The Direct Method In Soliton Theory

By L. Vogler. 2016

1,480 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/882c9aedfc5f35f21f3ed24628229065949a9de0

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Citations in the corpus

N-soliton solutions and the Hirota conditions in (2+1)-dimensions (2020, methodology): “The concept of bilinear derivatives is the key tool in the method, and Hirota bilinear forms are crucial in presenting soliton solutions.”
Lump solutions to the Kadomtsev–Petviashvili equation (2015, methodology): “Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla Lump solutions to the Kadomtsev–Petviashvili equation Wen-Xiu Ma 1”
Multi-soliton, multi-breather and higher order rogue wave solutions to the complex short pulse equation (2015, methodology): “The general N-soliton solution including multi-loop andmulti-breather ones was given in [25,26] by using Hirota’s bilinear method [27].”
Lump solutions to nonlinear partial differential equations via Hirota bilinear forms (2016, background): “The Korteweg–de Vries (KdV) equation and the Kadomtsev–Petviashvili (KP) equation are nonlinear integrable differential equations, and their Hirota bilinear forms play a crucial role in generating their soliton solutions, a kind of exponentially localized solutions, describing diverse nonlinear phen”
On the Integrability of a Generalized Variable‐Coefficient Forced Korteweg‐de Vries Equation in Fluids (2014, methodology): “Successful methods include inverse scattering transformation [1], Darboux and BT [2–4], Hirota method [5] and so on, Among them, the Hirota one is an analytic tool for researching NLEEs and relevant soliton problems.”
Consistent Riccati Expansion for Integrable Systems (2015, methodology): “To solve this problem, some elegant approaches have been established such as the inverse scattering transformation [1], Darboux–Bäcklund transformations [2], Hirota’s bilinear direct method [3], and nonlinearizations of Lax pairs [4] have been established.”
Long-Time Asymptotics of a Three-Component Coupled mKdV System (2019, methodology): “direct method [59], the generalized bilinear technique [60], the Wronskian technique [61,62] and the 198”
Riemann–Hilbert problems andN-soliton solutions for a coupled mKdV system (2018, methodology): “There are many other approaches to soliton solutions in the field of integrable systems, which include the bilinear method [31,32], the Wronskian technique [33,34] and the Darboux transformation [35].”
Lump Solutions and Interaction Phenomenon for (2+1)-Dimensional Sawada-Kotera Equation (2017, unknown): “(1) as (D(6) x + 5D 3 xDy − 5D(2) y +DxDt)(f · f) = 0 , (6) where D(6) x, D 3 xDy, D 2 y and DxDt are all the bilinear derivative operators([30]) defined by D xD β yD γ t (f ·g) = ( ∂ ∂x − ∂ ∂x′ )α( ∂ ∂y − ∂ ∂y′ )β( ∂ ∂t − ∂ ∂t′ )γ ×f(x, y, t)g(x′, y′, t′) |x′=x,y′=y,t′=t .”
The inverse scattering transform and soliton solutions of a combined modified Korteweg–de Vries equation (2019, methodology): “Many other approaches to soliton solutions are available in the field of integrable equations, among which are the Hirota direct method [15], the generalized bilinear technique [26], the Wronskian technique [9,32] and the Darboux transformation [33,36].”
Rational and Semirational Solutions of the Nonlocal Davey–Stewartson Equations (2017, methodology): “By the Hirota’s bilinear method with the perturbation expansion [36], and set f and g be the forms of”
Application of the Riemann–Hilbert approach to the multicomponent AKNS integrable hierarchies (2019, methodology): “available in the field of integrable equations, among which are the Hirota direct method [50], the generalized bilinear technique [51,52], the Wronskian technique [53,54] and the Darboux transformation [55].”
Bilinear Bäcklund transformation, soliton and periodic wave solutions for a $$\varvec{(3 + 1)}$$(3+1)-dimensional variable-coefficient generalized shallow water wave equation (2017, unknown): “⎝ 2B[1] 0 (t)−B[2] 1 (t) −8α1(t)νπ2 B[1] 0 (t) ⎞”