THE DIRECT METHOD IN SOLITON THEORY (Cambridge Tracts in Mathematics 155)

By Peter A. Clarkson. 2006

432 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Mechanical Engineering

View Paper: https://www.semanticscholar.org/paper/d9c0e73bfd31753ba67d90722ffd628103500339

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Mechanical Engineering

Citations in the corpus

Bilinear equations, Bell polynomials and linear superposition principle (2013, methodology): “Hirota presented a direct method to solve a kind of specific bilinear differential equations [1]; and soliton solutions are, despite their diversity, a universal phenomenon that Hirota bilinear equations describe [2].”
KP solitons and total positivity for the Grassmannian (2011, methodology): “They found a large variety of soliton solutions which were previously overlooked by those using the Hirota method of a perturbation expansion [12].”
General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation (2011, methodology): “This derivation utilizes the bilinear method in the soliton theory [15].”
Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations (2016, background): “(1) When c1c4 6= 0, but c1 and c2 are arbitrary, we have”
A novel (G′/G)-expansion method and its application to the Boussinesq equation (2014, background): “(3) can be expressed in powers of ψ(ξ ) as follows:”
General N-soliton solution to a vector nonlinear Schrödinger equation (2014, methodology): “The proof of the above equations can be shown by using the Grammian technique [30, 31], which is omitted here.”
KP solitons in shallow water (2010, background): “where the τ -function is assumed to be the Wronskian determinant with N functions fi’s (see for examples [40, 25, 14, 16]),”
Bilinearization and new multisoliton solutions for the (4+1)-dimensional Fokas equation (2016, background): “where Dx , Dt , Dy1 and Dy2 are the Hirota’s differential operators [19] defined by D t D n xf (t, x) · g(t, x) = (∂t − ∂ ′ t )(∂x − ∂ ′ x)nf (t, x)g(t ′, x ′) ∣∣ t ′=t,x ′=x .”
Soliton Solutions of the KP Equation and Application to Shallow Water Waves (2009, methodology): “5 (the following proof is the same as that in [13]): Proof.”
Lump solutions to the BKP equation by symbolic computation (2016, methodology): “This set of parameters, in turn, generates positive quadratic function solutions to the bilinear BKP equation (6): f = ✓ a1x+ a2y + 5(a1a2(2) a1a6(2) + 2a2a5a6) a1 2 + a5(2) t+ a4 ◆2 + ✓ a5x+ a6y + 5(2a1a2a6 a2(2)a5 + a5a6(2)) a1 2 + a5(2) t+ a8 ◆2 3(a1a2 + a5a6)(a1 2 + a5(2))(2) (a1a6 a2a5)(2) , (1”
General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions (2017, methodology): “The proof of above equations can be done by using the determinant technique [71], which is omitted here.”