Betchov-Type Hydrodynamic Formulation

• Betchov-type hydrodynamic formulation expresses the nonlinear Schrödinger equation using a real density and phase-gradient velocity via the Madelung transformation, yielding continuity and momentum conservation laws.
• It bridges geometric fluid mechanics and financial modeling by structurally linking the Ivancevic option-pricing equation with vortex filament dynamics through analogous conservation laws.
• The approach highlights the interplay between nonlinear pressure, convective flow, and dispersive terms while noting limitations in regions where the density vanishes, affecting soliton and dark-soliton solutions.

Searching arXiv for the primary paper and closely related Betchov-type formulations to ground the article. A Betchov-type hydrodynamic formulation is a conservation-law representation of a nonlinear wave equation in terms of a real density and a real phase-gradient velocity, constructed so that its structure mirrors Betchov’s intrinsic hydrodynamic equations for vortex filament dynamics. In the setting of the Ivancevic option-pricing model, the constant-coefficient Ivancevic equation

$$i\psi_t + \frac{\sigma}{2}\psi_{ss} + \beta|\psi|^2\psi = 0$$

is treated as a cubic nonlinear Schrödinger equation (NLS), and the Madelung variables $\rho=|\psi|^2$ and $u=\sigma\theta_s$ yield a continuity equation together with a momentum-type conservation law. The resulting correspondence is explicitly presented as the scalar Ivancevic–Betchov correspondence and is intended as a structural bridge between nonlinear option-pricing waves and geometric fluid mechanics (Kumar, 11 Jun 2026).

1. NLS origin and the meaning of “Betchov-type”

In Ivancevic’s adaptive-wave option-pricing model, the unknown is a complex-valued option-price wave function $\psi(s,t)$, where $s$ denotes the underlying asset price and $t$ denotes time. Under the constant-coefficient assumptions used in the formulation—constant volatility $\sigma$ and constant adaptive coefficient $\beta$—the governing equation is the cubic NLS

$$i\psi_t + \frac{\sigma}{2}\psi_{ss} + \beta|\psi|^2\psi = 0,$$

or, more generally,

$$i\psi_t + D\psi_{ss} + \beta|\psi|^2\psi = 0,\qquad D\in\mathbb{R}\setminus\{0\},$$

with Ivancevic normalization $\rho=|\psi|^2$0. The sign of $\rho=|\psi|^2$1 distinguishes focusing ($\rho=|\psi|^2$2) from defocusing ($\rho=|\psi|^2$3) behavior, and the constant coefficients are described as crucial because they permit a clean, translation-invariant hydrodynamic formulation with conservation laws closely matching the Betchov structure from vortex filament theory (Kumar, 11 Jun 2026).

The expression “Betchov-type” is used by analogy with the Hasimoto–Betchov description of the vortex filament equation

$\rho=|\psi|^2$4

In that context, Hasimoto’s transform packages curvature $\rho=|\psi|^2$5 and torsion $\rho=|\psi|^2$6 into

$\rho=|\psi|^2$7

and Madelung variables give

$\rho=|\psi|^2$8

Betchov’s intrinsic equations then become the hydrodynamic system

$\rho=|\psi|^2$9

$u=\sigma\theta_s$0

In the Ivancevic setting, a Betchov-type formulation means that the NLS is rewritten in exactly this structural form: a density transported by a continuity equation and a momentum-like variable governed by a conservation law with convective, nonlinear-pressure, and dispersive terms (Kumar, 11 Jun 2026).

This usage aligns with a broader Betchov tradition in hydrodynamics, although different papers instantiate it differently. In homogeneous turbulence, for example, exact algebraic cascade laws have been written in terms of mixed second-order structure functions involving the Lamb vector $u=\sigma\theta_s$1, which has been described as a Betchov-type hydrodynamic formulation of scaling relations (Banerjee et al., 2016). In that broader literature, “Betchov-type” therefore denotes not a single equation, but a family of structurally exact reformulations that emphasize conserved or constrained hydrodynamic variables.

