{\delta}{\delta u^a}(v^bF_b)=0,$5

and defines approximate nonlinear self-adjointness through a substitution

$F_a^*=\frac{\delta}{\delta u^a}(v^bF_b)=0,$6

which makes the adjoint equations valid up to $F_a^*=\frac{\delta}{\delta u^a}(v^bF_b)=0,$7 (Zhang, 2011). A key theorem shows that if the unperturbed system is nonlinearly self-adjoint, then the perturbed system is approximately nonlinearly self-adjoint (Zhang, 2011). For scalar equations, approximate nonlinear self-adjointness of the perturbed equation is equivalent to nonlinear self-adjointness of the unperturbed one (Zhang, 2013).

This framework is applied to perturbed nonlinear wave equations of the form

$F_a^*=\frac{\delta}{\delta u^a}(v^bF_b)=0,$8

where the unperturbed equation

$F_a^*=\frac{\delta}{\delta u^a}(v^bF_b)=0,$9

is nonlinearly self-adjoint with

$L^*=L$0

Hence the entire perturbed class is approximately nonlinearly self-adjoint (Zhang, 2013). Approximate conserved vectors are then produced from the known unperturbed conservation laws.

Another line of refinement is Zhang’s equivalence between differential substitutions and adjoint symmetries (Zhang, 2014). This result sharpens a recurrent ambiguity in the early literature: whether nonlinear self-adjointness should be viewed as a special ad hoc substitution method or as an expression of an existing symmetry structure. The equivalence supports the latter interpretation.

A different but conceptually related use of “self-adjointness” appears in the theory of spectral-meromorphic operators and singular integrable systems. Grinevich and Novikov study linear ordinary differential operators with meromorphic coefficients whose eigenfunctions remain meromorphic at poles, and show that symmetric $L^*=L$1-meromorphic operators are self-adjoint with respect to an indefinite inner product on a Krein/Pontryagin-type space (Grinevich et al., 2014). This is not nonlinear self-adjointness in Ibragimov’s sense, but a generalized self-adjointness for linear spectral operators arising from nonlinear integrable systems. The distinction is explicit in the paper: there is no formal Lagrangian or adjoint nonlinear PDE, but there is a generalized spectral self-adjoint structure (Grinevich et al., 2014).

An even more separate usage appears in the Hilbert-space theory of locally self-adjoint extensions of nonlinear smooth operators. There, “nonlinear self-adjointness” refers to nonlinear operators $L^*=L$2 whose derivatives $L^*=L$3 are self-adjoint, with the graph of $L^*=L$4 forming a Lagrangian submanifold in a symplectic Hilbert manifold (Zelenko, 2020). This belongs to nonlinear operator extension theory rather than PDE conservation-law theory, though the shared terminology reflects a common extension of linear adjoint ideas to nonlinear settings (Zelenko, 2020).

These distinctions matter because “nonlinear self-adjointness” has at least three technically different meanings in the cited corpus: Ibragimov’s PDE adjoint-substitution framework, indefinite-metric spectral self-adjointness for singular integrable operators, and local self-adjointness of nonlinear operators in Hilbert spaces. A common misconception is to treat these as interchangeable; they are not.

7. Significance, methodology, and limitations

The principal significance of nonlinear self-adjointness in the PDE sense is methodological. It provides a symmetry-compatible route to local conservation laws for equations that lack a standard Lagrangian. The workflow is consistent across the literature:

1. Write the PDE or system as $L^*=L$5.
2. Form the formal Lagrangian $L^*=L$6.
3. Compute the adjoint equation $L^*=L$7.
4. Seek a substitution $L^*=L$8 such that $L^*=L$9 or lies in the differential ideal generated by $\mathcal{L}=v^\alpha F^\alpha$00.
5. Use a Lie symmetry and Ibragimov’s conservation formula to construct a current.
6. Substitute $\mathcal{L}=v^\alpha F^\alpha$01 to obtain a conservation law in the original variables (Ibragimov, 2011).

The advantages are clear in non-variational settings such as KP, anisotropic heat conduction, Burgers- and KdV-type equations, and perturbed wave models (Ibragimov, 2011, Ibragimov et al., 2012, Freire, 2011, Zhang, 2011). The method also unifies strict, quasi, weak, and differential versions of self-adjointness under a single formal-Lagrangian umbrella (Freire, 2012).

At the same time, the limitations are also explicit in the cited work. The required substitution $\mathcal{L}=v^\alpha F^\alpha$02 may not exist, or may exist only in differential form, which complicates classification and implementation (Ibragimov, 2011). Conservation laws produced by the method can be trivial or equivalent to already known ones, and the resulting families may involve redundancies (Zhang, 2014). Zhang further emphasizes that Ibragimov’s conservation-law formula is not complete in the sense of generating all conservation laws of a given PDE system (Zhang, 2014).

A plausible implication is that nonlinear self-adjointness is best understood as one effective conservation-law technology among several—particularly close to the adjoint-symmetry and multiplier approaches—rather than as a universal characterization of integrability. The equivalence results in (Zhang, 2014) reinforce this interpretation.

In the PDE literature around 2011–2014, nonlinear self-adjointness became a central organizing concept for extending Noether-type constructions to non-variational equations. Its enduring value lies less in a single canonical definition than in a robust formal mechanism: the adjoint system, once related back to the original PDE by an appropriate substitution, turns symmetries into conservation laws in settings where classical variational methods do not directly apply (Ibragimov, 2011).