The Singular Manifold Method: Darboux Transformations and Nonclassical Symmetries

Estévez, P.G.; Gordoa, Pilar R.

doi:10.2991/jnmp.1995.2.3-4.14

Cited by 16 publications

(9 citation statements)

References 38 publications

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“…In this section, the singular manifold method is applied to find new families of solutions for the (3 + 1)‐dimensional generalized KP equation . According to the singular manifold method (SMM) based on the Painlevé analysis carried out by Weiss, the solution of equation is written in a series form:

ψ ((), η) = \sum_{j = 0}^{\infty} ψ_{j} φ {(η)}^{j - α}

where φ is an Eigenfunction weighting the poles and α is determined through a dominant behavior analysis and represent the possible order of the poles.Case solution of Equation …”

Section: Investigating Solutions Using the Singular Manifold Methodsmentioning

confidence: 97%

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New exact solutions of (3 + 1)‐dimensional generalized Kadomtsev‐Petviashvili equation using a combination of lie symmetry and singular manifold methods

Saleh

Rashed

2019

Math Methods in App Sciences

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A new combination of Lie symmetry and Singular Manifold methods has been employed to study (3 + 1)-dimensional generalized Kadomtsev-Petviashvili (KP). Infinite-dimensional space of Lie vectors has been established. Single and dual linear combinations of Lie vectors are used after appropriate calculations of the arbitrary functions to reduce the equation to an ordinary differential equation (ODE). The resulting ODE is then analytically solved through the singular manifold method which resulted in a Bäcklund truncated series with seminal analysis leading to a Schwarzian differential equation in the Eigenfunction φ (η). Solving this differential equation leads to new analytical solutions. KEYWORDS generalized Kadomtsev-Petviashvili equation, Lie infinitesimals, singular manifolds method, Schwarzian derivative MSC CLASSIFICATION 35Q53; 76M60; 76B15

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ψ ((), η) = \sum_{j = 0}^{\infty} ψ_{j} φ {(η)}^{j - α}

where φ is an Eigenfunction weighting the poles and α is determined through a dominant behavior analysis and represent the possible order of the poles.Case solution of Equation …”

Section: Investigating Solutions Using the Singular Manifold Methodsmentioning

confidence: 97%

“…KP equation admits infinite‐dimensional Lie space with 15 arbitrary functions which are determined using an optimization process. The singular manifold method is used to solve these ODEs. A Schwarzian derivative of the Eigenfunction φ is derived.…”

Section: Introductionmentioning

confidence: 97%

New exact solutions of (3 + 1)‐dimensional generalized Kadomtsev‐Petviashvili equation using a combination of lie symmetry and singular manifold methods

Saleh

Rashed

2019

Math Methods in App Sciences

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“…In this section, the Singular Manifold Method is applied to find the BT and Lax pair for the (2+1) dimensional CBS equation (1). Singular Manifold Method is an inverse solution [30][31][32] of nonlinear partial differential equations having a series form;…”

Section: Singular Manifolds Methodsmentioning

confidence: 99%

Similarity Solution of (2+1)-Dimensional Calogero-Bogoyavlenskii-Schiff Equation Lax Pair

Salem¹,

Kassem²,

Mabrouk³

2019

AJAM

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In this paper, we discussed and studied the solutions of the (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation. The Calogero-Bogoyavlenskii-Schiff equation describes the propagation of Riemann waves along the y-axis, with long wave propagating along the x-axis. Lax pair and Bäcklund transformation of the Calogero-Bogoyavlenskii-Schiff equation are derived by using the singular manifold method (SMM). The optimal Lie infinitesimals of the Lax pair are obtained. The detected Lie infinitesimals contain eight unknown functions. These functions are optimized through the commutator table.The eight unknown functions are evaluated through the solution of a set of linear differential equations, in which solutions lead to optimal Lie vectors. The CBS Lax pair is reduced by using the optimal Lie vectors to a system of ordinary differential equations (ODEs). The solitary wave solutions of Calogero-Bogoyavlenskii-Schiff equation Lax pair's show soliton and kink waves. The obtained similarity solutions are plotted for different arbitrary functions and compared with previous analytical solutions. The comparison shows that we derive new solutions of Calogero-Bogoyavlenskii-Schiff equation by using the combination of two methods, which is different from the previous findings.

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“…• The truncation (1.3) of the Painlevé series has itself the meaning of an auto-Bäcklund transformation between two solutions of a PDE [37], [12].…”

Section: The Singular Manifold Methodsmentioning

confidence: 99%

Unified approach to Miura, Bäcklund and Darboux Transformations for Nonlinear Partial Differential Equations

Estévez¹,

Conde²,

Gordoa³

1998

JNMP

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This paper is an attempt to present and discuss at some length the Singular Manifold Method. This Method is based upon the Painlevé Property systematically used as a tool for obtaining clear cut answers to almost all the questions related with Nonlinear Partial Differential Equations: Lax pairs, Miura, Bäcklund or Darboux Transformations as well as τ -functions, in a unified way. Besides to present the basics of the Method we exemplify this approach by applying it to four equations in (1 + 1)-dimensions. Two of them are related with the other two through Miura transformations that are also derived by using the Singular Manifold Method.

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The Singular Manifold Method: Darboux Transformations and Nonclassical Symmetries

Cited by 16 publications

References 38 publications

New exact solutions of (3 + 1)‐dimensional generalized Kadomtsev‐Petviashvili equation using a combination of lie symmetry and singular manifold methods

New exact solutions of (3 + 1)‐dimensional generalized Kadomtsev‐Petviashvili equation using a combination of lie symmetry and singular manifold methods

Similarity Solution of (2+1)-Dimensional Calogero-Bogoyavlenskii-Schiff Equation Lax Pair

Unified approach to Miura, Bäcklund and Darboux Transformations for Nonlinear Partial Differential Equations

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