2. Madelung variables and the conservation-law system

The central transformation is the Madelung representation

$u=\sigma\theta_s$2

with

$u=\sigma\theta_s$3

On regions where $u=\sigma\theta_s$4, the Ivancevic NLS is equivalent to the pair

$u=\sigma\theta_s$5

$u=\sigma\theta_s$6

The first is a continuity equation, while the second is a momentum-type conservation law. In compact form,

$u=\sigma\theta_s$7

$u=\sigma\theta_s$8

The paper identifies this system as the Betchov-type conservation system associated with the Ivancevic equation and states that the equivalence is the scalar Ivancevic–Betchov correspondence (Kumar, 11 Jun 2026).

The terms in the flux $u=\sigma\theta_s$9 have a direct structural interpretation. The term $\psi(s,t)$0 is the convective contribution, $\psi(s,t)$1 is the nonlinear pressure, and $\psi(s,t)$2 is the quantum-pressure or dispersion term. For the general constant-coefficient NLS with dispersion $\psi(s,t)$3, the velocity is defined as

$\psi(s,t)$4

and the general momentum flux is

$\psi(s,t)$5

Under Ivancevic scaling $\psi(s,t)$6, this reduces to the preceding formula (Kumar, 11 Jun 2026).

A notable feature of this representation is that it is conservation-law based rather than purely phase-equation based. This places it in a family of Betchov-style formulations that recast nonlinear dynamics into exact structural balances. A plausible implication is that the Ivancevic model can be studied with tools developed for hydrodynamic, geometric, and soliton systems, provided the constant-coefficient NLS interpretation is accepted.

3. Derivation, identities, and domain of validity

Substituting

$\psi(s,t)$7

into the Ivancevic equation and separating real and imaginary parts yields two PDEs. The imaginary part gives

$\psi(s,t)$8

which becomes the continuity equation once $\psi(s,t)$9 is introduced. The real part gives the phase equation

$s$0

Differentiating this in $s$1 produces

$s$2

After multiplication by $s$3 and use of the continuity equation, one obtains the divergence form of the momentum balance (Kumar, 11 Jun 2026).

The key algebraic identity used in the derivation is

$s$4

valid for smooth positive $s$5. This identity converts the dispersive contribution into the logarithmic-density form appearing in the momentum flux. The resulting conservation laws therefore depend not only on the Madelung substitution, but also on the ability to recast the quantum-potential term into divergence form (Kumar, 11 Jun 2026).

The domain of validity is explicitly restricted. The density–velocity representation is pointwise valid only where $s$6. If $s$7 has zeros, then $s$8 blows up and $s$9 becomes singular. In such cases the continuity equation should be kept in current form,

$t$0

and the velocity field should be understood only on non-vacuum regions where $t$1 (Kumar, 11 Jun 2026).

This restriction is conceptually important. The Betchov-type formulation is not a globally regular reparameterization of arbitrary NLS solutions; it is a hydrodynamic rewriting valid on positive-density domains. That feature is typical of Madelung-type formulations more generally, and in the Ivancevic case it governs the admissibility of dark-soliton cores and other zero-amplitude structures.

4. Structural interpretation and relation to geometric fluid mechanics

The formulation is presented as a financial analogue of the Hasimoto–Betchov hydrodynamic system. In the vortex-filament setting, $t$2 is curvature density and $t$3 is torsion-based velocity. In the Ivancevic setting, the corresponding quantities are

$t$4

with $t$5 interpreted as an option-price probability density in the asset-price variable and $t$6 interpreted as a phase-gradient velocity transporting that density along the price axis. The conservation system then plays the role of Betchov’s equations, with probability density replacing curvature density and phase-gradient transport replacing torsion transport (Kumar, 11 Jun 2026).

The coefficient dependence is explicit. For the general NLS, the nonlinear-pressure coefficient is $t$7 and the dispersive coefficient is $t$8; under Ivancevic scaling these become $t$9 and $\sigma$0. The paper states that this makes explicit how volatility and adaptive potential control the fluid-like pressure and dispersive terms. It also notes that although the full Hamiltonian is not written in hydrodynamic variables, the underlying NLS is Hamiltonian with density

$\sigma$1

and that the Betchov-type system inherits a Hamiltonian structure (Kumar, 11 Jun 2026).

