scholarly journals Wronskian determinant solutions of the (2 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation

2013 ◽  
Vol 1 (1) ◽  
Author(s):  
Malihe Najafi ◽  
Mohammad Najafi ◽  
Somayeh Arbabi
Keyword(s):  
Wronskian Determinant

scholarly journals Wronskian Addition Formula and Darboux-Pöschl-Teller Potentials

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Pierre Gaillard ◽  
Vladimir Matveev
Keyword(s):  
Explicit Description ◽  
Addition Formula ◽  
Kdv Hierarchy ◽  
Wronskian Determinant ◽  
New Formula

For the famous Darboux-Pöschl-Teller equation, we present new wronskian representation both for the potential and the related eigenfunctions. The simplest application of this new formula is the explicit description of dynamics of the DPT potentials and the action of the KdV hierarchy. The key point of the proof is some evaluation formulas for special wronskian determinant.

Use of the Wronskian determinant in the study of multicomponent systems

1991 ◽  
Vol 5 (1) ◽  
pp. 21-35 ◽  
Author(s):  
S. P. Koinis ◽  
A. T. Tsatsas ◽  
D. F. Katakis
Keyword(s):  
Multicomponent Systems ◽  
Wronskian Determinant

Nonautonomous solitons in terms of the double Wronskian determinant for a variable-coefficient Gross–Pitaevskii equation in the Bose–Einstein condensate

2016 ◽  
Vol 30 (09) ◽  
pp. 1650103 ◽  
Author(s):  
Chuan-Qi Su ◽  
Yi-Tian Gao ◽  
Qi-Min Wang ◽  
Jin-Wei Yang ◽  
Da-Wei Zuo
Keyword(s):  
Bound State ◽  
Einstein Condensate ◽  
Pitaevskii Equation ◽  
Bilinear Forms ◽  
Spectral Parameters ◽  
Potential Coefficient ◽  
Wronskian Determinant ◽  
Bose Einstein Condensate ◽  
Double Wronskian ◽  
Bose Einstein

Under investigation in this paper is a variable-coefficient Gross–Pitaevskii equation which describes the Bose–Einstein condensate. Lax pair, bilinear forms and bilinear Bäcklund transformation for the equation under some integrable conditions are derived. Based on the Lax pair and bilinear forms, double Wronskian solutions are constructed and verified. The [Formula: see text]th-order nonautonomous solitons in terms of the double Wronskian determinant are given. Propagation and interaction for the first- and second-order nonautonomous solitons are discussed from three cases. Amplitudes of the first- and second-order nonautonomous solitons are affected by a real parameter related to the variable coefficients, but independent of the gain-or-loss coefficient [Formula: see text] and linear external potential coefficient [Formula: see text]. For Case 1 [Formula: see text], [Formula: see text] leads to the accelerated propagation of nonautonomous solitons. Parabolic-, cubic-, exponential- and cosine-type nonautonomous solitons are exhibited due to the different choices of [Formula: see text]. For Case 2 [Formula: see text], if the real part of the spectral parameter equals 0, stationary soliton can be formed. If we take the harmonic external potential coefficient [Formula: see text] as a positive constant and let the real parts of the two spectral parameters be the same, bound-state-like structures can be formed, but there are only one attractive and two repulsive procedures. For Case 3 [[Formula: see text] and [Formula: see text] are taken as nonzero constants], head-on interaction, overtaking interaction and bound-state structure can be formed based on the signs of the two spectral parameters.

POSITON, NEGATON, SOLITON AND COMPLEXITON SOLUTIONS TO A FOUR-DIMENSIONAL NONLINEAR EVOLUTION EQUATION

2009 ◽  
Vol 23 (25) ◽  
pp. 2971-2991 ◽  
Author(s):  
ZHAQILAO ◽  
ZHI-BIN LI
Keyword(s):  
Evolution Equation ◽  
Exact Solutions ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Direct Method ◽  
Sufficient Conditions ◽  
Comprehensive Approach ◽  
Wronskian Determinant ◽  
Interaction Solutions ◽  
Complexiton Solutions

A generalized Wronskian formulation is presented for a four-dimensional nonlinear evolution equation. The representative systems are explicitly solved by selecting a broad set of sufficient conditions which make the Wronskian determinant a solution to the bilinearized four-dimensional nonlinear evolution equation. The obtained solution formulas provide us with a comprehensive approach to construct explicit exact solutions to the four-dimensional nonlinear evolution equation, by which positons, negatons, solitons and complexitons are computed for the four-dimensional nonlinear evolution equation. Applying the Hirota's direct method, multi-soliton, non-singular complexiton, and their interaction solutions of the four-dimensional nonlinear evolution equation are also obtained.

scholarly journals Two Different Classes of Wronskian Conditions to a (3 + 1)-Dimensional Generalized Shallow Water Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Yaning Tang ◽  
Pengpeng Su
Keyword(s):  
Shallow Water ◽  
Shallow Water Equation ◽  
Sufficient Conditions ◽  
Wronskian Technique ◽  
Rational Solutions ◽  
Bilinear Method ◽  
Wronskian Determinant ◽  
Linear Partial Differential Equations ◽  
Hirota Bilinear Equation ◽  
Hirota Bilinear

Based on the Hirota bilinear method and Wronskian technique, two different classes of sufficient conditions consisting of linear partial differential equations system are presented, which guarantee that the Wronskian determinant is a solution to the corresponding Hirota bilinear equation of a (3+1)-dimensional generalized shallow water equation. Our results show that the nonlinear equation possesses rich and diverse exact solutions such as rational solutions, solitons, negatons, and positons.

Bäcklund transformation and soliton solutions for two (3+1)-dimensional nonlinear evolution equations

2016 ◽  
Vol 30 (24) ◽  
pp. 1650309
Author(s):  
Lin Wang ◽  
Qixing Qu ◽  
Liangjuan Qin
Keyword(s):  
Bilinear Form ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Backlund Transformation ◽  
Wronskian Determinant ◽  
Bilinear Bäcklund Transformation ◽  
Bilinear Equations

In this paper, two (3[Formula: see text]+[Formula: see text]1)-dimensional nonlinear evolution equations (NLEEs) are under investigation by employing the Hirota’s method and symbolic computation. We derive the bilinear form and bilinear Bäcklund transformation (BT) for the two NLEEs. Based on the bilinear form, we obtain the multi-soliton solutions for them. Furthermore, multi-soliton solutions in terms of Wronskian determinant for the first NLEE are constructed, whose validity is verified through direct substitution into the bilinear equations.

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