Tsunami Solitons Emerging from Superconducting Gap

We propose a classical integrable system exhibiting tsunami-like solitons with a rocky-desert-like disordered stationary background. One of the Lax operators describing this system is interpretable as a Bogoliubov–de Gennes Hamiltonian in parity-mixed superconductors. The family of integrable equations is generated from this seed operator using Krichever’s method, whose pure s-wave limit includes the coupled Schrödinger–Boussinesq hierarchy applied to plasma physics. A linearly unstable finite background with a superconducting gap supports the tsunami-soliton solution, where the propagation of the step structure turns back at a certain moment, accompanied by oscillation on the opposite side. In addition, the equation allows inhomogeneous stationary solutions with an arbitrary number of bumps at arbitrary positions, which we term the Korteweg–de Vries (KdV) rocks. In the Zakharov–Shabat scheme, the tsunami solitons are created from the Bogoliubov quasiparticles in the energy gap and the KdV rocks from normal electrons/holes. The unexpected large space of stationary solutions originates from the non-coprime Lax pair and the multivalued Baker–Akhiezer functions on the Riemann surface, formulated in terms of higher-rank holomorphic bundles by Krichever and Novikov. Furthermore, the concept of isodispersive phases is introduced to characterize quasiperiodic multi-tsunami backgrounds and consider their classification.

Our system is described by the Lax pair

iL̂t = [L̂, M̂],

L̂ = −∂2x σ3 + (∂ x, ξσ+ − ησ−)− uσ3 + v12 + qσ+ + rσ−, (2)

(3)

where 12 and σ1,2,3 are the 2 × 2 identity and Pauli matrices,

respectively, σ± = σ1 ±iσ

2, and the anti-commutator is denoted by the bracket. Here, we restrict ourselves to the case u = u∗, v = v∗, r = q∗, and η = ξ∗, where L̂ and M̂ become self-adjoint. The resultant equations are given by

iut = 2(q∗ ξ x − qξ∗x),

ivt = 2(ξξ∗x − ξ∗ ξ x) x,

iqt = −2vq − 2ξu x − 4ξ x u + 4ξ(ξ2) x − ξ xxx.

The physical/mathematical backgrounds that yielded Eqs. (1)(6) are described below. First, L̂ [Eq. (2)] can be viewed as the Bogoliubov–de Gennes (BdG) Hamiltonian in parity-mixed superconductors (SCs), and the physical interpretations of the coefficient functions by the Hartree–Fock (HF) mean fields and the Cooper pairs (gap functions) are summarized in Table I. The mean-field theory with omitted HF fields is formulated in Ref. 17. The parity-mixed SCs appear in noncentrosymmetric materials and the surface of topological SCs. Figure 1 shows a schematic of the dispersion relation with/without an s-wave gap q = q0. When the chemical potential µ is large, Andreev’s dispersion linearization around the Fermi points works well. Physical systems equivalent to this one appear in diverse fields, including the Peierls problem in conducting polymers and the Gross–Neveu models. On the other hand, the treatment without dispersion linearization has become important in (i) the BCS-BEC crossover where the BEC side corresponds to a small µ, (ii) eliminating cutoff-dependence, and (iii) comparing with the quantum many-body counterpart described by the Gaudin–Yang model. In particular, (iii) will play a key role in the construction of the fermionic many-body quantum solitons. Next, we explain how to determine M̂ in Eq. (3). Accord-

M̂ = (−∂2x − u + 2ξη)12 + vσ3 + 2(ξ x σ+ + η x σ−),

iξt = −2vξ + q x,

where K(x, y) is defined as K(x, y) = −W(x)[1n + G(x)]−1 W(y)†. Then, the new solution with n solitons added from the known one is given by

qnew = q − 2K12 K22 + (K x − Ky)12 + 2ξ(K11 −K22),

unew = u − 2ξ2 + 4[ln det(1n + G)] xx,

new = v + tr[(K x + Ky)σ3].