The structural analogy should be distinguished from empirical interpretation. The paper emphasizes that the hydrodynamic/Betchov interpretation is structural and model-dependent. Structural means that it follows from the generic Madelung transformation of NLS and has the same mathematical form as the Hasimoto–Betchov formulation. Model-dependent means that the financial reading of $\sigma$2, $\sigma$3, current, and flux depends entirely on accepting the Ivancevic NLS as a model of option-price dynamics; no empirical validation is claimed (Kumar, 11 Jun 2026).

A broader hydrodynamic context sharpens this point. In turbulence theory, Betchov-type relations are exact kinematic or statistical identities constraining energy and helicity transfer mechanisms, such as

$\sigma$4

for homogeneous incompressible turbulence (Banerjee et al., 2016), or the exact helicity-flux relation

$\sigma$5

in homogeneous turbulence (Capocci et al., 2023). The Ivancevic formulation is different in content—it is a scalar NLS hydrodynamization rather than a turbulence identity—but similar in that it reorganizes the dynamics into exact balance laws with explicit structural roles for nonlinear and dispersive terms.

5. Soliton realizations in hydrodynamic variables

The Betchov-type system is illustrated on several classes of Ivancevic-type exact solutions. For the scalar bright soliton

$\sigma$6

the induced hydrodynamic variables are

$\sigma$7

Thus the density is a localized probability bump moving at speed $\sigma$8, while the velocity is constant in space and time. For this profile,

$\sigma$9

so the nonlinear-pressure and dispersive terms cancel exactly in the momentum flux (Kumar, 11 Jun 2026).

For the scalar dark soliton

$\beta$0

with $\beta$1, one has

$\beta$2

This is a density dip propagating on a nonzero background. However, $\beta$3 vanishes at the soliton core, so $\beta$4 is singular সেখানে; accordingly, the hydrodynamic formulation applies only away from the zero set (Kumar, 11 Jun 2026).

The coupled volatility–option model can be written as the two-component Manakov system

$\beta$5

Under a fixed-polarization ansatz

$\beta$6

the natural hydrodynamic density is the total intensity

$\beta$7

and the velocity is

$\beta$8

Corollary 2.2 states that $\beta$9 satisfy the same Betchov-type system, now with flux

$$i\psi_t + \frac{\sigma}{2}\psi_{ss} + \beta|\psi|^2\psi = 0,$$0

For the explicit two-component bright soliton

$$i\psi_t + \frac{\sigma}{2}\psi_{ss} + \beta|\psi|^2\psi = 0,$$1

one finds

$$i\psi_t + \frac{\sigma}{2}\psi_{ss} + \beta|\psi|^2\psi = 0,$$2

again giving a localized density packet moving at constant speed (Kumar, 11 Jun 2026).

The formulation also extends to Hirota $$i\psi_t + \frac{\sigma}{2}\psi_{ss} + \beta|\psi|^2\psi = 0,$$3-bright-soliton solutions. If

$$i\psi_t + \frac{\sigma}{2}\psi_{ss} + \beta|\psi|^2\psi = 0,$$4

then

$$i\psi_t + \frac{\sigma}{2}\psi_{ss} + \beta|\psi|^2\psi = 0,$$5

and

$$i\psi_t + \frac{\sigma}{2}\psi_{ss} + \beta|\psi|^2\psi = 0,$$6

These satisfy the Betchov-type continuity and momentum equations on regions where $$i\psi_t + \frac{\sigma}{2}\psi_{ss} + \beta|\psi|^2\psi = 0,$$7. For $$i\psi_t + \frac{\sigma}{2}\psi_{ss} + \beta|\psi|^2\psi = 0,$$8, the density exhibits two interacting localized peaks, and the current $$i\psi_t + \frac{\sigma}{2}\psi_{ss} + \beta|\psi|^2\psi = 0,$$9 concentrates mainly along the moving peak; the interaction is nonlinear rather than a linear superposition (Kumar, 11 Jun 2026).

