Generalized PT-symmetric nonlinear Dirac equation: exact solitary waves solutions, stability and conservation laws

Abstract: We derive an exact solitary wave solution for the $\PTb$-symmetric nonlinear Dirac equation with a scalar-scalar interaction. We consider a power-law nonlinearity of the form $|\barΨ\,Ψ|^{k}\,Ψ$ for positive values of $k$. The system's energy is conserved despite the presence of a gain-loss term, which is quantified by the parameter $Λ$. We show that the $\PTb$-transition point is defined by the solution's existence condition and is independent of the nonlinearity exponent $k$. Furthermore, momentum is conserved, although neither the canonical momentum nor the charge is a conserved quantity. A notable result is that the stationary solution, obtained from the continuity equations, exhibits nonzero momentum in its rest frame. We also derive a moving soliton solution, where the gain-loss parameter allows the soliton's velocity to be precisely chosen so that the moving soliton possesses zero momentum. Finally, we establish that the presence of a gain-loss mechanism and higher-order nonlinearity restrict the stability domain of the solutions.

• The paper presents exact solitary wave solutions using a generalized PT-symmetric nonlinear Dirac framework with power-law self-interactions.
• It identifies a universal soliton existence threshold and a novel momentum tuning mechanism enabled by the non-Hermitian gain-loss term.
• Spectral stability analysis reveals marginal stability for k < 2 and critical instabilities for k ≥ 2, setting guidelines for parameter selection.

Generalized $\mathit{PT}$-Symmetric Nonlinear Dirac Equation: Exact Soliton Solutions, Stability, and Conservation Laws

Introduction and Theoretical Framework

This work systematically investigates a generalized class of $\mathit{PT}$-symmetric nonlinear Dirac equations (NLD) in $(1+1)$ dimensions with scalar-scalar self-interactions, focusing on power-law nonlinearities of the type $|\bar{\Psi} \Psi|^k \Psi$ for arbitrary positive $k$. The model extends classical field-theoretic frameworks such as the Thirring and Gross-Neveu equations, serving as a versatile platform for exploring solitary wave phenomena subject to non-Hermitian $\mathit{PT}$-symmetric gain-loss mechanisms. The analysis is motivated by the observation that certain non-Hermitian (but $\mathit{PT}$-symmetric) quantum models can admit real spectra and physically relevant solitons, and aims to construct a robust, real-valued energy functional that circumvents known inconsistencies in previous formulations with complex nonlinear interactions.

Model Formulation and Symmetry Properties

The primary system studied is a two-component Dirac spinor with a gain-loss term proportional to $\gamma^5$ and controlled by a real parameter $\Lambda$. Importantly, the nonlinear self-interaction is formulated as $|\bar{\Psi} \Psi|^k \Psi$ with $\mathit{PT}$0, generalizing the standard Gross-Neveu ($\mathit{PT}$1) case. The spinor dynamics preserve $\mathit{PT}$2 symmetry, Lorentz covariance, and admit a dissipative structure with nontrivial implications for conservation laws.

The equations can be reformulated in terms of new complex fields for symmetric analysis, revealing explicit invariance under $\mathit{PT}$3 operations and Lorentz boosts. This representation exposes how the gain-loss parameter $\mathit{PT}$4 modulates the system and simplifies the identification of stationary and moving soliton solutions.

Exact Soliton Solutions

The principal result is the construction of a family of exact solitary wave solutions (both stationary and boosted) for arbitrary nonlinearity exponent $\mathit{PT}$5. The stationary ansatz yields coupled ODEs for the amplitudes and phases of the spinor components, which are solved exactly to express the field profiles in terms of hyperbolic functions and parameter combinations involving $\mathit{PT}$6 (mass), $\mathit{PT}$7, $\mathit{PT}$8 (internal frequency), and $\mathit{PT}$9. Explicit analytical expressions are obtained for the amplitude profiles, phases, and composite scalar invariants, with exact criteria for the existence and structure (e.g., single- versus double-hump) of soliton profiles as functions of $(1+1)$0 and $(1+1)$1.

A notable finding is that the existence threshold (the $(1+1)$2-transition point) is determined by $(1+1)$3, independent of $(1+1)$4. This universal constraint governs the domain of soliton existence in parameter space.

By applying Lorentz boosts, the moving soliton solutions are constructed, preserving the $(1+1)$5-symmetric structure and yielding explicit forms for charge ($(1+1)$6), energy ($(1+1)$7), and momentum ($(1+1)$8) as functions of velocity and $(1+1)$9. Particularly, the presence of $|\bar{\Psi} \Psi|^k \Psi$0 allows for dynamical control: for any nonzero $|\bar{\Psi} \Psi|^k \Psi$1, the moving soliton velocity can be tuned so that the conserved momentum vanishes, a situation reminiscent of minimal electromagnetic coupling but achieved here without any gauge field, strictly due to the non-Hermitian gain-loss mechanism.

