Step-like initial value problem and Whitham modulation in fluid dynamics to a generalized derivative nonlinear Schrödinger equation
In this paper, we study the step-like initial value problem for a generalized derivative nonlinear Schrödinger equation using the Whitham modulation theory. First, we utilize the finite-gap integration method to obtain the periodic solutions and the relevant Whitham equations for the 0-, 1-, and 2-g...
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| Published in | Physics of fluids (1994) Vol. 36; no. 6 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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01.06.2024
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| Abstract | In this paper, we study the step-like initial value problem for a generalized derivative nonlinear Schrödinger equation using the Whitham modulation theory. First, we utilize the finite-gap integration method to obtain the periodic solutions and the relevant Whitham equations for the 0-, 1-, and 2-genus cases used to characterize dispersive shock waves (DSWs). Second, we investigate four fundamental waves: two rarefaction waves (RWs) and two DSWs with step-like initial data. On this basis, we show the effect of varying certain parameters on the dynamics of the fluid model. We find that the boundary value, amplitude, and shape of the wave pattern in the fluid dynamics model will be significantly impacted by these parameters. Third, under two step-like initial data, waves are divided into six cases, which are actually combinations of DSWs and RWs. Finally, the dam break problem is explored to prove the effectiveness of the Whitham modulation theory in physical applications. |
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| AbstractList | In this paper, we study the step-like initial value problem for a generalized derivative nonlinear Schrödinger equation using the Whitham modulation theory. First, we utilize the finite-gap integration method to obtain the periodic solutions and the relevant Whitham equations for the 0-, 1-, and 2-genus cases used to characterize dispersive shock waves (DSWs). Second, we investigate four fundamental waves: two rarefaction waves (RWs) and two DSWs with step-like initial data. On this basis, we show the effect of varying certain parameters on the dynamics of the fluid model. We find that the boundary value, amplitude, and shape of the wave pattern in the fluid dynamics model will be significantly impacted by these parameters. Third, under two step-like initial data, waves are divided into six cases, which are actually combinations of DSWs and RWs. Finally, the dam break problem is explored to prove the effectiveness of the Whitham modulation theory in physical applications. |
| Author | Liu, Bingyu Li, Xinyue Zhao, Qiulan |
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| Cites_doi | 10.1103/PhysRevE.96.062202 10.1016/0167-2789(95)00147-V 10.1016/j.cnsns.2022.106612 10.1006/jmaa.1998.6212 10.3367/UFNe.0180.201001b.0055 10.1016/j.apnum.2022.12.007 10.1016/j.aml.2024.109025 10.1103/PhysRevA.101.053827 10.1088/1751-8121/acb117 10.1002/cpa.3160360302 10.1098/rspa.2017.0233 10.1016/j.chaos.2024.114731 10.1016/j.chaos.2024.114539 10.1017/S0022112065000745 10.1007/s11071-023-08459-x 10.1175/1520-0493(1981)109%3C1726:TMGOTG%3E2.0.CO;2 10.1016/j.physd.2016.04.006 10.1111/sapm.12246 10.1063/1.523737 10.1103/PhysRevE.108.024222 10.1088/0031-8949/20/3-4/026 10.3390/math12060927 10.1103/PhysRevE.98.052220 10.1007/s11005-024-01780-5 10.1002/cpa.3160330605 |
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| References | Ma (c27) 2024 Geng, Dai (c28) 1999 Kaup, Newell (c21) 1978 Chen, Lee, Liu (c22) 1979 Ma (c20) 2024 Ma (c19) 2024 Ma (c29) 2024 Ivanov, Kamchatnov, Congy (c18) 2017 Biondini (c14) 2018 El, Hoefer (c1) 2016 Bai, Li, Zhao (c11) 2024 Hoefer, Smyth, Sprenger (c12) 2019 Gerdjikov, Ivanov (c23) 1983 Abeya, Biondini, Hoefer (c15) 2023 Gurevich, Pitaevskii (c6) 1974 Scotti, Pineda (c3) 2004 Luke (c10) 1966 Flaschka, Forest, McLaughlin (c9) 1980 Whitham (c5) 1965 Clarke, Smith, Reid (c2) 1981 Li, Han, Zhao (c26) 2022 Ma (c30) 2017 Lax, Levermore (c7) 1983 Zhang, Hao (c13) 2023 El, Geogjaev, Gurevich, Krylov (c16) 1995 Whitham (c8) 1965 Shukla, Eliasson (c4) 2010 Ivanov (c25) 2020 Geng, Jia, Xue, Zhai (c31) 2024 Gao, Wang (c24) 2023 (2024060611420152200_c7) 1983; 36 (2024060611420152200_c25) 2020; 101 (2024060611420152200_c13) 2023; 111 (2024060611420152200_c23) 1983; 10 (2024060611420152200_c27) 2024; 12 (2024060611420152200_c30) 2017; 473 (2024060611420152200_c1) 2016; 333 (2024060611420152200_c29) 2024; 153 (2024060611420152200_c17) 1977 (2024060611420152200_c11) 2024; 182 (2024060611420152200_c4) 2010; 53 (2024060611420152200_c3) 2004; 31 (2024060611420152200_c19) 2024; 180 (2024060611420152200_c8) 1965; 22 (2024060611420152200_c15) 2023; 56 (2024060611420152200_c5) 1965; 283 (2024060611420152200_c24) 2023; 108 (2024060611420152200_c12) 2019; 142 (2024060611420152200_c31) 2024; 114 (2024060611420152200_c28) 1999; 233 (2024060611420152200_c2) 1981; 109 (2024060611420152200_c10) 1966; 292 (2024060611420152200_c18) 2017; 96 (2024060611420152200_c22) 1979; 20 (2024060611420152200_c21) 1978; 19 (2024060611420152200_c14) 2018; 98 (2024060611420152200_c16) 1995; 87 (2024060611420152200_c20) 2024; 199 (2024060611420152200_c26) 2022; 114 (2024060611420152200_c9) 1980; 33 (2024060611420152200_c6) 1974; 38 |
| References_xml | – start-page: 11 year: 2016 ident: c1 article-title: Dispersive shock waves and modulation theory publication-title: Physica D – start-page: 025701 year: 2023 ident: c15 article-title: Whitham modulation theory for the defocusing nonlinear Schrödinger equation in two and three spatial dimensions publication-title: J. Phys. A – start-page: 106612 year: 2022 ident: c26 article-title: Interactions of localized wave and dynamics analysis in generalized derivative nonlinear Schrödinger equation publication-title: Commun. Nonlinear Sci. Numer. Simul. – start-page: 109025 year: 2024 ident: c29 article-title: A four-component hierarchy of combined integrable equations with bi-Hamiltonian formulations publication-title: Appl. Math. Lett. – start-page: 114731 year: 2024 ident: c11 article-title: Evolution of dispersive shock waves to the complex modified Korteweg–de Vries equation with higher-order effects publication-title: Chaos, Solitons Fractals – start-page: 052220 year: 2018 ident: c14 article-title: Riemann problems and dispersive shocks in self-focusing media publication-title: Phys. Rev. E – start-page: 273 year: 1965 ident: c8 article-title: A general approach to linear and non-linear dispersive waves using a Lagrangian publication-title: J. Fluid Mech. – start-page: 105 year: 2024 ident: c20 article-title: Type (λ ,λ) reduced nonlocal integrable AKNS equations and their soliton solutions publication-title: Appl. Numer. Math. – start-page: 114539 year: 2024 ident: c19 article-title: Binary Darboux transformation of vector nonlocal reverse-time integrable NLS equations publication-title: Chaos, Solitons Fractals – start-page: 024222 year: 2023 ident: c24 article-title: Optical undular bores in Riemann problem of photon fluid with quintic nonlinearity publication-title: Phys. Rev. E – start-page: 053827 year: 2020 ident: c25 article-title: Riemann problem for the light pulses in optical fibers for the generalized Chen-Lee-Liu equation publication-title: Phys. Rev. A – start-page: 490 year: 1979 ident: c22 article-title: Integrability of nonlinear Hamiltonian systems by inverse scattering method publication-title: Phys. Scr. – start-page: 062202 year: 2017 ident: c18 article-title: Solution of the Riemann problem for polarization waves in a two-component Bose-Einstein condensate publication-title: Phys. Rev. E – start-page: 238 year: 1965 ident: c5 article-title: Nonlinear dispersive waves publication-title: Proc. R. Soc. A – start-page: 0233 year: 2017 ident: c30 article-title: Trigonal curves and algebro-geometric solutions to soliton hierarchies II publication-title: Proc. R. Soc. A – start-page: L22307 year: 2004 ident: c3 article-title: Observation of very large and steep internal waves of elevation near the Massachusetts coast publication-title: Geophys. Res. Lett. – start-page: 219 year: 2019 ident: c12 article-title: Modulation theory solution for nonlinearly resonant, fifth-order Korteweg–de Vries, nonclassical, traveling dispersive shock waves publication-title: Stud. Appl. Math. – start-page: 51 year: 2010 ident: c4 article-title: Nonlinear aspects of quantum plasma physics publication-title: Phys. Usp. – start-page: 12461 year: 2023 ident: c13 article-title: Whitham modulation theory and periodic solutions for the fifth-order nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain publication-title: Nonlinear Dyn. – start-page: 130 year: 1983 ident: c23 article-title: The quadratic bundle of general form and the nonlinear evolution equations publication-title: Bulg. J. Phys. – start-page: 291 year: 1974 ident: c6 article-title: Nonstationary structure of a collisionless shock wave publication-title: Sov. Phys. JETP – start-page: 798 year: 1978 ident: c21 article-title: An exact solution for a derivative nonlinear Schrödinger equation publication-title: J. Math. Phys. – start-page: 927 year: 2024 ident: c27 article-title: A generalized hierarchy of combined integrable bi-Hamiltonian equations from a specific fourth-order matrix spectral problem publication-title: Mathematics – start-page: 26 year: 1999 ident: c28 article-title: Nonlinearization of the 33 matrix eigenvalue problem related to coupled nonlinear Schrödinger equations publication-title: J. Math. Anal. Appl. – start-page: 739 year: 1980 ident: c9 article-title: Multiphase averaging and the inverse spectral solution of the Korteweg–de Vries equation publication-title: Commun. Pure Appl. Math. – start-page: 253 year: 1983 ident: c7 article-title: The small dispersion limit of the Korteweg-de Vries equation publication-title: Commun. Pure Appl. Math. – start-page: 1726 year: 1981 ident: c2 article-title: The morning glory of the Gulf of Carpentaria: An atmospheric undular bore publication-title: Mon. Weather Rev. – start-page: 403 year: 1966 ident: c10 article-title: A perturbation method for nonlinear dispersive wave problems publication-title: Proc. R. Soc. A – start-page: 186 year: 1995 ident: c16 article-title: Decay of an initial discontinuity in the defocusing NLS hydrodynamics publication-title: Physica D – start-page: 30 year: 2024 ident: c31 article-title: Application of tetragonal curves to coupled Boussinesq equations publication-title: Lett. Math. Phys. – volume: 292 start-page: 403 year: 1966 ident: 2024060611420152200_c10 article-title: A perturbation method for nonlinear dispersive wave problems publication-title: Proc. R. Soc. A – volume: 96 start-page: 062202 year: 2017 ident: 2024060611420152200_c18 article-title: Solution of the Riemann problem for polarization waves in a two-component Bose-Einstein condensate publication-title: Phys. Rev. E doi: 10.1103/PhysRevE.96.062202 – volume: 87 start-page: 186 year: 1995 ident: 2024060611420152200_c16 article-title: Decay of an initial discontinuity in the defocusing NLS hydrodynamics publication-title: Physica D doi: 10.1016/0167-2789(95)00147-V – volume: 114 start-page: 106612 year: 2022 ident: 2024060611420152200_c26 article-title: Interactions of localized wave and dynamics analysis in generalized derivative nonlinear Schrödinger equation publication-title: Commun. Nonlinear Sci. Numer. Simul. doi: 10.1016/j.cnsns.2022.106612 – volume: 233 start-page: 26 year: 1999 ident: 2024060611420152200_c28 article-title: Nonlinearization of the 33 matrix eigenvalue problem related to coupled nonlinear Schrödinger equations publication-title: J. Math. Anal. Appl. doi: 10.1006/jmaa.1998.6212 – volume: 10 start-page: 130 year: 1983 ident: 2024060611420152200_c23 article-title: The quadratic bundle of general form and the nonlinear evolution equations publication-title: Bulg. J. Phys. – volume: 53 start-page: 51 year: 2010 ident: 2024060611420152200_c4 article-title: Nonlinear aspects of quantum plasma physics publication-title: Phys. Usp. doi: 10.3367/UFNe.0180.201001b.0055 – volume: 199 start-page: 105 year: 2024 ident: 2024060611420152200_c20 article-title: Type (λ*,λ) reduced nonlocal integrable AKNS equations and their soliton solutions publication-title: Appl. Numer. Math. doi: 10.1016/j.apnum.2022.12.007 – volume: 38 start-page: 291 year: 1974 ident: 2024060611420152200_c6 article-title: Nonstationary structure of a collisionless shock wave publication-title: Sov. Phys. JETP – volume: 153 start-page: 109025 year: 2024 ident: 2024060611420152200_c29 article-title: A four-component hierarchy of combined integrable equations with bi-Hamiltonian formulations publication-title: Appl. Math. Lett. doi: 10.1016/j.aml.2024.109025 – volume: 101 start-page: 053827 year: 2020 ident: 2024060611420152200_c25 article-title: Riemann problem for the light pulses in optical fibers for the generalized Chen-Lee-Liu equation publication-title: Phys. Rev. A doi: 10.1103/PhysRevA.101.053827 – volume: 56 start-page: 025701 year: 2023 ident: 2024060611420152200_c15 article-title: Whitham modulation theory for the defocusing nonlinear Schrödinger equation in two and three spatial dimensions publication-title: J. Phys. A doi: 10.1088/1751-8121/acb117 – volume: 36 start-page: 253 year: 1983 ident: 