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Are there Soliton Solutions for Maxwell's Equations?
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Some non-linear differential equations (such as Korteweg–de Vries and Kadomtsev–Petviashvili equations) have "solitary waves" solutions (solitons).
Does the set of partial differential equations known as "Maxwell's equations" theoretically admit such kind of solutions?
In that case, should these solutions appear in the form of "stationary shells" of electromagnetic field? By "stationary", I mean do the solutions maintain their shape?
Thanks for your comments!
pde mathematical-physics nonlinear-system electromagnetism soliton-theory
asked Aug 1 at 13:01
Andrea Prunotto
569114
 |Â
show 3 more comments
up vote
8
down vote
favorite
Some non-linear differential equations (such as Korteweg–de Vries and Kadomtsev–Petviashvili equations) have "solitary waves" solutions (solitons).
Does the set of partial differential equations known as "Maxwell's equations" theoretically admit such kind of solutions?
In that case, should these solutions appear in the form of "stationary shells" of electromagnetic field? By "stationary", I mean do the solutions maintain their shape?
Thanks for your comments!
pde mathematical-physics nonlinear-system electromagnetism soliton-theory
asked Aug 1 at 13:01
Andrea Prunotto
569114
What means "solitary wave solution" for you? Something like compact support and "not changing form"?
– Severin Schraven
Aug 1 at 13:10
@SeverinSchraven Yes! Something like this. I would say, the equivalent of the solitons in the (shallow) water.
– Andrea Prunotto
Aug 1 at 13:14
@andrea.prunotto: Could you please clarify what you mean by "stationary shells"?
– Adrian Keister
Aug 1 at 13:31
@SeverinSchraven With this term I mean "a bubble" of electromagnetic field, sorry if I cannot be more precise, the 3D equivalent of the 1-D soliton in the water (i.e. a wave that does not involve crests and troughs, but is made only of a crest, or a trough).
– Andrea Prunotto
Aug 1 at 14:38
@AdrianKeister Sorry, the previous comment was for you!
– Andrea Prunotto
Aug 1 at 14:48
 |Â
show 3 more comments
up vote
8
down vote
favorite
up vote
8
down vote
favorite
Some non-linear differential equations (such as Korteweg–de Vries and Kadomtsev–Petviashvili equations) have "solitary waves" solutions (solitons).
Does the set of partial differential equations known as "Maxwell's equations" theoretically admit such kind of solutions?
In that case, should these solutions appear in the form of "stationary shells" of electromagnetic field? By "stationary", I mean do the solutions maintain their shape?
Thanks for your comments!
pde mathematical-physics nonlinear-system electromagnetism soliton-theory
asked Aug 1 at 13:01
Andrea Prunotto
569114
Some non-linear differential equations (such as Korteweg–de Vries and Kadomtsev–Petviashvili equations) have "solitary waves" solutions (solitons).
Does the set of partial differential equations known as "Maxwell's equations" theoretically admit such kind of solutions?
In that case, should these solutions appear in the form of "stationary shells" of electromagnetic field? By "stationary", I mean do the solutions maintain their shape?
Thanks for your comments!
pde mathematical-physics nonlinear-system electromagnetism soliton-theory
asked Aug 1 at 13:01
Andrea Prunotto
569114
edited Aug 1 at 21:19
asked Aug 1 at 13:01
Andrea Prunotto
569114
asked Aug 1 at 13:01
Andrea Prunotto
569114
asked Aug 1 at 13:01
Andrea Prunotto
569114
569114
What means "solitary wave solution" for you? Something like compact support and "not changing form"?
– Severin Schraven
Aug 1 at 13:10
@SeverinSchraven Yes! Something like this. I would say, the equivalent of the solitons in the (shallow) water.
– Andrea Prunotto
Aug 1 at 13:14
@andrea.prunotto: Could you please clarify what you mean by "stationary shells"?
– Adrian Keister
Aug 1 at 13:31
@SeverinSchraven With this term I mean "a bubble" of electromagnetic field, sorry if I cannot be more precise, the 3D equivalent of the 1-D soliton in the water (i.e. a wave that does not involve crests and troughs, but is made only of a crest, or a trough).
– Andrea Prunotto
Aug 1 at 14:38
@AdrianKeister Sorry, the previous comment was for you!
– Andrea Prunotto
Aug 1 at 14:48
 |Â
show 3 more comments
What means "solitary wave solution" for you? Something like compact support and "not changing form"?
