GEOMETRICAL CONSTRUCTION OF THE HIROTA BILINEAR FORM OF THE MODIFIED KORTEWEG-DE VRIES EQUATION ON A THIN ELASTIC ROD: BOSONIC CLASSICAL THEORY

1995 ◽  
Vol 10 (22) ◽  
pp. 3109-3123 ◽  
Author(s):  
SHIGEKI MATSUTANI
Keyword(s):  
Bilinear Form ◽  
Vries Equation ◽  
Mkdv Equation ◽  
Arc Length ◽  
Geometrical Construction ◽  
Elastic Rod ◽  
Hirota Bilinear Form ◽  
De Vries ◽  
Thin Elastic Rod ◽  
Hirota Bilinear

Recently there have been several studies of a nonrelativistic elastic rod in R2 whose dynamics is governed by the modified Korteweg-de Vries (MKdV) equation. Goldstein and Petrich found the MKdV hierarchy through its dynamics [Phys. Rev. Lett. 69, 555 (1992).] In this article, we will show the physical meaning of the Hirota bilinear form along the lines of the elastica problem after we formally complexify its arc length.

PHYSICAL REALIZATION OF THE JIMBO-MIWA THEORY OF THE MODIFIED KORTEWEG-DE VRIES EQUATION ON A THIN ELASTIC ROD: FERMIONIC THEORY

1995 ◽  
Vol 10 (22) ◽  
pp. 3091-3107 ◽  
Author(s):  
SHIGEKI MATSUTANI
Keyword(s):  
Dimensional Space ◽  
Mkdv Equation ◽  
Dirac Field ◽  
Elastic Rod ◽  
Bilinear Method ◽  
Physical Relation ◽  
De Vries ◽  
Lax Operator ◽  
Thin Elastic Rod ◽  
Hirota Bilinear

Recently we found that the Dirac operator on a thin elastic rod is identical with the Lax operator of the modified Korteweg-de Vries (MKdV) equation while the thin elastic rod is governed by the MKdV equation. In this article, we will show the physical relation between the Hirota bilinear method and the Dirac field in a thin rod on two-dimensional space, along the lines of the Jimbo-Miwa construction of the MKdV soliton.

scholarly journals The soliton solutions for semidiscrete complex mKdV equation

2020 ◽  
Vol 34 ◽  
pp. 03002
Author(s):  
Corina N. Babalic
Keyword(s):  
Bilinear Form ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Mkdv Equation ◽  
Complete Integrability ◽  
Nonlinear Schrödinger ◽  
Bilinear Formalism ◽  
De Vries ◽  
Hirota Bilinear

The semidiscrete complex modified Korteweg–de Vries equation (semidiscrete cmKdV), which is the second member of the semidiscrete nonlinear Schrődinger hierarchy (Ablowitz–Ladik hierarchy), is solved using the Hirota bilinear formalism. Starting from the focusing case of semidiscrete form of cmKdV, proposed by Ablowitz and Ladik, we construct the bilinear form and build the multi-soliton solutions. The complete integrability of semidiscrete cmKdV, focusing case, is proven and results are discussed.

Lump-type solutions, interaction solutions and periodic wave solutions of a (3+1)-dimensional Korteweg–de Vries equation

2019 ◽  
Vol 33 (27) ◽  
pp. 1950319 ◽  
Author(s):  
Hongfei Tian ◽  
Jinting Ha ◽  
Huiqun Zhang
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Periodic Wave ◽  
Periodic Wave Solutions ◽  
Interaction Solutions ◽  
Dynamical Behaviors ◽  
Wave Solutions ◽  
Hirota Bilinear Form ◽  
De Vries ◽  
Hirota Bilinear

Based on the Hirota bilinear form, lump-type solutions, interaction solutions and periodic wave solutions of a (3[Formula: see text]+[Formula: see text]1)-dimensional Korteweg–de Vries (KdV) equation are obtained. The interaction between a lump-type soliton and a stripe soliton including two phenomena: fission and fusion, are illustrated. The dynamical behaviors are shown more intuitively by graphics.

