Nonlinear Schrödinger Equation on:
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In
s
:
:
Unlike_its_linear_counterpart,_it_never_describes_the_time_evolution_of_a_quantum_state.
The_case_with_negative_κ_is_called_focusing_and_allows_for_ bright_soliton_solutions_(localized_in_space,_and_having_spatial_attenuation_towards_infinity)_as_well_as_ breather_solutions._It_can_be_solved_exactly_by_use_of_the_
_
For_complex_
Complex_commonly_refers_to:
*_Complexity,_the_behaviour_of_a_system_whose_components_interact_in_multiple_ways_so_possible_interactions_are_difficult_to_describe
**_Complex_system,_a_system_composed_of_many_components_which_may_interact_with_each__...
_field_''ψ'',_as_appearing_in_the_nonlinear_Schrödinger_equation,_is_related_to_the_amplitude_and_phase_of_the_water_waves._Consider_a_slowly_modulated_
In theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
variation of the Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose–Einstein condensate
In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67&n ...
s confined to highly anisotropic cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves
In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere ...
on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves Plasma oscillations, also known as Langmuir waves (after Irving Langmuir), are rapid oscillations of the electron density in conducting media such as plasmas or metals in the ultraviolet region. The oscillations can be described as an instability i ...
in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets
of quasi-monochromatic waves in weakly nonlinear media that have dispersion
Dispersion may refer to:
Economics and finance
*Dispersion (finance), a measure for the statistical distribution of portfolio returns
*Price dispersion, a variation in prices across sellers of the same item
*Wage dispersion, the amount of variatio ...
. Unlike the linear Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
.
In quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the 1D NLSE is a special case of the classical nonlinear Schrödinger field
In quantum mechanics and quantum field theory, a Schrödinger field, named after Erwin Schrödinger, is a quantum field which obeys the Schrödinger equation. While any situation described by a Schrödinger field can also be described by a many-bo ...
, which in turn is a classical limit of a quantum Schrödinger field. Conversely, when the classical Schrödinger field is canonically quantized, it becomes a quantum field theory (which is linear, despite the fact that it is called ″quantum ''nonlinear'' Schrödinger equation″) that describes bosonic point particles with delta-function interactions — the particles either repel or attract when they are at the same point. In fact, when the number of particles is finite, this quantum field theory is equivalent to the Lieb–Liniger model
The Lieb–Liniger model describes a gas of particles moving in one dimension and satisfying Bose–Einstein statistics.
Introduction
A model of a gas of particles moving in one dimension and satisfying Bose–Einstein statistics was introduced in ...
. Both the quantum and the classical 1D nonlinear Schrödinger equations are integrable. Of special interest is the limit of infinite strength repulsion, in which case the Lieb–Liniger model becomes the Tonks–Girardeau gas In physics, a Tonks–Girardeau gas is a Bose gas in which the repulsive interactions between bosonic particles confined to one dimension dominate the system's physics. It is named after physicists Marvin D. Girardeau and Lewi Tonks. It is not a B ...
(also called the hard-core Bose gas, or impenetrable Bose gas). In this limit, the bosons may, by a change of variables that is a continuum generalization of the Jordan–Wigner transformation
The Jordan–Wigner transformation is a transformation that maps spin operators onto fermionic creation and annihilation operators. It was proposed by Pascual Jordan and Eugene Wigner for one-dimensional lattice models, but now two-dimensional ana ...
, be transformed to a system one-dimensional noninteracting spinless fermions.
The nonlinear Schrödinger equation is a simplified 1+1-dimensional form of the Ginzburg–Landau equation introduced in 1950 in their work on superconductivity, and was written down explicitly by in their study of optical beams.
Multi-dimensional version replaces the second spatial derivative by the Laplacian. In more than one dimension, the equation is not integrable, it allows for a collapse and wave turbulence.
Equation
The nonlinear Schrödinger equation is anonlinear partial differential equation
In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear terms. They describe many different physical systems, ranging from gravitation to fluid dynamics, and have been used in mathe ...
, applicable to classical and quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
.
Classical equation
The classical field equation (indimensionless
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
form) is:. Originally in: ''Teoreticheskaya i Matematicheskaya Fizika'' 19(3): 332–343. June 1974.
for the complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
field ''ψ''(''x'',''t'').