– Severin Schraven
Aug 1 at 13:10
@SeverinSchraven Yes! Something like this. I would say, the equivalent of the solitons in the (shallow) water.
– Andrea Prunotto
Aug 1 at 13:14
@andrea.prunotto: Could you please clarify what you mean by "stationary shells"?
– Adrian Keister
Aug 1 at 13:31
@SeverinSchraven With this term I mean "a bubble" of electromagnetic field, sorry if I cannot be more precise, the 3D equivalent of the 1-D soliton in the water (i.e. a wave that does not involve crests and troughs, but is made only of a crest, or a trough).
– Andrea Prunotto
Aug 1 at 14:38
@AdrianKeister Sorry, the previous comment was for you!
– Andrea Prunotto
Aug 1 at 14:48
What means "solitary wave solution" for you? Something like compact support and "not changing form"?
– Severin Schraven
Aug 1 at 13:10
What means "solitary wave solution" for you? Something like compact support and "not changing form"?
– Severin Schraven
Aug 1 at 13:10
@SeverinSchraven Yes! Something like this. I would say, the equivalent of the solitons in the (shallow) water.
– Andrea Prunotto
Aug 1 at 13:14
@SeverinSchraven Yes! Something like this. I would say, the equivalent of the solitons in the (shallow) water.
– Andrea Prunotto
Aug 1 at 13:14
@andrea.prunotto: Could you please clarify what you mean by "stationary shells"?
– Adrian Keister
Aug 1 at 13:31
@andrea.prunotto: Could you please clarify what you mean by "stationary shells"?
– Adrian Keister
Aug 1 at 13:31
@SeverinSchraven With this term I mean "a bubble" of electromagnetic field, sorry if I cannot be more precise, the 3D equivalent of the 1-D soliton in the water (i.e. a wave that does not involve crests and troughs, but is made only of a crest, or a trough).
– Andrea Prunotto
Aug 1 at 14:38
@SeverinSchraven With this term I mean "a bubble" of electromagnetic field, sorry if I cannot be more precise, the 3D equivalent of the 1-D soliton in the water (i.e. a wave that does not involve crests and troughs, but is made only of a crest, or a trough).
– Andrea Prunotto
Aug 1 at 14:38
@AdrianKeister Sorry, the previous comment was for you!
– Andrea Prunotto
Aug 1 at 14:48
@AdrianKeister Sorry, the previous comment was for you!
– Andrea Prunotto
Aug 1 at 14:48
 |Â
show 3 more comments
1 Answer
1
active
oldest
votes
up vote
13
down vote
accepted
The answer is yes to both questions. If you cast Maxwell's Equations in cylindrical coordinates for a fiber optic cable, and you take birefringence into account, you get the coupled nonlinear Schrödinger equations. You can then solve those by means of the Inverse Scattering Transform, which takes the original system of nonlinear pde's (nonlinear because of the coordinate system), transforms them into a coupled system of linear ode's (the Manakov system) which are straight-forward to solve, and then, by means of the Gel'fand-Levitan-Marchenko integral equation, you arrive at the soliton solutions of the original pde's. For references, see C. Menyuk, Application of multiple-length-scale methods to the study of optical fiber transmission, Journal of Engineering Mathematics 36: 113-136, 1999, Kluwer Academic Publishers, Netherlands, and my own dissertation, which includes other references of interest. In particular, Shaw's book Mathematical Principles of Optical Fiber Communication has most of these derivations in it.
The resulting soliton solutions behave mostly like waves, but they also interact in a particle-like fashion; for example, in a collision, they can alter each others' phase - a decidedly non-wave-like behavior. Solitons do not stay in one place; in the case above, they would travel down the fiber cable (indeed, solitons are the reason fiber is the backbone of the Internet!), and self-correct their shape as they go. And, as Maxwell's equations are all about electromagnetic fields, the solutions are, indeed, stationary (in your sense) "shells" of electromagnetic fields.
answered Aug 1 at 13:29
Adrian Keister
3,49321533
Thanks for the detailed and illuminating answer, Adrian! A further question for you: Do you know if there are macroscopic conditions in which such phenomenon occurs? I mean, a case in which such solitons can have dimensions of meters, rather than the dimensions of the diameter of a fiber cable?