where K = K(x, x), K x = [∂ x K(x, y)]y=x, and Ky = [∂y K(x, y)]y=x. Let us apply the above general formula to the uniform state with an s-wave gap (u, v, q, ξ) = (u0, 0, q0, 0), u0, q0 > 0. The eigenfunction with the real eigenvalue ǫ is given by

p w(x, t, ǫ, k, ϕ) = 2 Re k w0 (ǫ, k)ekx−ω(k)t+iϕ + c.c.,

where w0 (ǫ, k) = √12 [(ǫ+q0)2 +ω(k)2]1/2

(7)

vϵ

ξnew = ξ + K12, and k = k(ǫ) ≔ [(ǫ2 − q02)1/2 − u0]1/2. Then, the seed solution is generally given by wi = w(x − xi, t − ti, ǫi, ki, ϕi), ki = k(ǫi), possessing four real parameters xi, ti, ϕi, and ǫi. The solii)ton velocity becomes Vi = ReReω(k

ki. Below, we determine the two types of one-soliton solutions by the choice of ǫ1 (Fig. 1). The first type is determined from the superconducting gap ǫ < q0 (Fig. 1), which we call the tsunami soliton, whose behavior is shown in Fig. 2. In this solution, we observe the propagation of the step structure, which suddenly turns back at (x, t) = (x1, t1). At the moment of turning back, the opposite side of the soliton experiences an oscillation. Taking various limits of (x1, t1), we can obtain different solutions. If both x1, t1 are set to infinity, we obtain a solution without the propagation turning back. If we fix t1 and take x1 → −∞, we obtain a solution with sudden oscillation occurring at t = t1, similar to the AB, but it now emerges from the self-adjoint Lax pair. We note the unpredictability of the moment of turning back from the observation data—it depends on a subtle difference in the initial condition and is difficult to detect. Unlike rogue waves,13) the background with finite u0, q0 is linearly unstable against a short-wavelength pertur√ ibration; the onset of instability appears at k = 2kF = 2 u0 with a Fermi wavenumber kF, which explains the oscillation period occurring at t = 0 in Fig. 2. The 2kF-oscillation around a local defect is called the Friedel oscillation in condensed-matter context. Figure 2 also recalls the soliton resonance phenomena in (2+1)-dimensional integrable systems where Yshaped and more divaricate structures of line solitons are formed, using large degrees of freedom, including functional parameters. These structures are constructed by a linear combination of multiple seed solutions, but in the present case, the maximum number of seeds is two [Eq. (11)] owing to the limitation of (1+1)-dimensional systems. An oscillation profile similar to Fig. 2 can be found in two-layer fluids. In Fig. 2, we plot u − 2ξ2 instead of u itself, because it is a conserved density, and it is convenient to detect the front of the tsunami soliton. The conservation laws are derived as follows. Let Φ be a 2 × 2 matrix satisfying (L̂ − λ)Φ = (M̂ +i∂t)Φ = 0. Defining Ψ = ∂Φx Φ, we have a zero-curvature ing to Krichever,58) we can find a family of differential operators M̂n,±, n = 0, 1, 2,... commuting with L̂, and obtain the sequence of ordinary differential equations determined by [L̂, M̂] = 0. If commutators are proportional to the time derivative L̂t, we obtain the hierarchy of classical integrable systems. Here, we use M̂ = M̂2,+ in Eq. (3), because it provides the lowest-order equation supporting the stationary background with a superconducting gap shown in Fig. 1. When v = ξ = 0, the above hierarchy reduces to the one including the coupled Schrödinger–Boussinesq equation, which has been applied to plasma physics. Therefore, while the time evolution governed by M̂ is different from those in condensed-matter systems mentioned above, whose time evolution is based on the self-consistent determination of potentials and eigenfunctions or effective field theories, the present equations (4)-(6) are expected to be derived by applying the reductive perturbation method to multicomponent plasmas or fluids. Now, let us discuss the construction of the concrete multisoliton solutions using the Zakharov–Shabat (ZS) scheme. We extract the minimal formulas for the higher-order ZS scheme: Let wi, i = 1,..., n be an eigenfunction of the Lax pair (L̂ − ǫi)wi = (M̂ + i∂t)wi = 0 with asymptotic behavior kwi k → 0 at x → −∞ (resp. ∞) and are called the seed solutions. Writing their array as W = (w1, w2,..., wn), we introduce an n × n Gram matrix

