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Klein Gordon equation - references
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The Klein Gordon equation of the form:
$Delta u+ lambda u^p=0$
is been studied for $p = 2$?
(i.e.$Delta u+ lambda u^2=0$)
If yes are there references?
ap.analysis-of-pdes
asked 4 hours ago
exxxit8
425
add a comment |Â
up vote
2
down vote
favorite
The Klein Gordon equation of the form:
$Delta u+ lambda u^p=0$
is been studied for $p = 2$?
(i.e.$Delta u+ lambda u^2=0$)
If yes are there references?
ap.analysis-of-pdes
asked 4 hours ago
exxxit8
425
1
Klein-Gordon ? Did you google that name ?
– Denis Serre
2 hours ago
1
By the way, I did my PhD thesis on this equation (plus a source term). That was in 1978.
– Denis Serre
2 hours ago
Thank you for answer, I found this equation (in this form) in a list of nonlinear pde on wikipedia en.m.wikipedia.org/wiki/… and I asked myself if it had been studied for $p = 2$
– exxxit8
1 hour ago
1
By the way, this is probably not what you are asking, but in $5+1$ dimensions, this is the classical equation of the so-called $phi^3$ theory. The resulting quantum field theory is (was?) a popular toy model because it displays asymptotic freedom, just like (nonabelian) Yang-Mills in $3+1$, which is harder to study.
– José Figueroa-O'Farrill
32 mins ago
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
The Klein Gordon equation of the form:
$Delta u+ lambda u^p=0$
is been studied for $p = 2$?
(i.e.$Delta u+ lambda u^2=0$)
If yes are there references?
ap.analysis-of-pdes
asked 4 hours ago
exxxit8
425
The Klein Gordon equation of the form:
$Delta u+ lambda u^p=0$
is been studied for $p = 2$?
(i.e.$Delta u+ lambda u^2=0$)
If yes are there references?
ap.analysis-of-pdes
ap.analysis-of-pdes
asked 4 hours ago
exxxit8
425
asked 4 hours ago
exxxit8
425
asked 4 hours ago
exxxit8
425
asked 4 hours ago
exxxit8
425
asked 4 hours ago
exxxit8
425
425
1
Klein-Gordon ? Did you google that name ?
– Denis Serre
2 hours ago
1
By the way, I did my PhD thesis on this equation (plus a source term). That was in 1978.
– Denis Serre
2 hours ago
Thank you for answer, I found this equation (in this form) in a list of nonlinear pde on wikipedia en.m.wikipedia.org/wiki/… and I asked myself if it had been studied for $p = 2$
– exxxit8
1 hour ago
1
By the way, this is probably not what you are asking, but in $5+1$ dimensions, this is the classical equation of the so-called $phi^3$ theory. The resulting quantum field theory is (was?) a popular toy model because it displays asymptotic freedom, just like (nonabelian) Yang-Mills in $3+1$, which is harder to study.
– José Figueroa-O'Farrill
32 mins ago
add a comment |Â
1
Klein-Gordon ? Did you google that name ?
– Denis Serre
2 hours ago
1
By the way, I did my PhD thesis on this equation (plus a source term). That was in 1978.
– Denis Serre
2 hours ago
Thank you for answer, I found this equation (in this form) in a list of nonlinear pde on wikipedia en.m.wikipedia.org/wiki/… and I asked myself if it had been studied for $p = 2$
– exxxit8
1 hour ago
1
By the way, this is probably not what you are asking, but in $5+1$ dimensions, this is the classical equation of the so-called $phi^3$ theory. The resulting quantum field theory is (was?) a popular toy model because it displays asymptotic freedom, just like (nonabelian) Yang-Mills in $3+1$, which is harder to study.
– José Figueroa-O'Farrill
32 mins ago
1
1
Klein-Gordon ? Did you google that name ?
– Denis Serre
2 hours ago
Klein-Gordon ? Did you google that name ?