This equation arises from the Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
:inverse_scattering_transform In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to solv ...
,_as_shown_by__(see_ below)._The_other_case,_with_κ_positive,_is_the_defocusing_NLS_which_has_ dark_soliton_solutions_(having_constant_amplitude_at_infinity,_and_a_local_spatial_dip_in_amplitude).
__Quantum_mechanics_
To_get_the_ quantized_version,_simply_replace_the_Poisson_brackets_by_commutators : and_normal_order
In quantum field theory a product of quantum fields, or equivalently their creation and annihilation operators, is usually said to be normal ordered (also called Wick order) when all creation operators are to the left of all annihilation operator ...
_the_Hamiltonian
:
The_quantum_version_was__solved_by_ Bethe_ansatz_by_ Lieb_and_Liniger._Thermodynamics_was_described_by_Chen-Ning_Yang
Yang Chen-Ning or Chen-Ning Yang (; born 1 October 1922), also known as C. N. Yang or by the English name Frank Yang, is a Chinese Theoretical physics, theoretical physicist who made significant contributions to statistical mechanics, integrab ...
._Quantum_correlation_functions_also_were_evaluated_by_Korepin_in_1993.__Solving_the_equation_
The_nonlinear_Schrödinger_equation_is_integrable_in_1d:__solved_it_with_the_inverse_scattering_transform In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to solv ...
._The_corresponding_linear_system_of_equations_is_known_as_the_ Zakharov–Shabat_system:
:
where
:
The_nonlinear_Schrödinger_equation_arises_as_compatibility_condition_of_the_Zakharov–Shabat_system:
:
By_setting_''q''_=_''r''*_or_''q''_=_−_''r''*_the_nonlinear_Schrödinger_equation_with_attractive_or_repulsive_interaction_is_obtained.
An_alternative_approach_uses_the_Zakharov–Shabat_system_directly_and_employs__the_following_Darboux_transformation Darboux is a surname. Notable people with the surname include:
* Jean Gaston Darboux (1842–1917), French mathematician
* Lauriane Doumbouya (née Darboux), the current First Lady of Guinea since 5 September 2021
* Paul Darboux (1919–1982), ...
:
:
which_leaves_the_system_invariant.
Here,_''φ''_is_another_invertible_matrix_solution_(different_from_''ϕ'')_of_the_Zakharov–Shabat_system_with_spectral_parameter_Ω:
:
Starting_from_the_trivial_solution_''U''_=_0_and_iterating,_one_obtains_the_solutions_with_''n''_soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium ...
s.
The_NLS_equation_is_a_partial_differential_equation_like_the_Gross–Pitaevskii_equation The Gross–Pitaevskii equation (GPE, named after Eugene P. Gross and Lev Petrovich Pitaevskii) describes the ground state of a quantum system of identical bosons using the Hartree–Fock approximation and the pseudopotential interaction model.
A ...
.___Usually_it__does_not_have_analytic_solution_and_the_same_numerical_methods_used_to_solve_the_Gross–Pitaevskii_equation,_such_as_the_split-step_ Crank–Nicolson_and_ Fourier_spectral_methods,_are_used_for_its_solution.__There_are_different_Fortran_and_C_programs_for_its_solution.
__Galilean_invariance_
The_nonlinear_Schrödinger_equation_is_ Galilean_invariant_in_the_following_sense: Given_a_solution_''ψ''(''x,_t'')_a_new_solution_can_be_obtained_by_replacing_''x''_with_''x''_+_''vt''_everywhere_in_ψ(''x,_t'')_and_by_appending_a_phase_factor_of_: :__The_nonlinear_Schrödinger_equation_in_fiber_optics_
In_optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
,_the_nonlinear_Schrödinger_equation_occurs_in_the_ Manakov_system,_a_model_of_wave_propagation_in_fiber_optics._The_function_ψ_represents_a_wave_and_the_nonlinear_Schrödinger_equation_describes_the_propagation_of_the_wave_through_a_nonlinear_medium._The_second-order_derivative_represents_the_dispersion,_while_the_''κ''_term_represents_the_nonlinearity._The_equation_models_many_nonlinearity_effects_in_a_fiber,_including_but_not_limited_to_self-phase_modulation Self-phase modulation (SPM) is a nonlinear optical effect of light–matter interaction.