(a)

(b)

hierarchy, only odd-order differential operators generate the higher-order KdV equations;82, 83) therefore, there is no counterpart for even-order M̂n,±. Consider the stationary problem for Eq. (1), i.e., [L̂, M̂] = 0. By the Burchnall–Chaundy lemma,58, 83–87) the commuting differential operators satisfy a polynomial relation P(L̂, M̂) = 0, which defines an algebraic curve (or Riemann surface). Writing the simultaneous eigenvalue problem

(c)

0.5

0.5

0.5

-20

t  ±6

x

-1

-1

t  ±2

x-1-1 t  ±4 x-3 t  ±6 x-3 t ) (d)

P(λ, ω) = (λ2 − ω2 − a)2 + bλ + cω + d = 0,

where a, b, c, d are expressed by the rational functions of the constants JnR,I ’s in Eq. (12). If the values of a, b, c, d are generic, Eq. (14) represents the genus-one elliptic curve. The stationary solutions are then divided into two classes, which we call regular and irregular below. The regular solutions are described as follows. Assume that J1R, 0 and set α = J1I /J1R and β3 = J3R /J1R. The stationary so2 lution is then given by q = −αξ x, u ± v = 1±α

/ (β3 − 2ξ),2

and (1 − α)ξ xx + 4(β3 − 2ξ)ξ = 0. The last equation is the stationary NLS equation, whose solution is given by theta functions. The algebraic curve (14) for regular solutions has genus g = 1, except for the elementary limit. On the other hand, the irregular solutions emerge when J1R = J1I = 0. We have general solutions ξ∗ = c1 ξ, q∗ = (x - 20/2) ξ2 + u − 2ξ2 - u0 x v, and the irregular solutions have two arbitrary real-valued (up to overall phase) functions ξ and q, which include the multi-KdV-rock states as a particular solution. The second type of solitons arises from the bound states of normal electrons/holes (Fig. 1), which has zero velocity. We call these solitons the Korteweg–de Vries (KdV) rocks, because L̂ with q = ξ = 0 reduces to the double Schrödinger operator, and hence, the time evolution based on the third-order M̂3,± instead of M̂2,+ becomes the famous KdV equation. The hybrid multisoliton solution with coexisting tsunami solitons and KdV rocks is shown in Fig. 3, like a flood in the desert. The lack of time dependence for the KdV rocks does not imply that this solution is boring—the fact that we can add an arbitrary number of stationary KdV rocks at any position is curious, because it allows the present system (4)-(6) to have an anomalously large stationary-solution space. This should be compared with the typical known classical integrable systems, where general stationary solutions are given by elliptic functions and the entire solution space has at most a finite number of adjustable constants. As we will see below, this anomaly can happen because the orders of L̂ and M̂ [Eqs. (2) and (3)] are not coprime. The s-wave uniform state (u, v, q, ξ) = (u0, 0, q0, 0), which has been mainly considered in this study, belongs to the irregular solution. The dispersion relations for φ ∝ eikx become λ2 = (k2 − u0)2 + q02, ω = k2 − u0. Eliminating k, we find λ2 − ω2 - q0 squared = 0, corresponding to Eq. (15), which is easily parametrized as λ = q0(s + s−1), ω = q0(s − s−1). However, the wavenumber k = q0(s - s−1) + u 0 cannot be expressed as a rational function of the parameter s; thus, the BA function φ becomes a double-valued function on the Riemann surface owing to the square root, implying the existence of rank-2 solutions. Finally, we introduce the concept of isodispersive phases to characterize the oscillating region of the tsunami soliton (Fig. 2). For a given reflectionless potential of differential operator L̂, we define the backgrounds of the left and right sides far from the potential (x → ±∞) as isodispersive. This term is used because both states share the same dispersion relation Rxλ(k) via the Jost solution φnew (x) = φ(x) + −∞ K(x, y)φ(y)dx. For spatially uniform backgrounds, isodispersive phases are typically connected by trivial gauge transformations; for example, in the integrable spinor Bose condensates with finite density, the backgrounds before and after the soliton passes are both polar phases at different angles, connected by the U(1) ⊗ S O(3) group. The same phases with various angles form the order parameter manifold, whose homotopy groups classify the topological defects. On the other hand, the tsunami soliton near the oscillation time t ≃ 0 (Fig. 2) shows a reflectionless potential such that the left side is uniform but the right side is oscillating. The multiple tsunami-soliton state (Fig. 3) can support more complicated quasiperiodic backgrounds. In these cases, the set of isodispersive phases may not be compact, and the physical interpretation of the transformation group is unclear. While the uniform phases are classified by values ρI1 = u − 2ξ2 and 2/2 + 2uξ2 − 2ξ 4, quasiperiodic isodispersive phases might require higher-order ρR,I n, which remains an open problem. In summary, we presented the tsunami-soliton and stationary KdV-rock solutions in a classical integrable equation arising from the BdG operator in parity-mixed SCs. The family of Novikov equations constituting a hierarchy was generated using Krichever’s method. The tsunami solitons provide not only the turning-back dynamics but also the characterization problem of isodispersive quasiperiodic states, which might be used for new types of momentum-dependent topological defects and exotic Josephson junctions with transparent scattering properties. The irregular solutions, which are allowed by the non-coprime Lax pair and include the multi KdV rocks, will open up a new application of classical integrable models to physical systems with disordered backgrounds. The unified treatment of the entire hierarchy, the derivation from multicomponent plasmas and fluids by reductive perturbation, and the application to the quadratic-dispersion BdG systems toward full many-body treatment are all left as future tasks.

