Spectral Problems and Darboux Transformation

2009 ◽  
pp. 93-99
Keyword(s):  
Darboux Transformation ◽  
Spectral Problems

Explicit Solutions and Conservation Laws of a Coupled Burgers’ Equation

2017 ◽  
Vol 72 (9) ◽  
pp. 789-793
Author(s):  
Bo Xue ◽  
Fang Li ◽  
Yihao Li ◽  
Mingming Sun
Keyword(s):  
Conservation Laws ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Burgers Equation ◽  
Integrable Equation ◽  
Explicit Solutions ◽  
Spectral Problems ◽  
Coupled Burgers Equation

AbstractBased on the gauge transformation between the corresponding 3×3 matrix spectral problems, N-fold Darboux transformation for a coupled Burgers’ equation is constructed. Considering the N=1 case of the derived Darboux transformation, explicit solutions for the coupled Burgers’ equation are given and their figures are plotted. Moreover, conservation laws of this integrable equation are deduced.

Darboux Transformation for Coupled Non-Linear Schrödinger Equation and Its Breather Solutions

2017 ◽  
Vol 72 (1) ◽  
pp. 9-15 ◽  
Author(s):  
Lili Feng ◽  
Fajun Yu ◽  
Li Li
Keyword(s):  
Schrödinger Equation ◽  
Spectral Problem ◽  
Darboux Transformation ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Soliton Equations ◽  
Spectral Problems ◽  
Non Linear ◽  
And Mathematics

AbstractStarting from a 3×3 spectral problem, a Darboux transformation (DT) method for coupled Schrödinger (CNLS) equation is constructed, which is more complex than 2×2 spectral problems. A scheme of soliton solutions of an integrable CNLS system is realised by using DT. Then, we obtain the breather solutions for the integrable CNLS system. The method is also appropriate for more non-linear soliton equations in physics and mathematics.

DARBOUX TRANSFORMATION AND VARIOUS SOLUTIONS FOR A NONLINEAR EVOLUTION EQUATION IN (3 + 1)-DIMENSIONS

2008 ◽  
Vol 22 (30) ◽  
pp. 2945-2966 ◽  
Author(s):  
ZHAQILAO ◽  
ZHI-BIN LI
Keyword(s):  
Evolution Equation ◽  
Soliton Solution ◽  
Darboux Transformation ◽  
Nonlinear Evolution Equation ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Spectral Problems ◽  
Complexiton Solution

A new (3 + 1)-dimensional nonlinear evolution equation can be decomposed into three (1 + 1)-dimensional nonlinear evolution equations. In this paper, N-soliton solution, resonant solution and complexiton solution of the (3 + 1)-dimensional nonlinear evolution equation are obtained via an N-fold Darboux transformation of the Ablowitz–Kaup–Newell–Segur spectral problems.

Triple Wronskian Representation of the Multi-Soliton Solutions for a Coupled GMNLS Equations

2013 ◽  
Vol 860-863 ◽  
pp. 2940-2945
Author(s):  
Lei Wang
Keyword(s):  
Darboux Transformation ◽  
Soliton Solutions ◽  
Previous Method ◽  
Soliton Equations ◽  
Spectral Problems ◽  
Bilinear Equation

In this paper, we mainly extend the notion of the Wronskian to a coupled GMNLS equations. Based on the N-fold Darboux transformation (DT), we present the triple Wronskian representation of the multi-soliton solutions for the coupled GMNLS equations. Di erent fromthe previous method, the triple Wronskian solutions can be obtained from the N -fold DT without substituting it into the bilinear equation. A signi cant advantage of such method is that it avoids guessing the Wroskian representation and Wronskian condition. Our approach could be applied other soliton equations with 3×3 spectral problems.

N-fold Darboux transformation and conservation laws of the modified Volterra lattice

2018 ◽  
Vol 32 (33) ◽  
pp. 1850409 ◽  
Author(s):  
Jian-Ping Yu ◽  
Wen-Xiu Ma ◽  
Yong-Li Sun ◽  
Chaudry Masood Khalique
Keyword(s):  
Conservation Laws ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Nonlinear Phenomena ◽  
Explicit Solutions ◽  
Spectral Problems

In this work, we study the modified Volterra lattice. Applying the gauge transformation of the associated [Formula: see text] matrix spectral problems, we establish the N-fold Darboux transformation (DT), and then construct a few explicit solutions in terms of determinants upon using the obtained DT. Moreover, all the results are illustrated by the graphs of the solitonic evolution profiles of the aforementioned solutions. Finally, infinitely many conservation laws for the modified Volterra lattice are proposed. The obtained results of this research might be applied to the research on nonlinear phenomena in physics or engineering areas.

scholarly journals Darboux Transformation and Explicit Solutions for a Generalized Sawada-Kotera Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Guo-Liang He ◽  
Ting Su
Keyword(s):  
Periodic Solutions ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Explicit Solutions ◽  
Rational Solutions ◽  
Lax Pairs ◽  
Spectral Problems

A generalized Sawada-Kotera equation and its Lax pairs are proposed. With the help of the gauge transformation between spectral problems, a Darboux transformation for the generalized SK equation is constructed. As an application of the Darboux transformation, we give some explicit solutions of the generalized SK equation such as the rational solutions, soliton solutions, and periodic solutions.

SPECTRAL PROBLEMS ARISING IN THE STABILIZATION PROBLEM FOR THE LOADED HEAT EQUATION: A TWO-DIMENSIONAL AND MULTI-POINT CASES

2019 ◽  
Vol 7 (1) ◽  
pp. 23-37
Author(s):  
Jenaliyev M.T. ◽  
◽  
Imanberdiyev K.B. ◽  
Kassymbekova A.S. ◽  
Sharipov K.S. ◽  
...  
Keyword(s):  
Heat Equation ◽  
Two Dimensional ◽  
Stabilization Problem ◽  
Spectral Problems
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