An ultrashort pulse of light, when travelling in a medium, will induce a varying refractive index of the medium due to the optical Kerr effect. This variatio ...
,_four-wave_mixing Four-wave mixing (FWM) is an intermodulation phenomenon in nonlinear optics, whereby interactions between two or three wavelengths produce two or one new wavelengths. It is similar to the third-order intercept point in electrical systems. Four-wave ...
,_second-harmonic_generation
Second-harmonic generation (SHG, also called frequency doubling) is a nonlinear optical process in which two photons with the same frequency interact with a nonlinear material, are "combined", and generate a new photon with twice the energy of ...
,_stimulated_Raman_scattering
Raman scattering or the Raman effect () is the inelastic scattering of photons by matter, meaning that there is both an exchange of energy and a change in the light's direction. Typically this effect involves vibrational energy being gained by a ...
,_optical_solitons
In optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and linear effects in the medium. There are two main kinds of solitons:
* spatial solitons: th ...
,
ultrashort_pulses,_etc.
__The_nonlinear_Schrödinger_equation_in_water_waves_
water_waves
In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of ...
,_the_nonlinear_Schrödinger_equation_describes_the_evolution_of_the_envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a sh ...
_of_modulated
In electronics and telecommunications, modulation is the process of varying one or more properties of a periodic waveform, called the ''carrier signal'', with a separate signal called the ''modulation signal'' that typically contains informatio ...
_wave_groups._In_a_paper_in_1968,_ Vladimir_E._Zakharov_describes_the_Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
_structure_of_water_waves._In_the_same_paper_Zakharov_shows,_that_for_slowly_modulated_wave_groups,_the_wave_amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
_satisfies_the_nonlinear_Schrödinger_equation,_approximately._The_value_of_the_nonlinearity_parameter_''к''_depends_on_the_relative_water_depth._For_deep_water,_with_the_water_depth_large_compared_to_the_wave_length
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
_of_the_water_waves,_''к''_is_negative_and_envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a sh ...
_soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium ...
s_may_occur._Furthermore,_these_envelope_solitons_may_be_accelerated_under_external_time_dependent_water_flow.
For_shallow_water,_with_wavelengths_longer_than_4.6_times_the_water_depth,_the_nonlinearity_parameter_''к''_is_positive_and_''wave_groups''_with_''envelope''_solitons_do_not_exist._In_shallow_water_''surface-elevation''_solitons_or_ waves_of_translation_do_exist,_but_they_are_not_governed_by_the_nonlinear_Schrödinger_equation.
The_nonlinear_Schrödinger_equation_is_thought_to_be_important_for_explaining_the_formation_of_rogue_wave
Rogue waves (also known as freak waves, monster waves, episodic waves, killer waves, extreme waves, and abnormal waves) are unusually large, unpredictable, and suddenly appearing surface waves that can be extremely dangerous to ships, even to lar ...
s.
The_carrier_wave
In telecommunications, a carrier wave, carrier signal, or just carrier, is a waveform (usually sinusoidal) that is modulated (modified) with an information-bearing signal for the purpose of conveying information. This carrier wave usually has a ...
_with_water_surface_elevation
The elevation of a geographic location is its height above or below a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface (see Geodetic datum § Vert ...
_''η''_of_the_form:
:
where_''a''(''x''0,_''t''0)_and_''θ''(''x''0,_''t''0)_are_the_slowly_modulated_amplitude_and_phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
._Further_''ω''0_and_''k''0_are_the_(constant)_angular_frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
_and_wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
_of_the_carrier_waves,_which_have_to_satisfy_the_dispersion
Dispersion may refer to:
Economics and finance
*Dispersion (finance), a measure for the statistical distribution of portfolio returns
*Price dispersion, a variation in prices across sellers of the same item
*Wage dispersion, the amount of variatio ...
_relation_''ω''0_=_Ω(''k''0)._Then
:
So_its_ modulus_, ''ψ'', _is_the_wave_amplitude_''a'',_and_its_argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
_arg(''ψ'')_is_the_phase_''θ''.
