Riemann-Hilbert approach to TD equation with nonzero boundary condition

Abstract

We extend the Riemann-Hilbert approach to the TD equation, which is a highly nonlinear differential integrable equation. Zero boundary condition at infinity for the TD equation is not suitable. Inverse scattering transform for this equation involves the singular Riemann-Hilbert problem, which means that the sectionally analytic functions have singularities on the boundary curve. Regularization procedures of the singular Riemann-Hilbert problem for two cases, the general case and the case for reflectionless potentials, are considered. Solitonic solutions to the TD equation are given.

Key words： Riemann-Hilbert problem; singular; TD equation

References

1 Ablowitz M J, Biondini G, Prinari B. Inverse scattering transform for the integrable discrete nonlinear Schrödinger equation with nonvanishing boundary conditions. Inverse Problems, 2007, 23: 1711–1758

2 Asano N, Kato Y. Non-self-adjoint Zakharov-Shabat operator with a potential of the finite asymptotic values: I. Direct spectral and scattering problems. J Math Phys, 1981, 22: 2780–2793

3 Bikbaev R F. Influence of viscosity on the structure of shock waves in the mKdV model. J Math Sci, 1995, 77: 3042–3045

4 Biondini G, Kovačič G. Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions. J Math Phys, 2014, 55: 031506

5 Biondini G, Prinari B. On the spectrum of the Dirac operator and the existence of discrete eigenvalues for the defocusing nonlinear Schrödinger equation. Stud Appl Math, 2014, 132: 138–159

6 Chen X J, Lam W K. Inverse scattering transform for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Phys Rev E, 2004, 69: 066604

7 Deift P, Zhou X. A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation. Ann Math, 1993, 137: 295–368

8 Demontis F, Prinari B, van der Mee C, Vitale F. The inverse scattering transform for the defocusing nonlinear Schrödinger equations with nonzero boundary conditions. Stud Appl Math, 2013, 131: 1–40

9 Faddeev L D, Takhtajan L A. Hamiltonian Methods in the Theory of Solitons. Berlin: Springer, 1987

10 Frolov I S. Inverse scattering problem for the Dirac system on the whole line. Sov Math Dokl, 1972, 13: 1468–1472

11 Gardner C S, Greene J M, Kruskal M D, Miura R M. Method for solving the Kortewegde Vries equation. Phys Rev Lett, 1967, 19: 1095–1097

12 Garnier J, Kalimeris K. Inverse scattering perturbation theory for the nonlinear Schrödinger equation with nonvanishing background. J Phys A: Math Gen, 2012, 45: 035202

13 Gelash A A, Zakharov V E. Superregular solitonic solutions: a novel scenario for the nonlinear stage of modulation instability. Nonlinearity, 2014, 27: R1–R39

14 Geng X G, Wu L H, He G L. Algebro-geometric constructions of the modified Boussinesq flows and quasi-periodic solutions. Phys D, 2011, 240: 1262–1288

15 Geng X G, Zeng X, Xue B. Algebro-geometric solutions of the TD hierarchy. Math Phys Anal Geom, 2013, 16: 229–251

16 Gerdjikov V S, Kulish P P. Completely integrable Hamiltonian systems connected with the non self-adjoint Dirac operator. Bulg J Phys, 1978, 5: 337–349 (in Russian)

17 Gu C H, Hu H S, Zhou Z X. Darboux Transformations in Integrable Systems: Theory and their Applications to Geometry. Dordrecht: Springer, 2005

18 Hirota H. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys Rev Lett, 1971, 27: 1192–1194

19 Hirota R. A new form of Bäcklund transformation and its relation to the inverse scattering problem. Prog Theor Phys, 1974, 52: 1498–1512

20 Ieda J, Uchiyama M, Wadati M. Inverse scattering method for square matrix nonlinear Schrödinger equation under nonvanishing boundary conditions. J Math Phys, 2007, 48: 013507

21 Kawata T, Inoue H. Inverse scattering method for nonlinear evolution equations under nonvanishing conditions. J Phys Soc Japan, 1978, 44: 1722–1729

22 Kawata T, Inoue H. Exact solutions of the derivative nonlinear Schrödinger equation under the nonvanishing conditions. J Phys Soc Japan, 1978, 44: 1968–1976

23 Kotlyarov V, Minakov A. Riemann-Hilbert problem to the modified Korteveg-de Vries equation: Long-time dynamics of the steplike initial data. J Math Phys, 2010, 51: 093506

24 Kulish P P, Manakov S V, Faddeev L D. Comparison of the exact quantum and quasiclassical results for a nonlinear Schrödinger equation. Theoret and Math Phys, 1976, 28: 615–620