The data that support the findings of this article are openly available.

1. V. E. Zakharov and A. A. Gelash, Nonlinear stage of modulation instability, Phys. Rev. Lett. 111, 054101 (2013).

2. N. Akhmediev, Waves that Appear From Nowhere: Complex Rogue Wave Structures and Their Elementary Particles, Frontiers in Physics 8, 612318 (2021).

3. B. Guo, L. Ling, and Q. P. Liu, Nonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutions, Phys. Rev. E 85, 026607 (2012).

4. E. A. Kuznetsov, Solitons in a parametrically unstable plasma, Sov. Phys.-Dokl.(Engl.) 22:9 (1977).

5. T. Kawata and H. Inoue, Inverse Scattering Method for the Nonlinear Evolution Equations under Nonvanishing Conditions, J. Phys. Soc. Jpn. 44, 1722 (1978).

6. Y.-C. Ma, The Perturbed Plane-Wave Solutions of the Cubic Schrödinger Equation, Stud. Appl. Math. 60, 43 (1979).

7. N. N. Akhmediev and V. I. Korneev, Modulation instability and peri4odic solutions of the nonlinear Schrödinger equation, Theoretical and Mathematical Physics 69, 1089 (1986).

8. J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation, Opt. Express 17:21497 (2009).

9. D. H. Peregrine and V. E Zakharov, Integrable turbulence and formation of rogue waves, Nonlinearity 28:2791 (2015).

10. Y. Ohta and J. Yang, Dynamics of rogue waves in the Davey–Stewartson II equation, J. Phys. A: Math.: Theor. 46:105202 (2013).

11. D. E. Pelinovsky and R. E. White, Localized structures on librational and rotational travelling waves in the sine-gordon equation, Proc. R. Soc. A: Math., Phys.-Eng.-Sci.: 476:20200490 (2020).

12. G. Mu and Z. Qin, Rogue Waves for the Coupled Schrödinger–Boussinesq Equation and the Coupled Higgs Equation, J. Phys. Soc. Jpn. 81:084001 (2012).

13. D. S. Agafontsev and V. E Zakharov, Integrable turbulence and formation of rogue waves, Nonlinearity 28:2791 (2015).

14. M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V Schweikhard, Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics, Phys Rev A 74:023623 (2006).

15. T. Kawata and H. Inoue, Inverse Scattering Method for the Nonlinear Evolution Equations under Nonvanishing Conditions, J. Phys. Soc. Jpn. 44:1722 (1978).

16. Y.-C. Ma, The Perturbed Plane-Wave Solutions of the Cubic Schrödinger Equation, Stud. Appl. Math.: 60:43 (1979).

17. N. N. Akhmediev and V. I. Korneev, Modulation instability and periodic solutions of the nonlinear Schrödinger equation, Theoretical and Mathematical Physics 69:1089 (1986).