– Denis Serre
2 hours ago
1
1
By the way, I did my PhD thesis on this equation (plus a source term). That was in 1978.
– Denis Serre
2 hours ago
By the way, I did my PhD thesis on this equation (plus a source term). That was in 1978.
– Denis Serre
2 hours ago
Thank you for answer, I found this equation (in this form) in a list of nonlinear pde on wikipedia en.m.wikipedia.org/wiki/… and I asked myself if it had been studied for $p = 2$
– exxxit8
1 hour ago
Thank you for answer, I found this equation (in this form) in a list of nonlinear pde on wikipedia en.m.wikipedia.org/wiki/… and I asked myself if it had been studied for $p = 2$
– exxxit8
1 hour ago
1
1
By the way, this is probably not what you are asking, but in $5+1$ dimensions, this is the classical equation of the so-called $phi^3$ theory. The resulting quantum field theory is (was?) a popular toy model because it displays asymptotic freedom, just like (nonabelian) Yang-Mills in $3+1$, which is harder to study.
– José Figueroa-O'Farrill
32 mins ago
By the way, this is probably not what you are asking, but in $5+1$ dimensions, this is the classical equation of the so-called $phi^3$ theory. The resulting quantum field theory is (was?) a popular toy model because it displays asymptotic freedom, just like (nonabelian) Yang-Mills in $3+1$, which is harder to study.
– José Figueroa-O'Farrill
32 mins ago
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
This is a nonlinear Klein-Gordon equation of the type
$$ Delta u = f(u) $$
which, at least in $1+1$ dimensions has been studied for several functions $f$; although the most interesting seem to be when $f$ are exponential or trigonometric, e.g., sine-Gordon equation or approximations thereof.
The case where $f(u) = u^n$ appears in §7.1.1.1 of the Handbook of nonlinear PDEs by Polyanin and Zaitsev.
@book MR2865542,
AUTHOR = Polyanin, Andrei D. and Zaitsev, Valentin F.,
TITLE = Handbook of nonlinear partial differential equations,
EDITION = Second,
PUBLISHER = CRC Press, Boca Raton, FL,
YEAR = 2012,
PAGES = xxxvi+876,
ISBN = 978-1-4200-8723-9,
MRCLASS = 35-00 (35C05),
MRNUMBER = 2865542,
answered 58 mins ago
José Figueroa-O'Farrill
25k370148
Thank you very much, your answer was very helpful
– exxxit8
17 mins ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
This is a nonlinear Klein-Gordon equation of the type
$$ Delta u = f(u) $$
which, at least in $1+1$ dimensions has been studied for several functions $f$; although the most interesting seem to be when $f$ are exponential or trigonometric, e.g., sine-Gordon equation or approximations thereof.
The case where $f(u) = u^n$ appears in §7.1.1.1 of the Handbook of nonlinear PDEs by Polyanin and Zaitsev.
@book MR2865542,
AUTHOR = Polyanin, Andrei D. and Zaitsev, Valentin F.,
TITLE = Handbook of nonlinear partial differential equations,
EDITION = Second,
PUBLISHER = CRC Press, Boca Raton, FL,
YEAR = 2012,
PAGES = xxxvi+876,
ISBN = 978-1-4200-8723-9,
MRCLASS = 35-00 (35C05),
MRNUMBER = 2865542,
answered 58 mins ago
José Figueroa-O'Farrill
25k370148
Thank you very much, your answer was very helpful
– exxxit8
17 mins ago
add a comment |Â
up vote
2
down vote
accepted
This is a nonlinear Klein-Gordon equation of the type
$$ Delta u = f(u) $$
which, at least in $1+1$ dimensions has been studied for several functions $f$; although the most interesting seem to be when $f$ are exponential or trigonometric, e.g., sine-Gordon equation or approximations thereof.
The case where $f(u) = u^n$ appears in §7.1.1.1 of the Handbook of nonlinear PDEs by Polyanin and Zaitsev.