– Andrea Prunotto
Aug 1 at 14:40
@andrea.prunotto: Thank you for your kind words. An interesting question, that. I don't know the answer; my hunch/intuition would be no, but I don't have much of anything to back that up.
– Adrian Keister
Aug 1 at 14:46
1
@andrea.prunotto I googled: High power, pulsed soliton generation at radio and microwave frequencies.
– Keith McClary
Aug 1 at 17:30
1
Also en.wikipedia.org/wiki/Ball_lightning#Soliton_hypothesis.
– Adrian Keister
Aug 1 at 17:33
@Adrian,Keith Thanks for your comments!
– Andrea Prunotto
Aug 1 at 22:18
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
13
down vote
accepted
The answer is yes to both questions. If you cast Maxwell's Equations in cylindrical coordinates for a fiber optic cable, and you take birefringence into account, you get the coupled nonlinear Schrödinger equations. You can then solve those by means of the Inverse Scattering Transform, which takes the original system of nonlinear pde's (nonlinear because of the coordinate system), transforms them into a coupled system of linear ode's (the Manakov system) which are straight-forward to solve, and then, by means of the Gel'fand-Levitan-Marchenko integral equation, you arrive at the soliton solutions of the original pde's. For references, see C. Menyuk, Application of multiple-length-scale methods to the study of optical fiber transmission, Journal of Engineering Mathematics 36: 113-136, 1999, Kluwer Academic Publishers, Netherlands, and my own dissertation, which includes other references of interest. In particular, Shaw's book Mathematical Principles of Optical Fiber Communication has most of these derivations in it.
The resulting soliton solutions behave mostly like waves, but they also interact in a particle-like fashion; for example, in a collision, they can alter each others' phase - a decidedly non-wave-like behavior. Solitons do not stay in one place; in the case above, they would travel down the fiber cable (indeed, solitons are the reason fiber is the backbone of the Internet!), and self-correct their shape as they go. And, as Maxwell's equations are all about electromagnetic fields, the solutions are, indeed, stationary (in your sense) "shells" of electromagnetic fields.
answered Aug 1 at 13:29
Adrian Keister
3,49321533
Thanks for the detailed and illuminating answer, Adrian! A further question for you: Do you know if there are macroscopic conditions in which such phenomenon occurs? I mean, a case in which such solitons can have dimensions of meters, rather than the dimensions of the diameter of a fiber cable?
– Andrea Prunotto
Aug 1 at 14:40
@andrea.prunotto: Thank you for your kind words. An interesting question, that. I don't know the answer; my hunch/intuition would be no, but I don't have much of anything to back that up.
– Adrian Keister
Aug 1 at 14:46
1
@andrea.prunotto I googled: High power, pulsed soliton generation at radio and microwave frequencies.
– Keith McClary
Aug 1 at 17:30
1
Also en.wikipedia.org/wiki/Ball_lightning#Soliton_hypothesis.
– Adrian Keister
Aug 1 at 17:33
@Adrian,Keith Thanks for your comments!
– Andrea Prunotto
Aug 1 at 22:18
add a comment |Â
up vote
13
down vote
accepted
The answer is yes to both questions. If you cast Maxwell's Equations in cylindrical coordinates for a fiber optic cable, and you take birefringence into account, you get the coupled nonlinear Schrödinger equations. You can then solve those by means of the Inverse Scattering Transform, which takes the original system of nonlinear pde's (nonlinear because of the coordinate system), transforms them into a coupled system of linear ode's (the Manakov system) which are straight-forward to solve, and then, by means of the Gel'fand-Levitan-Marchenko integral equation, you arrive at the soliton solutions of the original pde's. For references, see C. Menyuk, Application of multiple-length-scale methods to the study of optical fiber transmission, Journal of Engineering Mathematics 36: 113-136, 1999, Kluwer Academic Publishers, Netherlands, and my own dissertation, which includes other references of interest. In particular, Shaw's book Mathematical Principles of Optical Fiber Communication has most of these derivations in it.
The resulting soliton solutions behave mostly like waves, but they also interact in a particle-like fashion; for example, in a collision, they can alter each others' phase - a decidedly non-wave-like behavior. Solitons do not stay in one place; in the case above, they would travel down the fiber cable (indeed, solitons are the reason fiber is the backbone of the Internet!), and self-correct their shape as they go. And, as Maxwell's equations are all about electromagnetic fields, the solutions are, indeed, stationary (in your sense) "shells" of electromagnetic fields.
answered Aug 1 at 13:29
Adrian Keister
3,49321533
Thanks for the detailed and illuminating answer, Adrian! A further question for you: Do you know if there are macroscopic conditions in which such phenomenon occurs? I mean, a case in which such solitons can have dimensions of meters, rather than the dimensions of the diameter of a fiber cable?