25 Lakshmanan M. Continuum spin system as an exactly solvable dynamical system. Phys Lett A, 1977, 61: 53–54

26 Leon J. The Dirac inverse spectral transform: kinks and boomerons. J Math Phys, 1980, 21: 2572–2578

27 Ma Y C. The perturbed plane-wave solutions of the cubic Schrödinger equation. Stud Appl Math, 1079, 60: 43–58

28 Matveev V B, Salle M A. Darboux Transformation and Solitions. Berlin: Springer, 1991

29 Mjølhus E. Nonlinear Alfvén waves and the DNLS equation: oblique aspects. Physica Scripta, 1989, 40: 227–237

30 Prinari B, Ablowitz M J, Biondini G. Inverse scattering transform for vector nonlinear Schrödinger equation with non-vanishing boundary conditions. J Math Phys, 2006, 47: 063508

31 Prinari B, Biondini G, Trubatch A D. Inverse scattering transform for the multicomponent nonlinear Schrödinger equation with nonzero boundary conditions. Stud Appl Math, 2010, 126: 245–302

32 Prinari B, Vitale F, Biondini G. Dark-bright soliton solutions with nontrivial polarization interactions for the three-component defocusing nonlinear Schrödinger equation with nonzero boundary conditions. J Math Phys, 2015, 56: 071505

33 Prinari B. Vitale F. Inverse scattering transform for the focusing Ablowitz-Ladik system with nonzero boundary conditions. Stud Appl Math, 2016, 137: 28–52

34 Qiao Z J. A new completely integrable Liouville’s system produced by the Kaup-Newell eigenvalue problem. J Math Phys, 1993, 34: 3110–3120

35 Qiao Z J. A finite-dimensional integrable system and the involutive solutions of the higher-order Heisenberg spin chain equations. Phys Lett A, 1994, 186: 97–102

36 Qiao Z J. Non-dynamical r-matrix and algebraic-geometric solution for a discrete system. Chin Sci Bull, 1998, 43: 1149–1153

37 Rogers C, Schief W K. Bäcklund and Darboux Transformations—Geometry and Modern Applications in Soliton Theory. Cambridge: Cambridge Univ Press, 2002

38 Steudel H. The hierarchy of multi-soliton solutions of the derivative nonlinear Schrödinger equation. J Phys A: Math Gen, 2003, 36: 1931–1946

39 Takhtajan L A. Integration of the continuous Heisenberg spin chain through the inverse scattering method. Phys Lett A, 1977, 64: 235–237

40 Tu G Z. The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. J Math Phys, 1989, 30: 330–338

41 Tu G Z, Meng D Z. The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. II. Acta Math Appl Sin, 1989, 5: 89–96

42 Vekslerchik V E, Konotop V V. Discrete nonlinear Schrödinger equation under nonvanishing boundary conditions. Inverse Probl, 1992, 8: 889–909

43 Wahlquist H D, Estabrook F B. Bäcklund transformation for solutions of the Kortewegde

44 Vries equation. Phys Rev Lett, 1973, 23: 1386–1389

45 Wang S K, Guo H Y, Wu K. Inverse scattering transform and regular Riemann-Hilbert problem. Commun Theor Phys (Beijing), 1983, 2: 1169–1173

46 Wang S K, Guo H Y, Wu K. Principal Riemann-Hilbert problem and N-fold charged Kerr solution. Classical Quantum Gravity, 1984, 1: 378–384

47 Zakharov V E, Gelash A A. Nonlinear stage of modulation instability. Phys Rev Lett, 2013, 111: 054101

48 Zakharov V E, Shabat A B. Interaction between solitons in a stable medium. Sov Phys-JETP, 1973, 37: 823–828

49 Zakharov V E, Shabat A B. Integration of nonlinear equations of mathematical physics by the method of the inverse scattering. II. Funct Anal Appl, 1979, 13: 166–174

50 Zhou R G. The finite-band solution of the Jaulent-Miodek equation. J Math Phys, 1997, 38: 2535–2546

51 Zhou R G. A new (2+ 1)-dimensional integrable system and its algebro-geometric solution. Nuovo Cimento B, 2002, 117: 925–939

52 Zhu J Y, Geng X G. Miura transformation for the TD hierarchy. Chin Phys Lett, 2006, 23: 1–3

53 Zhu J Y, Wang L L. Kuznetsov-Ma solution and Akhmediev breather for TD equation. Commun Nonlinear Sci Numer Simul, 2019, 67: 555–567