國立高雄大學圖資館 |

Language: English

Back

Transverse instability of solitary wavesmultisymplectic dirac operators and the Evans function /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Transverse instability of solitary wavesby Timothy J. Burchell, Thomas J. Bridges.
Reminder of title:	multisymplectic dirac operators and the Evans function /
Author:	Burchell, Timothy J.
other author:	Bridges, Thomas J.
Published:	Singapore :Springer Nature Singapore :2025.
Description:	xii, 164 p. :ill., digital ;24 cm.
Contained By:	Springer Nature eBook
Subject:	Hamiltonian systems.
Online resource:	https://doi.org/10.1007/978-981-95-2046-6
ISBN:	9789819520466$q(electronic bk.)

Transverse instability of solitary wavesmultisymplectic dirac operators and the Evans function /
Burchell, Timothy J.

Transverse instability of solitary wavesmultisymplectic dirac operators and the Evans function /[electronic resource] :by Timothy J. Burchell, Thomas J. Bridges. - Singapore :Springer Nature Singapore :2025. - xii, 164 p. :ill., digital ;24 cm. - Mathematical physics studies,2352-3905. - Mathematical physics studies..

Chapter 1 Introduction -- Chapter 2 Literature Review -- Chapter 3 Multisymplectic Wave Equations and Dirac Operators -- Chapter 4 Solitary Wave Solutions and Their Properties -- Chapter 5 Linearisation about Solitary Waves -- Chapter 6 Spectral Stability and the Evans Function -- Chapter 7 Derivatives of the Evans Function -- Chapter 8 Summary of Hypotheses Used -- Chapter 9 Example: Nonlinear Wave Equation in 2 + 1 -- Chapter 10 Concluding Remarks.

This book presents a wide-ranging geometric approach to the stability of solitary wave solutions of Hamiltonian partial differential equations (PDEs). It blends original research with background material and a review of the literature. The overarching aim is to integrate geometry, algebra, and analysis into a theoretical framework for the spectral problem associated with the transverse instability of line solitary wave solutions-waves that travel uniformly in a horizontal plane and are embedded in two spatial dimensions. Rather than focusing on individual PDEs, the book develops an abstract class of Hamiltonian PDEs in two spatial dimensions and time, based on multisymplectic Dirac operators and their generalizations. This class models a broad range of nonlinear wave equations and benefits from a distinct symplectic structure associated with each spatial dimension and time. These structures inform both the existence theory (via variational principles, the Maslov index, and transversality conditions) and the linear stability analysis (through a multisymplectic partition of the Evans function). The spectral problem arising from linearization about a solitary wave is formulated as a dynamical system, with three symplectic structures contributing to the analysis. A two-parameter Evans function-depending on the spectral parameter and transverse wavenumber-is constructed from this system. This structure enables new results concerning the Evans function and the linear transverse instability of solitary waves. A key result is an abstract derivative formula for the Evans function in the regime of small stability exponents and transverse wavenumbers. To illustrate the theory, the book introduces a class of vector-valued nonlinear wave equations in 2+1 dimensions that are multisymplectic and admit explicit solitary wave solutions. In this example, the stable and unstable subspaces involved in the Evans function construction are each four-dimensional and can be explicitly computed. The example is used to demonstrate the geometric instability condition and to explore the inner workings of the theory in detail.

ISBN: 9789819520466$q(electronic bk.)

Standard No.: 10.1007/978-981-95-2046-6doiSubjects--Topical Terms:

207807
Hamiltonian systems.

LC Class. No.: QA614.83

Dewey Class. No.: 514.74

Transverse instability of solitary wavesmultisymplectic dirac operators and the Evans function /
LDR:03693nmm a2200337 a 4500 001 692293
003 DE-He213
005 20251124120459.0
006 m d
007 cr nn 008maaau
008 260604s2025 si s 0 eng d
020 $a 9789819520466$q(electronic bk.)
020 $a 9789819520459$q(paper)
024 7 $a 10.1007/978-981-95-2046-6 $2 doi
035 $a 978-981-95-2046-6
040 $a GP $c GP
041 0 $a eng
050 4 $a QA614.83
072 7 $a PHU $2 bicssc
072 7 $a SCI040000 $2 bisacsh
072 7 $a PHU $2 thema
082 0 4 $a 514.74 $2 23
090 $a QA614.83 $b .B947 2025
100 1 $a Burchell, Timothy J. $3 1008956
245 1 0 $a Transverse instability of solitary waves $h [electronic resource] : $b multisymplectic dirac operators and the Evans function / $c by Timothy J. Burchell, Thomas J. Bridges.
260 $a Singapore : $b Springer Nature Singapore : $b Imprint: Springer, $c 2025.
300 $a xii, 164 p. : $b ill., digital ; $c 24 cm.
490 1 $a Mathematical physics studies, $x 2352-3905
505 0 $a Chapter 1 Introduction -- Chapter 2 Literature Review -- Chapter 3 Multisymplectic Wave Equations and Dirac Operators -- Chapter 4 Solitary Wave Solutions and Their Properties -- Chapter 5 Linearisation about Solitary Waves -- Chapter 6 Spectral Stability and the Evans Function -- Chapter 7 Derivatives of the Evans Function -- Chapter 8 Summary of Hypotheses Used -- Chapter 9 Example: Nonlinear Wave Equation in 2 + 1 -- Chapter 10 Concluding Remarks.
520 $a This book presents a wide-ranging geometric approach to the stability of solitary wave solutions of Hamiltonian partial differential equations (PDEs). It blends original research with background material and a review of the literature. The overarching aim is to integrate geometry, algebra, and analysis into a theoretical framework for the spectral problem associated with the transverse instability of line solitary wave solutions-waves that travel uniformly in a horizontal plane and are embedded in two spatial dimensions. Rather than focusing on individual PDEs, the book develops an abstract class of Hamiltonian PDEs in two spatial dimensions and time, based on multisymplectic Dirac operators and their generalizations. This class models a broad range of nonlinear wave equations and benefits from a distinct symplectic structure associated with each spatial dimension and time. These structures inform both the existence theory (via variational principles, the Maslov index, and transversality conditions) and the linear stability analysis (through a multisymplectic partition of the Evans function). The spectral problem arising from linearization about a solitary wave is formulated as a dynamical system, with three symplectic structures contributing to the analysis. A two-parameter Evans function-depending on the spectral parameter and transverse wavenumber-is constructed from this system. This structure enables new results concerning the Evans function and the linear transverse instability of solitary waves. A key result is an abstract derivative formula for the Evans function in the regime of small stability exponents and transverse wavenumbers. To illustrate the theory, the book introduces a class of vector-valued nonlinear wave equations in 2+1 dimensions that are multisymplectic and admit explicit solitary wave solutions. In this example, the stable and unstable subspaces involved in the Evans function construction are each four-dimensional and can be explicitly computed. The example is used to demonstrate the geometric instability condition and to explore the inner workings of the theory in detail.
650 0 $a Hamiltonian systems. $3 207807
650 0 $a Index theorems. $3 750337
650 0 $a Differential equations, Linear. $3 185864
650 1 4 $a Mathematical Physics. $3 522725
700 1 $a Bridges, Thomas J. $3 1008957
710 2 $a SpringerLink (Online service) $3 273601
773 0 $t Springer Nature eBook
830 0 $a Mathematical physics studies. $3 684220
856 4 0 $u https://doi.org/10.1007/978-981-95-2046-6
950 $a Mathematics and Statistics (SpringerNature-11649)