– Andrea Prunotto
Aug 1 at 14:40
@andrea.prunotto: Thank you for your kind words. An interesting question, that. I don't know the answer; my hunch/intuition would be no, but I don't have much of anything to back that up.
– Adrian Keister
Aug 1 at 14:46
1
@andrea.prunotto I googled: High power, pulsed soliton generation at radio and microwave frequencies.
– Keith McClary
Aug 1 at 17:30
1
Also en.wikipedia.org/wiki/Ball_lightning#Soliton_hypothesis.
– Adrian Keister
Aug 1 at 17:33
@Adrian,Keith Thanks for your comments!
– Andrea Prunotto
Aug 1 at 22:18
add a comment |Â
up vote
13
down vote
accepted
up vote
13
down vote
accepted
The answer is yes to both questions. If you cast Maxwell's Equations in cylindrical coordinates for a fiber optic cable, and you take birefringence into account, you get the coupled nonlinear Schrödinger equations. You can then solve those by means of the Inverse Scattering Transform, which takes the original system of nonlinear pde's (nonlinear because of the coordinate system), transforms them into a coupled system of linear ode's (the Manakov system) which are straight-forward to solve, and then, by means of the Gel'fand-Levitan-Marchenko integral equation, you arrive at the soliton solutions of the original pde's. For references, see C. Menyuk, Application of multiple-length-scale methods to the study of optical fiber transmission, Journal of Engineering Mathematics 36: 113-136, 1999, Kluwer Academic Publishers, Netherlands, and my own dissertation, which includes other references of interest. In particular, Shaw's book Mathematical Principles of Optical Fiber Communication has most of these derivations in it.
The resulting soliton solutions behave mostly like waves, but they also interact in a particle-like fashion; for example, in a collision, they can alter each others' phase - a decidedly non-wave-like behavior. Solitons do not stay in one place; in the case above, they would travel down the fiber cable (indeed, solitons are the reason fiber is the backbone of the Internet!), and self-correct their shape as they go. And, as Maxwell's equations are all about electromagnetic fields, the solutions are, indeed, stationary (in your sense) "shells" of electromagnetic fields.
answered Aug 1 at 13:29
Adrian Keister
3,49321533
The answer is yes to both questions. If you cast Maxwell's Equations in cylindrical coordinates for a fiber optic cable, and you take birefringence into account, you get the coupled nonlinear Schrödinger equations. You can then solve those by means of the Inverse Scattering Transform, which takes the original system of nonlinear pde's (nonlinear because of the coordinate system), transforms them into a coupled system of linear ode's (the Manakov system) which are straight-forward to solve, and then, by means of the Gel'fand-Levitan-Marchenko integral equation, you arrive at the soliton solutions of the original pde's. For references, see C. Menyuk, Application of multiple-length-scale methods to the study of optical fiber transmission, Journal of Engineering Mathematics 36: 113-136, 1999, Kluwer Academic Publishers, Netherlands, and my own dissertation, which includes other references of interest. In particular, Shaw's book Mathematical Principles of Optical Fiber Communication has most of these derivations in it.
The resulting soliton solutions behave mostly like waves, but they also interact in a particle-like fashion; for example, in a collision, they can alter each others' phase - a decidedly non-wave-like behavior. Solitons do not stay in one place; in the case above, they would travel down the fiber cable (indeed, solitons are the reason fiber is the backbone of the Internet!), and self-correct their shape as they go. And, as Maxwell's equations are all about electromagnetic fields, the solutions are, indeed, stationary (in your sense) "shells" of electromagnetic fields.
answered Aug 1 at 13:29
Adrian Keister
3,49321533
edited Aug 1 at 15:56
answered Aug 1 at 13:29
Adrian Keister
3,49321533
answered Aug 1 at 13:29
Adrian Keister
3,49321533
answered Aug 1 at 13:29
Adrian Keister
3,49321533
3,49321533
Thanks for the detailed and illuminating answer, Adrian! A further question for you: Do you know if there are macroscopic conditions in which such phenomenon occurs? I mean, a case in which such solitons can have dimensions of meters, rather than the dimensions of the diameter of a fiber cable?