The_relation_between_the_physical_coordinates_(''x''0,_''t''0)_and_the_(''x,_t'')_coordinates,_as_used_in_the_ nonlinear_Schrödinger_equation_given_above,_is_given_by:
:
Thus_(''x,_t'')_is_a_transformed_coordinate_system_moving_with_the_group_velocity
The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space.
For example, if a stone is thrown into the middl ...
_Ω'(''k''0)_of_the_carrier_waves,
The_dispersion-relation_curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
_Ω"(''k''0)_–_representing_group_velocity_dispersion
In optics, group velocity dispersion (GVD) is a characteristic of a dispersive medium, used most often to determine how the medium will affect the duration of an optical pulse traveling through it. Formally, GVD is defined as the derivative of the ...
_–_is_always_negative_for_water_waves_under_the_action_of_gravity,_for_any_water_depth.
For_waves_on_the_water_surface_of_deep_water,_the_coefficients_of_importance_for_the_nonlinear_Schrödinger_equation_are:
:__so__
where_''g''_is_the_ acceleration_due_to_gravity_at_the_Earth's_surface.
In_the_original_(''x''0,_''t''0)_coordinates_the_nonlinear_Schrödinger_equation_for_water_waves_reads:
:
with__(i.e._the_complex_conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
_of_)_and__So__for_deep_water_waves.
__Gauge_equivalent_counterpart_
NLSE_(1)_is_gauge_equivalent_to__the_following_isotropic_ Landau-Lifshitz_equation_(LLE)_or_Heisenberg_ferromagnet
A spin wave is a propagating disturbance in the ordering of a magnetic material. These low-lying collective excitations occur in magnetic lattices with continuous symmetry. From the equivalent quasiparticle point of view, spin waves are known as ...
_equation
:
Note_that_this_equation__admits_several_integrable_and_non-integrable_generalizations_in_2_+_1_dimensions_like_the_Ishimori_equation The Ishimori equation is a partial differential equation proposed by the Japanese mathematician . Its interest is as the first example of a nonlinear spin-one field model in the plane that is integrable .
Equation
The Ishimori equation has the for ...
_and_so_on.
__Relation_to_vortices_
_showed_that_the_work_of__on_vortex_filaments_is_closely_related_to_the_nonlinear_Schrödinger_equation._Subsequently,__used_this_correspondence_to_show_that_breather_solutions_can_also_arise_for_a_vortex_filament.__See_also_
*_ AKNS_system *_Eckhaus_equation In mathematical physics, the Eckhaus equation – or the Kundu–Eckhaus equation – is a nonlinear partial differential equation within the nonlinear Schrödinger class:
:i \psi_t + \psi_ +2 \left( , \psi, ^2 \right)_x\, \psi + , \psi, ^4\, \psi ...
*_Quartic_interaction
In quantum field theory, a quartic interaction is a type of self-interaction in a scalar field. Other types of quartic interactions may be found under the topic of four-fermion interactions. A classical free scalar field \varphi satisfies the Kle ...
_for_a_related_model_in_quantum_field_theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
*_Soliton_(optics)
In optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and linear effects in the medium. There are two main kinds of solitons:
* spatial solitons: th ...
*_Logarithmic_Schrödinger_equation In theoretical physics, the logarithmic Schrödinger equation (sometimes abbreviated as LNSE or LogSE) is one of the nonlinear modifications of Schrödinger's equation. It is a classical wave equation with applications to extensions of quantum mech ...
__Notes_
__References_
__Notes_
__Other_
* * * * *__External_links_
*at_EqWorld:_The_World_of_Mathematical_Equations.
at_EqWorld:_The_World_of_Mathematical_Equations.
at_EqWorld:_The_World_of_Mathematical_Equations.
at_Dispersive_Wiki {{DEFAULTSORT:Nonlinear_Schrodinger_equation Partial_differential_equations Exactly_solvable_models Schrödinger_equation Integrable_systemshtml" ;"title="\partial_x\psi, ^2+, \psi, ^4\right] with the
Poisson bracket
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ...
s
:
:
Unlike its linear counterpart, it never describes the time evolution of a quantum state.
The case with negative κ is called focusing and allows for bright soliton solutions (localized in space, and having spatial attenuation towards infinity) as well as breather solutions. It can be solved exactly by use of the inverse scattering transform In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to solv ...
