Nonlocal symmetries and new interaction waves of the variable-coefficient modified Korteweg–de Vries equation in fluid-filled elastic tubes

Wu, Jian-Wen; He, Juntao; Lin, Ji

doi:10.1140/epjp/s13360-022-03033-7

Cited by 13 publications

(7 citation statements)

References 39 publications

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“…Hence, if f is the solution to Eq. ( 10), the u we obtained by transformation (9) is the single soliton solution to Eq. ( 8).…”

Section: The Bilinear Form Of the Vchsi Equationmentioning

confidence: 99%

“…Therefore, search for efficient solution methods has gained more and more attention. To date, there are several effective methods that have been commonly used by mathematicians, such as Darboux transformation, [1,2] Bäcklund transformation, [3][4][5] Lie group method, [6][7][8][9] inverse scattering method, [10] Hirota bilinear method, [11][12][13][14] and hyperbolic method. [15] Among them, the Hirota bilinear method proposed by Hirota is a straightforward method for finding exact solutions of NLEEs.…”

Section: Introductionmentioning

confidence: 99%

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Interaction solutions and localized waves to the (2+1)-dimensional Hirota–Satsuma–Ito equation with variable coefficient

2023

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This article investigates the Hirota-Satsuma-Ito equation with variable coefficient using the Hirota bilinear method and the long wave limit method. The equation is proved to be Painlevé integrable by Painlevé analysis. On the basis of the bilinear form, the forms of two-soliton solutions, three-soliton solutions and four-soliton solutions are studied specifically. The appropriate parameter values are chosen and the corresponding figures are presented. The breather waves solutions, lump solutions, periodic solutions and the interaction of breather waves solutions and soliton solutions, etc are given in this article. In addition, we also analysed the different effects of the parameters on the figures. The figures of the same set of parameters in different planes are presented to describe the dynamical behaviour of solutions. These are important for describing water waves in nature.

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“…Hence, if f is the solution to Eq. ( 10), the u we obtained by transformation (9) is the single soliton solution to Eq. ( 8).…”

Section: The Bilinear Form Of the Vchsi Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Interaction solutions and localized waves to the (2+1)-dimensional Hirota–Satsuma–Ito equation with variable coefficient

2023

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“…e nonlocal symmetry (23) in the original space x, t, u, v, w { } is localized to a Lie point symmetry (28) in the prolonged space x, t, u, v, w, ϕ, g 􏼈 􏼉.…”

Section: Nonlocal Symmetry and Bt For Cid Systemmentioning

confidence: 99%

“…ese interaction excitations cannot be obtained by the direct Lie point symmetry method. Recently, different types of excitations are valid by using the symmetry reductions related to nonlocal symmetry and a consistent tanh expansion (CTE) method [20][21][22][23][24][25][26][27][28]. e CTE method can be investigated not only for various different types of excitations but also for an integrable property of the nonlinear systems, including the supersymmetric extension of the nonlinear systems [29][30][31][32].…”

Section: Introductionmentioning

confidence: 99%

CTE Solvability, Nonlocal Symmetry, and Interaction Solutions of Coupled Integrable Dispersionless System

Ren

Liu

et al. 2022

Complexity

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The consistent tanh expansion (CTE) method is successfully applied to the coupled integrable dispersionless (CID) system. A nonauto-Bäcklund transformation (BT) theorem includes two fields f and v 1 is obtained by using the CTE method. One obtains the consistent condition in the nonauto-BT theorem by means of the relation between the fields f and v 1 . The CID system possesses the CTE solvability property by some detailed analysis. Many interactions between one soliton and multiple resonant solitons, and between one soliton and cnoidal waves are generated by using the nonauto-BT theorem. The types of bright and gray two front waves are shown by some figures. In the meanwhile, the nonlocal symmetry is obtained by the truncated Painlevé method and the Möbious invariant form. The initial value problem and an auto-BT are constructed by the localization procedure.

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“…Many other techniques for finding exact solutions have been also discovered: inverse scattering theory, Darboux transformation, Riemann-Hilbert method, etc. Recently, more exact solutions of soliton equations are found [10][11][12][13], and more dynamic behaviors are studied [14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Algebro-Geometric Solutions of a ( $2 + 1$ )-Dimensional Integrable Equation Associated with the Ablowitz-Kaup-Newell-Segur Soliton Hierarchy

Chen

2022

Advances in Mathematical Physics

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The ( 2 + 1 )-dimensional Lax integrable equation is decomposed into solvable ordinary differential equations with the help of known ( 1 + 1 )-dimensional soliton equations associated with the Ablowitz-Kaup-Newell-Segur soliton hierarchy. Then, based on the finite-order expansion of the Lax matrix, a hyperelliptic Riemann surface and Abel-Jacobi coordinates are introduced to straighten out the associated flows, from which the algebro-geometric solutions of the ( 2 + 1 )-dimensional integrable equation are proposed by means of the Riemann θ functions.

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Nonlocal symmetries and new interaction waves of the variable-coefficient modified Korteweg–de Vries equation in fluid-filled elastic tubes

Cited by 13 publications

References 39 publications

Interaction solutions and localized waves to the (2+1)-dimensional Hirota–Satsuma–Ito equation with variable coefficient

Interaction solutions and localized waves to the (2+1)-dimensional Hirota–Satsuma–Ito equation with variable coefficient

CTE Solvability, Nonlocal Symmetry, and Interaction Solutions of Coupled Integrable Dispersionless System

Algebro-Geometric Solutions of a ( $2 + 1$ )-Dimensional Integrable Equation Associated with the Ablowitz-Kaup-Newell-Segur Soliton Hierarchy

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Nonlocal symmetries and new interaction waves of the variable-coefficient modified Korteweg–de Vries equation in fluid-filled elastic tubes

Cited by 13 publications

References 39 publications

Interaction solutions and localized waves to the (2+1)-dimensional Hirota–Satsuma–Ito equation with variable coefficient

Interaction solutions and localized waves to the (2+1)-dimensional Hirota–Satsuma–Ito equation with variable coefficient

CTE Solvability, Nonlocal Symmetry, and Interaction Solutions of Coupled Integrable Dispersionless System

Algebro-Geometric Solutions of a ( 2 + 1 )-Dimensional Integrable Equation Associated with the Ablowitz-Kaup-Newell-Segur Soliton Hierarchy

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Algebro-Geometric Solutions of a ( $2 + 1$ )-Dimensional Integrable Equation Associated with the Ablowitz-Kaup-Newell-Segur Soliton Hierarchy