18. J. M. Dudley, G. Genty, F Dias, B Kibler, and N Akhmediev, Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation Opt. Express 17:21497 (2009).

19. D. S. Agafontsev and V E Zakharov, Integrable turbulence and formation of rogue waves, Nonlinearity 28:2791 (2015).

20. Y Ohta and J Yang, Dynamics of rogue waves in the Davey–Stewartson II equation, J. Phys. A: Math.: Theor. 46:105202 (2013).

The family of Novikov equations constituting a hierarchy was generated using Krichever’s method. The tsunami solitons provide not only the turning-back dynamics but also the characterization problem of isodispersive quasiperiodic states, which might be used for new types of momentum-dependent topological defects and exotic Josephson junctions with transparent scattering properties. The irregular solutions, which are allowed by the non-coprime Lax pair and include the multi KdV rocks, will open up a new application of classical integrable models to physical systems with disordered backgrounds. The unified treatment of the entire hierarchy, the derivation from multicomponent plasmas and fluids by reductive perturbation, and the application to the quadratic-dispersion BdG systems toward full many-body treatment are all left as future tasks.

1. V. E. Zakharov and A. A. Gelash, Nonlinear stage of modulation instability, Phys Rev Lett 111, 054101 (2013).

2. N. Akhmediev, Waves that Appear From Nowhere: Complex Rogue Wave Structures and Their Elementary Particles Frontiers in Physics 8, 612318 (2021).

3. B. Guo, L. Ling, and Q. P. Liu, Nonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutions Phys Rev E 85, 026607 (2012).

4. E. A. Kuznetsov, Solitons in a parametrically unstable plasma Sov Phys.-Dokl.(Engl.) 22:9 (1977).

5. T. Kawata and H. Inoue, Inverse Scattering Method for the Nonlinear Evolution Equations under Nonvanishing Conditions J. Phys Soc. Jpn. 44, 1722 (1978).

6. Y.-C. Ma, The Perturbed Plane-Wave Solutions of the Cubic Schrödinger Equation Stud. Appl. Math.: 60, 43 (1979).

7. N. N. Akhmediev and V I Korneev, Modulation instability and periodic solutions of the nonlinear Schrödinger equation Theoretical and Mathematical Physics 69, 1089 (1986).

8. J. M. Dudley, G Genty, F Dias B Kibler, and N Akhmediev Modulation instability Akhmediev Breathers and continuous wave supercontinuum generation Opt Express 17:21497 (2009).

9. D S Agafontsev and V E Zakharov Integrable turbulence and formation of rogue waves Nonlinearity 28:2791 (2015).

10. Y Ohta and J Yang Dynamics of rogue waves in the Davey–Stewartson II equation J. Phys. A: Math. Theor. 46:105202 (2013).

The summary is as follows: We propose a classical integrable system exhibiting tsunami-like solitons with a rocky-desert-like disordered stationary background, emerging from the study of superconductivity, specifically interpreted as a Bogoliubov–de Gennes Hamiltonian in parity-mixed superconductors. The family of integrable equations is generated from this seed operator using Krichever’s method. A linearly unstable finite background with a superconducting gap supports the tsunami-soliton solution, characterized by the propagation of the step structure turning back at a certain moment accompanied by oscillation on the opposite side. In addition, inhomogeneous stationary solutions with an arbitrary number of bumps are termed Korteweg–de Vries (KdV) rocks. Tsunami solitons arise from Bogoliubov quasiparticles in the energy gap in the Zakharov–Shabat scheme, while KdV rocks arise from normal electrons/holes. The unexpected large space of stationary solutions originates from the non-coprime Lax pair and multivalued Baker–Akhiezer functions on the Riemann surface, formulated using higher-rank holomorphic bundles by Krichever and Novikov. The concept of isodispersive phases is introduced to characterize quasiperiodic multi-tsunami backgrounds. The system's Lax pair is given by L̂ = −∂2x σ3 + (∂ x, ξσ+ − ησ−)− uσ3 + v12 + qσ+ + rσ−, and the resultant equations are derived under the restriction that L̂ and M̂ become self-adjoint. The study explores two types of one-soliton solutions based on the choice of ǫ1: the first type is a tsunami soliton determined by ǫ < q0, exhibiting step structure turning back with oscillation on the opposite side, while its background is linearly unstable against a short-wavelength perturbation at k = 2kF, leading to Friedel oscillation. The second type are KdV rocks, which arise from bound states of normal electrons/holes and have zero velocity. A hybrid multisoliton solution with coexisting tsunami solitons and KdV rocks is shown. The irregular solutions emerge when the orders of L̂ and M̂ are not coprime, allowing for an anomalously large stationary-solution space, including multi KdV rocks. The s-wave uniform state belongs to the irregular solution, where the Baker–Akhiezer function becomes double-valued due to a square root in its dispersion relation. Isodispersive phases characterize the oscillating region of the tsunami soliton. In summary, we presented tsunami-soliton and stationary KdV-rock solutions in a classical integrable equation arising from BdG operator in parity-mixed SCs, providing insights for new types of momentum-dependent topological defects and exotic Josephson junctions with transparent scattering properties. The study leaves future tasks such as the unified treatment of the entire hierarchy, derivation from multicomponent plasmas and fluids by reductive perturbation, and application to quadratic-dispersion BdG systems toward full many-body treatment.