, as shown by (see below). The other case, with κ positive, is the defocusing NLS which has dark soliton solutions (having constant amplitude at infinity, and a local spatial dip in amplitude).
Quantum mechanics
To get the quantized version, simply replace the Poisson brackets by commutators : andnormal order
In quantum field theory a product of quantum fields, or equivalently their creation and annihilation operators, is usually said to be normal ordered (also called Wick order) when all creation operators are to the left of all annihilation operator ...
the Hamiltonian
:
The quantum version was solved by Bethe ansatz by Lieb and Liniger. Thermodynamics was described by Chen-Ning Yang
Yang Chen-Ning or Chen-Ning Yang (; born 1 October 1922), also known as C. N. Yang or by the English name Frank Yang, is a Chinese Theoretical physics, theoretical physicist who made significant contributions to statistical mechanics, integrab ...
. Quantum correlation functions also were evaluated by Korepin in 1993. The model has higher conservation laws - Davies and Korepin in 1989 expressed them in terms of local fields.
Solving the equation
The nonlinear Schrödinger equation is integrable in 1d: solved it with theinverse scattering transform In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to solv ...
. The corresponding linear system of equations is known as the Zakharov–Shabat system:
:
where
:
The nonlinear Schrödinger equation arises as compatibility condition of the Zakharov–Shabat system:
:
By setting ''q'' = ''r''* or ''q'' = − ''r''* the nonlinear Schrödinger equation with attractive or repulsive interaction is obtained.
An alternative approach uses the Zakharov–Shabat system directly and employs the following Darboux transformation Darboux is a surname. Notable people with the surname include:
* Jean Gaston Darboux (1842–1917), French mathematician
* Lauriane Doumbouya (née Darboux), the current First Lady of Guinea since 5 September 2021
* Paul Darboux (1919–1982), ...
:
:
which leaves the system invariant.
Here, ''φ'' is another invertible matrix solution (different from ''ϕ'') of the Zakharov–Shabat system with spectral parameter Ω:
:
Starting from the trivial solution ''U'' = 0 and iterating, one obtains the solutions with ''n'' soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium ...
s.
The NLS equation is a partial differential equation like the Gross–Pitaevskii equation The Gross–Pitaevskii equation (GPE, named after Eugene P. Gross and Lev Petrovich Pitaevskii) describes the ground state of a quantum system of identical bosons using the Hartree–Fock approximation and the pseudopotential interaction model.
A ...
. Usually it does not have analytic solution and the same numerical methods used to solve the Gross–Pitaevskii equation, such as the split-step Crank–Nicolson and Fourier spectral methods, are used for its solution. There are different Fortran and C programs for its solution.
Galilean invariance
The nonlinear Schrödinger equation is Galilean invariant in the following sense: Given a solution ''ψ''(''x, t'') a new solution can be obtained by replacing ''x'' with ''x'' + ''vt'' everywhere in ψ(''x, t'') and by appending a phase factor of : :The nonlinear Schrödinger equation in fiber optics
Inoptics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
, the nonlinear Schrödinger equation occurs in the Manakov system, a model of wave propagation in fiber optics. The function ψ represents a wave and the nonlinear Schrödinger equation describes the propagation of the wave through a nonlinear medium. The second-order derivative represents the dispersion, while the ''κ'' term represents the nonlinearity. The equation models many nonlinearity effects in a fiber, including but not limited to self-phase modulation Self-phase modulation (SPM) is a nonlinear optical effect of light–matter interaction.
An ultrashort pulse of light, when travelling in a medium, will induce a varying refractive index of the medium due to the optical Kerr effect. This variatio ...
, four-wave mixing Four-wave mixing (FWM) is an intermodulation phenomenon in nonlinear optics, whereby interactions between two or three wavelengths produce two or one new wavelengths. It is similar to the third-order intercept point in electrical systems. Four-wave ...
, second-harmonic generation
Second-harmonic generation (SHG, also called frequency doubling) is a nonlinear optical process in which two photons with the same frequency interact with a nonlinear material, are "combined", and generate a new photon with twice the energy of ...