2024060611420152200_c7 article-title: The small dispersion limit of the Korteweg-de Vries equation publication-title: Commun. Pure Appl. Math. doi: 10.1002/cpa.3160360302 – volume: 473 start-page: 0233 year: 2017 ident: 2024060611420152200_c30 article-title: Trigonal curves and algebro-geometric solutions to soliton hierarchies II publication-title: Proc. R. Soc. A doi: 10.1098/rspa.2017.0233 – volume: 182 start-page: 114731 year: 2024 ident: 2024060611420152200_c11 article-title: Evolution of dispersive shock waves to the complex modified Korteweg–de Vries equation with higher-order effects publication-title: Chaos, Solitons Fractals doi: 10.1016/j.chaos.2024.114731 – volume: 180 start-page: 114539 year: 2024 ident: 2024060611420152200_c19 article-title: Binary Darboux transformation of vector nonlocal reverse-time integrable NLS equations publication-title: Chaos, Solitons Fractals doi: 10.1016/j.chaos.2024.114539 – volume: 22 start-page: 273 year: 1965 ident: 2024060611420152200_c8 article-title: A general approach to linear and non-linear dispersive waves using a Lagrangian publication-title: J. Fluid Mech. doi: 10.1017/S0022112065000745 – volume: 111 start-page: 12461 year: 2023 ident: 2024060611420152200_c13 article-title: Whitham modulation theory and periodic solutions for the fifth-order nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain publication-title: Nonlinear Dyn. doi: 10.1007/s11071-023-08459-x – volume: 109 start-page: 1726 year: 1981 ident: 2024060611420152200_c2 article-title: The morning glory of the Gulf of Carpentaria: An atmospheric undular bore publication-title: Mon. Weather Rev. doi: 10.1175/1520-0493(1981)109%3C1726:TMGOTG%3E2.0.CO;2 – volume: 283 start-page: 238 year: 1965 ident: 2024060611420152200_c5 article-title: Nonlinear dispersive waves publication-title: Proc. R. Soc. A – volume: 333 start-page: 11 year: 2016 ident: 2024060611420152200_c1 article-title: Dispersive shock waves and modulation theory publication-title: Physica D doi: 10.1016/j.physd.2016.04.006 – volume: 142 start-page: 219 year: 2019 ident: 2024060611420152200_c12 article-title: Modulation theory solution for nonlinearly resonant, fifth-order Korteweg–de Vries, nonclassical, traveling dispersive shock waves publication-title: Stud. Appl. Math. doi: 10.1111/sapm.12246 – volume-title: Coherent Non-Linear Interaction of Wavesin Plasmas year: 1977 ident: 2024060611420152200_c17 – volume: 19 start-page: 798 year: 1978 ident: 2024060611420152200_c21 article-title: An exact solution for a derivative nonlinear Schrödinger equation publication-title: J. Math. Phys. doi: 10.1063/1.523737 – volume: 31 start-page: L22307 year: 2004 ident: 2024060611420152200_c3 article-title: Observation of very large and steep internal waves of elevation near the Massachusetts coast publication-title: Geophys. Res. Lett. – volume: 108 start-page: 024222 year: 2023 ident: 2024060611420152200_c24 article-title: Optical undular bores in Riemann problem of photon fluid with quintic nonlinearity publication-title: Phys. Rev. E doi: 10.1103/PhysRevE.108.024222 – volume: 20 start-page: 490 year: 1979 ident: 2024060611420152200_c22 article-title: Integrability of nonlinear Hamiltonian systems by inverse scattering method publication-title: Phys. Scr. doi: 10.1088/0031-8949/20/3-4/026 – volume: 12 start-page: 927 year: 2024 ident: 2024060611420152200_c27 article-title: A generalized hierarchy of combined integrable bi-Hamiltonian equations from a specific fourth-order matrix spectral problem publication-title: Mathematics doi: 10.3390/math12060927 – volume: 98 start-page: 052220 year: 2018 ident: 2024060611420152200_c14 article-title: Riemann problems and dispersive shocks in self-focusing media publication-title: Phys. Rev. E doi: 10.1103/PhysRevE.98.052220 – volume: 114 start-page: 30 year: 2024 ident: 2024060611420152200_c31 article-title: Application of tetragonal curves to coupled Boussinesq equations publication-title: Lett. Math. Phys. doi: 10.1007/s11005-024-01780-5 – volume: 33 start-page: 739 year: 1980 ident: 2024060611420152200_c9 article-title: Multiphase averaging and the inverse spectral solution of the Korteweg–de Vries equation publication-title: Commun. Pure Appl. Math. doi: 10.1002/cpa.3160330605 |
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| Title | Step-like initial value problem and Whitham modulation in fluid dynamics to a generalized derivative nonlinear Schrödinger equation |
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