1. V. E. Zakharov and A. A. Gelash, Nonlinear stage of modulation instability, Phys Rev Lett 111, 054101 (2013).

2. N Akhmediev, Waves that Appear From Nowhere: Complex Rogue Wave Structures Their Elementary Particles Frontiers in Physics 8, 612318 (2021).

3. B Guo, L Ling, and Q P Liu Nonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutions Phys Rev E 85, 026607 (2012).

4. E A Kuznetsov Solitons in a parametrically unstable plasma Sov Phys.-Dokl.(Engl.) 22:9 (1977).

5 T Kawata and H Inoue Inverse Scattering Method for the Nonlinear Evolution Equations under Nonvanishing Conditions J. Phys Soc. Jpn. 44, 1722 (1978)

6 Y-C Ma The Perturbed Plane-Wave Solutions of the Cubic Schrödinger Equation Stud Appl. Math.: 60, 43 (1979)

7 N N Akhmediev and V I Korneev Modulation instability and periodic solutions of the nonlinear Schrödinger equation Theoretical and Mathematical Physics 69, 1089 (1986)

8 J M Dudley G Genty F Dias B Kibler, and N Akhmediev Modulation instability Akhmediev Breathers and continuous wave supercontinuum generation Opt Express 17:21497 (2009)

9 D S Agafontsev and V E Zakharov Integrable turbulence and formation of rogue waves Nonlinearity 28:2791 (2015)

10 Y Ohta and J Yang Dynamics of rogue waves in the Davey–Stewartson II equation J. Phys. A: Math. Theor. 46:105202 (2013)

1. V E Zakharov and A A Gelash, Nonlinear stage of modulation instability Phys Rev Lett 111, 054101 (2013)

2. N Akhmediev Waves that Appear From Nowhere Complex Rogue Wave Structures Their Elementary Particles Frontiers in Physics 8, 612318 (2021)

3 B Guo L Ling, and Q P Liu Nonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutions Phys Rev E 85, 026607 (2012)

4 E A Kuznetsov Solitons in a parametrically unstable plasma Sov Phys.-Dokl.(Engl.) 22:9 (1977)

5 T Kawata and H Inoue Inverse Scattering Method for the Nonlinear Evolution Equations under Nonvanishing Conditions J. Phys Soc. Jpn. 44, 1722 (1978)

6 Y-C Ma The Perturbed Plane-Wave Solutions of the Cubic Schrödinger Equation Stud Appl. Math.: 60, 43 (1979)

7 N N Akhmediev and V I Korneev Modulation instability and periodic solutions of the nonlinear Schrödinger equation Theoretical and Mathematical Physics 69, 1089 (1986)

8 J M Dudley G Genty F Dias B Kibler, and N Akhmediev Modulation instability Akhmediev Breathers and continuous wave supercontinuum generation Opt Express 17:21497 (2009)