Conservation Laws and Nontrivial Phenomenology

A critical analysis of conservation laws reveals several peculiar and model-specific phenomena:

• Energy and Momentum Conservation: Despite the explicit gain-loss term, the total energy $|\bar{\Psi} \Psi|^k \Psi$2 and an appropriately defined momentum $|\bar{\Psi} \Psi|^k \Psi$3 (distinct from canonical momentum) are strictly conserved for soliton solutions due to the specific structure of the $|\bar{\Psi} \Psi|^k \Psi$4-symmetric nonlinearity. The canonical momentum, as well as the Dirac charge, are generally not conserved when $|\bar{\Psi} \Psi|^k \Psi$5.
• Nonzero Rest-Frame Momentum: The stationary (rest-frame) soliton solutions possess nonzero momentum whenever $|\bar{\Psi} \Psi|^k \Psi$6. This effect is directly traceable to the non-Hermitian mechanism and has no analog in conservative NLD models.
• Charge Oscillations: The Dirac charge is in general nonconserved for nonzero $|\bar{\Psi} \Psi|^k \Psi$7, exhibiting oscillatory behavior reflecting the exchange between the gain and loss channels.
• Relativistic Mass-Energy Relation: The energy and momentum of the moving soliton satisfy a generalized mass-energy relation of the form $|\bar{\Psi} \Psi|^k \Psi$8, where $|\bar{\Psi} \Psi|^k \Psi$9 (rest mass) depends nontrivially on $k$0, $k$1, and $k$2.

Linear Stability and Spectral Analysis

A rigorous spectral stability analysis is conducted based on the linearized evolution operator around soliton solutions. The operator's block-Jordan structure is elucidated, and the precise impact of $k$3 and $k$4 on the spectral properties is tracked.

Key findings include:

• For $k$5, all solitary wave solutions are marginally (spectrally) stable across the entire parameter range permitted by existence conditions.
• For $k$6, there exists a critical frequency $k$7: solitons are stable for $k$8 but undergo spectral instability (appearance of a positive real eigenvalue) for larger $k$9. Both increasing $\mathit{PT}$0 and $\mathit{PT}$1 lower $\mathit{PT}$2, reducing the stability domain.
• The Vakhitov-Kolokolov criterion is analytically recovered in the $\mathit{PT}$3 (conservative) limit; in the general non-Hermitian case, stability must be probed numerically due to intrinsic breaking of symplecticity.
• Internal modes associated with the system's symmetries (e.g., zero-modes from $\mathit{PT}$4 invariance and translation) are identified and their spectral location clarified.

Implications and Outlook

The analytic tractability of the constructed class of exact soliton solutions for arbitrary $\mathit{PT}$5, combined with the explicit implementation of a real energy functional in a $\mathit{PT}$6-symmetric, nonlinear, non-Hermitian field theory, makes this framework a rigorous prototype for exploring dissipative soliton physics in relativistic settings. The finding that energy and a modified momentum remain conserved despite explicit gain and loss mechanisms highlights the power of $\mathit{PT}$7-symmetric engineering to stabilize nonlinear structures outside the field of Hermitian quantum field theory.

Practically, these results suggest robust solitary-wave propagation in $\mathit{PT}$8-symmetric media with controllable nonlinearity exponents, relevant both for relativistic field models and their photonic analogs, e.g., nonlinear waveguides or metamaterials with balanced gain and loss. The ability to tune parameters such that the soliton's physical momentum vanishes at finite velocity is a striking feature with no parallel in conservative systems, potentially implying novel transport and localization effects.

Theoretically, the results raise further avenues for exploring exact solutions in higher-dimensional or more general non-Hermitian Dirac systems, the role of additional interactions, and quantization in the context of $\mathit{PT}$9-symmetric quantum field theories. The rigorous connection between $\mathit{PT}$0-symmetric field theory and dynamical stability of nonlinear excitations may inform both mathematical physics and the design of synthetic quantum systems.

Conclusion

This study provides a comprehensive analytical and numerical characterization of solitary wave solutions in generalized $\mathit{PT}$1-symmetric nonlinear Dirac equations with power-law self-interactions. Soliton existence, conservation laws, and spectral stability are unraveled for arbitrary positive nonlinearity exponents $\mathit{PT}$2, elucidating how non-Hermitian gain-loss mechanisms and higher-order nonlinearities affect the system's stability and dynamical properties. These findings supply a structured blueprint for the stable manipulation of nonlinear excitations in non-Hermitian relativistic field-theoretic and optical models (2604.19277).