6. Limitations, conceptual scope, and related Betchov formulations

The formulation is explicitly limited to constant-coefficient NLS. Time- or space-dependent coefficients and fully stochastic volatility fields are not analyzed hydrodynamically in the paper, except for the fixed-polarization Manakov reduction. The extension to full vector NLS with independent component phases is described as nontrivial and left for future work. The singular behavior at zeros of $$i\psi_t + D\psi_{ss} + \beta|\psi|^2\psi = 0,\qquad D\in\mathbb{R}\setminus\{0\},$$0 is a second fundamental limitation, since the pointwise velocity and logarithmic-density terms cease to be regular there (Kumar, 11 Jun 2026).

The conceptual scope is correspondingly narrow and precise. The paper does not claim empirical validation of the Ivancevic model, and it does not claim that the hydrodynamic variables are classical prices or returns. The velocity $$i\psi_t + D\psi_{ss} + \beta|\psi|^2\psi = 0,\qquad D\in\mathbb{R}\setminus\{0\},$$1 is specifically characterized as a phase-gradient transport variable, and the meaning of $$i\psi_t + D\psi_{ss} + \beta|\psi|^2\psi = 0,\qquad D\in\mathbb{R}\setminus\{0\},$$2 as a probability density belongs to Ivancevic’s interpretation of $$i\psi_t + D\psi_{ss} + \beta|\psi|^2\psi = 0,\qquad D\in\mathbb{R}\setminus\{0\},$$3, not to a model-independent theorem (Kumar, 11 Jun 2026).

At the same time, the formulation is presented as a bridge between nonlinear mathematical finance and geometric fluid mechanics. Suggested directions include a forced model with potential $$i\psi_t + D\psi_{ss} + \beta|\psi|^2\psi = 0,\qquad D\in\mathbb{R}\setminus\{0\},$$4,

$$i\psi_t + D\psi_{ss} + \beta|\psi|^2\psi = 0,\qquad D\in\mathbb{R}\setminus\{0\},$$5

for which the continuity equation is unchanged while the momentum law acquires forcing,

$$i\psi_t + D\psi_{ss} + \beta|\psi|^2\psi = 0,\qquad D\in\mathbb{R}\setminus\{0\},$$6

Other suggested directions are comparison with Black–Scholes hydrodynamics, calibration of soliton densities to empirical option-price or implied-volatility distributions, and a fuller hydrodynamic description of stochastic-volatility or vector-wave systems (Kumar, 11 Jun 2026).

In the broader Betchov literature, the phrase “Betchov-type” carries several distinct but related meanings. In incompressible isotropic turbulence, the classical Betchov homogeneity constraints

$$i\psi_t + D\psi_{ss} + \beta|\psi|^2\psi = 0,\qquad D\in\mathbb{R}\setminus\{0\},$$7

have been shown to be the only independent homogeneity constraints on single-point scalar functions of the velocity gradient $$i\psi_t + D\psi_{ss} + \beta|\psi|^2\psi = 0,\qquad D\in\mathbb{R}\setminus\{0\},$$8 (Carbone et al., 2021). In helicity-flux theory, Betchov-like relations constrain the mean contributions of physically distinct subfluxes (Capocci et al., 2023). In geometric curve-flow theory, the Betchov–Da Rios equation in $$i\psi_t + D\psi_{ss} + \beta|\psi|^2\psi = 0,\qquad D\in\mathbb{R}\setminus\{0\},$$9,

$\rho=|\psi|^2$00

has been used to build soliton surfaces whose geometric invariants classify flat, minimal, semi-umbilic, and Wintgen ideal evolutions (Kazan et al., 2024). The Ivancevic–Betchov correspondence belongs to this larger family of structural analogies, but its specific contribution is the transfer of a vortex-filament-style hydrodynamic decomposition into a constant-coefficient option-pricing NLS.

Taken together, these works indicate that a Betchov-type hydrodynamic formulation is best understood as an exact structural recoding of a nonlinear system into conservation laws or invariant relations centered on density, transport, and flux variables. In the Ivancevic case, that recoding yields a continuity equation and a momentum-type conservation law whose formal identity with the Hasimoto–Betchov pattern is the article’s central result (Kumar, 11 Jun 2026).