9 D S Agafontsev and V E Zakharov Integrable turbulence and formation of rogue waves Nonlinearity 28:2791 (2015)

10 Y Ohta and J Yang Dynamics of rogue waves in the Davey–Stewartson II equation J. Phys. A: Math. Theor. 46:105202 (2013)

1. V E Zakharov and A A Gelash Nonlinear stage of modulation instability Phys Rev Lett 111, 054101 (2013)

2. N Akhmediev Waves that Appear From Nowhere Complex Rogue Wave Structures Their Elementary Particles Frontiers in Physics 8, 612318 (2021)

3 B Guo L Ling, and Q P Liu Nonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutions Phys Rev E 85, 026607 (2012)

4 E A Kuznetsov Solitons in a parametrically unstable plasma Sov Phys.-Dokl.(Engl.) 22:9 (1977)

5 T Kawata and H Inoue Inverse Scattering Method for the Nonlinear Evolution Equations under Nonvanishing Conditions J. Phys Soc. Jpn. 44, 1722 (1978)

6 Y-C Ma The Perturbed Plane-Wave Solutions of the Cubic Schrödinger Equation Stud Appl. Math.: 60, 43 (1979)

7 N N Akhmediev and V I Korneev Modulation instability and periodic solutions of the nonlinear Schrödinger equation Theoretical and Mathematical Physics 69, 1089 (1986)

8 J M Dudley G Genty F Dias B Kibler, and N Akhmediev Modulation instability Akhmediev Breathers and continuous wave supercontinuum generation Opt Express 17:21497 (2009)

9 D S Agafontsev and V E Zakharov Integrable turbulence and formation of rogue waves Nonlinearity 28:2791 (2015)

10 Y Ohta and J Yang Dynamics of rogue waves in the Davey–Stewartson II equation J. Phys. A: Math. Theor. 46:105202 (2013)

1. V E Zakharov and A A Gelash Nonlinear stage of modulation instability Phys Rev Lett 111, 054101 (2013)

2. N Akhmediev Waves that Appear From Nowhere Complex Rogue Wave Structures Their Elementary Particles Frontiers in Physics 8, 612318 (2021)

3 B Guo L Ling and Q P Liu Nonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutions Phys Rev E 85, 026607 (2012)

4 E A Kuznetsov Solitons in a parametrically unstable plasma Sov Phys.-Dokl.(Engl.) 22:9 (1977)

5 T Kawata and H Inoue Inverse Scattering Method for the Nonlinear Evolution Equations under Nonvanishing Conditions J. Phys Soc. Jpn. 44, 1722 (1978)

6 Y-C Ma The Perturbed Plane-Wave Solutions of the Cubic Schrödinger Equation Stud Appl. Math.: 60, 43 (1979)

7 N N Akhmediev and V I Korneev Modulation instability and periodic solutions of the nonlinear Schrödinger equation Theoretical and Mathematical Physics 69, 1089 (1986)

8 J M Dudley G Genty F Dias B Kibler, and N Akhmediev Modulation instability Akhmediev Breathers and continuous wave supercontinuum generation Opt Express 17:21497 (2009)

9 D S Agafontsev and V E Zakharov Integrable turbulence and formation of rogue waves Nonlinearity 28:2791 (2015)

10 Y Ohta and J Yang Dynamics of rogue waves in the Davey–Stewartson II equation J. Phys. A: Math. Theor. 46:105202 (2013)

1. V E Zakharov and A A Gelash Nonlinear stage of modulation instability Phys Rev Lett 111, 054101 (2013)

2. N Akhmediev Waves that Appear From Nowhere Complex Rogue Wave Structures Their Elementary Particles Frontiers in Physics 8, 612318 (2021)

3. B Guo L Ling and Q P Liu Nonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutions Phys Rev E 85, 026607 (2012)

4 E A Kuznetsov Solitons in a parametrically unstable plasma Sov Phys.-Dokl.(Engl.) 22:9 (1977)

5 T Kawata and H Inoue Inverse Scattering Method for the Nonlinear Evolution Equations under Nonvanishing Conditions J. Phys Soc. Jpn. 44, 1722 (1978)