– Andrea Prunotto
Aug 1 at 14:40
@andrea.prunotto: Thank you for your kind words. An interesting question, that. I don't know the answer; my hunch/intuition would be no, but I don't have much of anything to back that up.
– Adrian Keister
Aug 1 at 14:46
1
@andrea.prunotto I googled: High power, pulsed soliton generation at radio and microwave frequencies.
– Keith McClary
Aug 1 at 17:30
1
Also en.wikipedia.org/wiki/Ball_lightning#Soliton_hypothesis.
– Adrian Keister
Aug 1 at 17:33
@Adrian,Keith Thanks for your comments!
– Andrea Prunotto
Aug 1 at 22:18
add a comment |Â
Thanks for the detailed and illuminating answer, Adrian! A further question for you: Do you know if there are macroscopic conditions in which such phenomenon occurs? I mean, a case in which such solitons can have dimensions of meters, rather than the dimensions of the diameter of a fiber cable?
– Andrea Prunotto
Aug 1 at 14:40
@andrea.prunotto: Thank you for your kind words. An interesting question, that. I don't know the answer; my hunch/intuition would be no, but I don't have much of anything to back that up.
– Adrian Keister
Aug 1 at 14:46
1
@andrea.prunotto I googled: High power, pulsed soliton generation at radio and microwave frequencies.
– Keith McClary
Aug 1 at 17:30
1
Also en.wikipedia.org/wiki/Ball_lightning#Soliton_hypothesis.
– Adrian Keister
Aug 1 at 17:33
@Adrian,Keith Thanks for your comments!
– Andrea Prunotto
Aug 1 at 22:18
Thanks for the detailed and illuminating answer, Adrian! A further question for you: Do you know if there are macroscopic conditions in which such phenomenon occurs? I mean, a case in which such solitons can have dimensions of meters, rather than the dimensions of the diameter of a fiber cable?
– Andrea Prunotto
Aug 1 at 14:40
Thanks for the detailed and illuminating answer, Adrian! A further question for you: Do you know if there are macroscopic conditions in which such phenomenon occurs? I mean, a case in which such solitons can have dimensions of meters, rather than the dimensions of the diameter of a fiber cable?
– Andrea Prunotto
Aug 1 at 14:40
@andrea.prunotto: Thank you for your kind words. An interesting question, that. I don't know the answer; my hunch/intuition would be no, but I don't have much of anything to back that up.
– Adrian Keister
Aug 1 at 14:46
@andrea.prunotto: Thank you for your kind words. An interesting question, that. I don't know the answer; my hunch/intuition would be no, but I don't have much of anything to back that up.
– Adrian Keister
Aug 1 at 14:46
1
1
@andrea.prunotto I googled: High power, pulsed soliton generation at radio and microwave frequencies.
– Keith McClary
Aug 1 at 17:30
@andrea.prunotto I googled: High power, pulsed soliton generation at radio and microwave frequencies.
– Keith McClary
Aug 1 at 17:30
1
1
Also en.wikipedia.org/wiki/Ball_lightning#Soliton_hypothesis.
– Adrian Keister
Aug 1 at 17:33
Also en.wikipedia.org/wiki/Ball_lightning#Soliton_hypothesis.
– Adrian Keister
Aug 1 at 17:33
@Adrian,Keith Thanks for your comments!
– Andrea Prunotto
Aug 1 at 22:18
@Adrian,Keith Thanks for your comments!
– Andrea Prunotto
Aug 1 at 22:18
add a comment |Â
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What means "solitary wave solution" for you? Something like compact support and "not changing form"?
– Severin Schraven
Aug 1 at 13:10
@SeverinSchraven Yes! Something like this. I would say, the equivalent of the solitons in the (shallow) water.
– Andrea Prunotto
Aug 1 at 13:14
@andrea.prunotto: Could you please clarify what you mean by "stationary shells"?
– Adrian Keister
Aug 1 at 13:31
@SeverinSchraven With this term I mean "a bubble" of electromagnetic field, sorry if I cannot be more precise, the 3D equivalent of the 1-D soliton in the water (i.e. a wave that does not involve crests and troughs, but is made only of a crest, or a trough).
– Andrea Prunotto
Aug 1 at 14:38
@AdrianKeister Sorry, the previous comment was for you!
– Andrea Prunotto
Aug 1 at 14:48