@book MR2865542,
AUTHOR = Polyanin, Andrei D. and Zaitsev, Valentin F.,
TITLE = Handbook of nonlinear partial differential equations,
EDITION = Second,
PUBLISHER = CRC Press, Boca Raton, FL,
YEAR = 2012,
PAGES = xxxvi+876,
ISBN = 978-1-4200-8723-9,
MRCLASS = 35-00 (35C05),
MRNUMBER = 2865542,
answered 58 mins ago
José Figueroa-O'Farrill
25k370148
Thank you very much, your answer was very helpful
– exxxit8
17 mins ago
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
This is a nonlinear Klein-Gordon equation of the type
$$ Delta u = f(u) $$
which, at least in $1+1$ dimensions has been studied for several functions $f$; although the most interesting seem to be when $f$ are exponential or trigonometric, e.g., sine-Gordon equation or approximations thereof.
The case where $f(u) = u^n$ appears in §7.1.1.1 of the Handbook of nonlinear PDEs by Polyanin and Zaitsev.
@book MR2865542,
AUTHOR = Polyanin, Andrei D. and Zaitsev, Valentin F.,
TITLE = Handbook of nonlinear partial differential equations,
EDITION = Second,
PUBLISHER = CRC Press, Boca Raton, FL,
YEAR = 2012,
PAGES = xxxvi+876,
ISBN = 978-1-4200-8723-9,
MRCLASS = 35-00 (35C05),
MRNUMBER = 2865542,
answered 58 mins ago
José Figueroa-O'Farrill
25k370148
This is a nonlinear Klein-Gordon equation of the type
$$ Delta u = f(u) $$
which, at least in $1+1$ dimensions has been studied for several functions $f$; although the most interesting seem to be when $f$ are exponential or trigonometric, e.g., sine-Gordon equation or approximations thereof.
The case where $f(u) = u^n$ appears in §7.1.1.1 of the Handbook of nonlinear PDEs by Polyanin and Zaitsev.
@book MR2865542,
AUTHOR = Polyanin, Andrei D. and Zaitsev, Valentin F.,
TITLE = Handbook of nonlinear partial differential equations,
EDITION = Second,
PUBLISHER = CRC Press, Boca Raton, FL,
YEAR = 2012,
PAGES = xxxvi+876,
ISBN = 978-1-4200-8723-9,
MRCLASS = 35-00 (35C05),
MRNUMBER = 2865542,
answered 58 mins ago
José Figueroa-O'Farrill
25k370148
answered 58 mins ago
José Figueroa-O'Farrill
25k370148
answered 58 mins ago
José Figueroa-O'Farrill
25k370148
answered 58 mins ago
José Figueroa-O'Farrill
25k370148
25k370148
Thank you very much, your answer was very helpful
– exxxit8
17 mins ago
add a comment |Â
Thank you very much, your answer was very helpful
– exxxit8
17 mins ago
Thank you very much, your answer was very helpful
– exxxit8
17 mins ago
Thank you very much, your answer was very helpful
– exxxit8
17 mins ago
add a comment |Â
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1
Klein-Gordon ? Did you google that name ?
– Denis Serre
2 hours ago
1
By the way, I did my PhD thesis on this equation (plus a source term). That was in 1978.
– Denis Serre
2 hours ago
Thank you for answer, I found this equation (in this form) in a list of nonlinear pde on wikipedia en.m.wikipedia.org/wiki/… and I asked myself if it had been studied for $p = 2$
– exxxit8
1 hour ago
1
By the way, this is probably not what you are asking, but in $5+1$ dimensions, this is the classical equation of the so-called $phi^3$ theory. The resulting quantum field theory is (was?) a popular toy model because it displays asymptotic freedom, just like (nonabelian) Yang-Mills in $3+1$, which is harder to study.
– José Figueroa-O'Farrill
32 mins ago