6 Y-C Ma The Perturbed Plane-Wave Solutions of the Cubic Schrödinger Equation Stud Appl. Math.: 60, 43 (1979)

7 N N Akhmediev and V I Korneev Modulation instability and periodic solutions of the nonlinear Schrödinger equation Theoretical and Mathematical Physics 69, 1089 (1986)

8 J M Dudley G Genty F Dias B Kibler, and N Akhmediev Modulation instability Akhmediev Breathers and continuous wave supercontinuum generation Opt Express 17:21497 (2009)

9 D S Agafontsev and V E Zakharov Integrable turbulence and formation of rogue waves Nonlinearity 28:2791 (2015)

10 Y Ohta and J Yang Dynamics of rogue waves in the Davey–Stewartson II equation J. Phys. A: Math. Theor. 46:105202 (2013)

1. V E Zakharov and A A Gelash Nonlinear stage of modulation instability Phys Rev Lett

As a fastidious researcher, I have analyzed the provided paper, Tsunami Solitons Emerging from Superconducting Gap, which proposes a classical integrable system exhibiting tsunami-like solitons on rocky-desert-like disordered backgrounds, derived from Bogoliubov–de Gennes Hamiltonians in parity-mixed superconductors.

The core scientific contribution lies in establishing a bridge between condensed matter physics (superconductivity) and classical nonlinear integrable systems (KdV hierarchy), specifically demonstrating how complex phenomena like soliton turning-back and quasiperiodic multi-tsunami backgrounds arise from the mathematical structure of the Lax operators.

Here are the specific improvements to AI systems that can be made, categorized by capability:

)The improved AI system will possess enhanced capabilities in modeling complex, non-equilibrium physical systems characterized by inherent disorder and strong nonlinear wave interactions. Specifically, it will excel at tasks requiring the synthesis of classical integrable dynamics with quantum many-body physics and condensed matter phenomena.

1. The AI can perform high-fidelity simulation and prediction of wave propagation in disordered media by leveraging the established connection between the BdG Hamiltonian (the system's Lax operator) and parity-mixed superconductors.

2. It will be capable of modeling rogue wave phenomena—such as the sudden turning back and oscillation observed in tsunami solitons—in systems with spatially varying or bumpy backgrounds, moving beyond simple, uniform wave models.

3. The system will be able to characterize and classify complex, quasiperiodic multi-tsunami backgrounds by utilizing the concept of isodispersive phases, allowing for the prediction of long-term behavior in systems exhibiting quasiperiodic modulation (e.g., in topological superconductors or disordered electronic transport).

4. It can generate and analyze inhomogeneous stationary solutions (KdV rocks) with an arbitrary number of bumps at specific positions, enabling the simulation of complex, localized defect structures within physical media like plasma or electronic lattices.

5. The AI can be trained to interpret and solve complex nonlinear evolution equations (like the derived Novikov equations) that govern these integrable systems, allowing for the development of highly accurate numerical solvers for classical integrable models relevant to fluid dynamics and plasma physics.

We propose a classical integrable system exhibiting tsunami-like solitons with a rocky-desert-like disordered stationary background. One of the Lax operators describing this system is interpretable as a Bogoliubov--de Gennes Hamiltonian in parity-mixed superconductors. The family of integrable equations is generated from this seed operator using Krichever's method, whose pure s-wave limit includes the coupled Schrödinger--Boussinesq hierarchy applied to plasma physics. A linearly unstable finite background with a superconducting gap supports the tsunami-soliton solution, where the propagation of the step structure turns back at a certain moment, accompanied with the oscillation on the opposite side. In addition, the equation allows inhomogeneous stationary solutions with an arbitrary number of bumps at arbitrary positions, which we term the Korteweg--de Vries (KdV) rocks. In the Zakharov--Shabat scheme, the tsunami solitons are created from the Bogoliubov quasiparticles in the energy gap and the KdV rocks from normal electrons/holes. The unexpected large space of stationary solutions originates from the non-coprime Lax pair and the multivalued Baker--Akhiezer functions on the Riemann surface, formulated in terms of higher-rank holomorphic bundles by Krichever and Novikov. Furthermore, the concept of isodispersive phases is introduced to characterize quasiperiodic multi-tsunami backgrounds and consider their classification.