THE RELATION OF LEMNISCATE AND A LOOP SOLITON AS 3/2 AND 1 SPIN FIELDS ALONG THE MODIFIED KORTEWEG-DE VRIES EQUATION

1995 ◽  
Vol 10 (08) ◽  
pp. 717-721 ◽  
Author(s):  
SHIGEKI MATSUTANI
Keyword(s):  
Vries Equation ◽  
Elastic Rod ◽  
De Vries ◽  
Spin Fields ◽  
Thin Elastic Rod

The modified Korteweg-de Vries equation on a thin elastic rod was studied recently. In this article, we show that the lemniscate is regarded as the elastica embedded in a space with a spin 3/2.

ANALYTIC ANALYSIS ON A GENERALIZED (2+1)-DIMENSIONAL VARIABLE-COEFFICIENT KORTEWEG–DE VRIES EQUATION USING SYMBOLIC COMPUTATION

2010 ◽  
Vol 24 (27) ◽  
pp. 5359-5370 ◽  
Author(s):  
CHENG ZHANG ◽  
BO TIAN ◽  
LI-LI LI ◽  
TAO XU
Keyword(s):  
Symbolic Computation ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Nonlinear Superposition ◽  
Hirota Bilinear Form ◽  
De Vries ◽  
The One ◽  
Dimensional Variable ◽  
Hirota Bilinear ◽  
Nonlinear Superposition Formula

With the help of symbolic computation, a generalized (2+1)-dimensional variable-coefficient Korteweg–de Vries equation is studied for its Painlevé integrability. Then, Hirota bilinear form is derived, from which the one- and two-solitary-wave solutions with the corresponding graphic illustration are presented. Furthermore, a bilinear auto-Bäcklund transformation is constructed and the nonlinear superposition formula and Lax pair are also obtained. Finally, the analytic solution in the Wronskian form is constructed and proved by direct substitution into the bilinear equation.

Abundant rogue wave solutions for the (2 + 1)-dimensional generalized Korteweg–de Vries equation

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huanhuan Lu ◽  
Yufeng Zhang
Keyword(s):  
Vries Equation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
Wave Solutions ◽  
De Vries ◽  
Rogue Wave Solutions ◽  
Bilinear Formulation ◽  
Hirota Bilinear

AbstractIn this paper, we analyse two types of rogue wave solutions generated from two improved ansatzs, to the (2 + 1)-dimensional generalized Korteweg–de Vries equation. With symbolic computation, the first-order rogue waves, second-order rogue waves, third-order rogue waves are generated directly from the first ansatz. Based on the Hirota bilinear formulation, another type of one-rogue waves and two-rogue waves can be obtained from the second ansatz. In addition, the dynamic behaviours of obtained rogue wave solutions are illustrated graphically.

Characteristics of the breather waves, lump waves and semi-rational solutions in a generalized (2+1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

2019 ◽  
Vol 33 (28) ◽  
pp. 1950350 ◽  
Author(s):  
Wei-Qi Peng ◽  
Shou-Fu Tian ◽  
Tian-Tian Zhang
Keyword(s):  
Bilinear Form ◽  
Rational Solutions ◽  
Bell Polynomial ◽  
Long Wave ◽  
Hirota Bilinear Form ◽  
Wave Limit ◽  
Parameter Constraints ◽  
Hirota Bilinear

In this work, we study a generalized (2[Formula: see text]+[Formula: see text]1)-dimensional asymmetrical Nizhnik–Novikov–Veselov (NNV) equation. Its Hirota bilinear form is constructed via the Bell polynomial. Based on the obtained bilinear form, the Nth-order breather waves are derived explicitly under certain parameter constraints. Moreover, we generate the nonsingular Nth-order lump waves through applying the long wave limit method. Additionally, we successfully present the semi-rational waves containing the combination of lump waves and single-soliton waves, the combination of lump waves and breather waves.

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