, stimulated Raman scattering
Raman scattering or the Raman effect () is the inelastic scattering of photons by matter, meaning that there is both an exchange of energy and a change in the light's direction. Typically this effect involves vibrational energy being gained by a ...
, optical solitons
In optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and linear effects in the medium. There are two main kinds of solitons:
* spatial solitons: th ...
,
ultrashort pulses, etc.
The nonlinear Schrödinger equation in water waves
For water waves
In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of ...
, the nonlinear Schrödinger equation describes the evolution of the envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a sh ...
of modulated
In electronics and telecommunications, modulation is the process of varying one or more properties of a periodic waveform, called the ''carrier signal'', with a separate signal called the ''modulation signal'' that typically contains informatio ...
wave groups. In a paper in 1968, Vladimir E. Zakharov describes the Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
structure of water waves. In the same paper Zakharov shows, that for slowly modulated wave groups, the wave amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
satisfies the nonlinear Schrödinger equation, approximately. The value of the nonlinearity parameter ''к'' depends on the relative water depth. For deep water, with the water depth large compared to the wave length
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
of the water waves, ''к'' is negative and envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a sh ...
soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium ...
s may occur. Furthermore, these envelope solitons may be accelerated under external time dependent water flow.
For shallow water, with wavelengths longer than 4.6 times the water depth, the nonlinearity parameter ''к'' is positive and ''wave groups'' with ''envelope'' solitons do not exist. In shallow water ''surface-elevation'' solitons or waves of translation do exist, but they are not governed by the nonlinear Schrödinger equation.
The nonlinear Schrödinger equation is thought to be important for explaining the formation of rogue wave
Rogue waves (also known as freak waves, monster waves, episodic waves, killer waves, extreme waves, and abnormal waves) are unusually large, unpredictable, and suddenly appearing surface waves that can be extremely dangerous to ships, even to lar ...
s.
The complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
field ''ψ'', as appearing in the nonlinear Schrödinger equation, is related to the amplitude and phase of the water waves. Consider a slowly modulated carrier wave
In telecommunications, a carrier wave, carrier signal, or just carrier, is a waveform (usually sinusoidal) that is modulated (modified) with an information-bearing signal for the purpose of conveying information. This carrier wave usually has a ...
with water surface elevation
The elevation of a geographic location is its height above or below a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface (see Geodetic datum § Vert ...
''η'' of the form:
:
where ''a''(''x''0, ''t''0) and ''θ''(''x''0, ''t''0) are the slowly modulated amplitude and phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
. Further ''ω''0 and ''k''0 are the (constant) angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
and wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
of the carrier waves, which have to satisfy the dispersion
Dispersion may refer to:
Economics and finance
*Dispersion (finance), a measure for the statistical distribution of portfolio returns
*Price dispersion, a variation in prices across sellers of the same item
*Wage dispersion, the amount of variatio ...
relation ''ω''0 = Ω(''k''0). Then
:
So its modulus , ''ψ'', is the wave amplitude ''a'', and its argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
arg(''ψ'') is the phase ''θ''.
The relation between the physical coordinates (''x''0, ''t''0) and the (''x, t'') coordinates, as used in the nonlinear Schrödinger equation given above, is given by:
:
Thus (''x, t'') is a transformed coordinate system moving with the group velocity
The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space.
For example, if a stone is thrown into the middl ...
Ω'(''k''0) of the carrier waves,
The dispersion-relation curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
Ω"(''k''0) – representing group velocity dispersion
In optics, group velocity dispersion (GVD) is a characteristic of a dispersive medium, used most often to determine how the medium will affect the duration of an optical pulse traveling through it. Formally, GVD is defined as the derivative of the ...
– is always negative for water waves under the action of gravity, for any water depth.
For waves on the water surface of deep water, the coefficients of importance for the nonlinear Schrödinger equation are:
: so
where ''g'' is the acceleration due to gravity at the Earth's surface.
In the original (''x''0, ''t''0) coordinates the nonlinear Schrödinger equation for water waves reads:
:
with (i.e. the complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of ) and So for deep water waves.
Gauge equivalent counterpart
NLSE (1) is gauge equivalent to the following isotropic Landau-Lifshitz equation (LLE) orHeisenberg ferromagnet
A spin wave is a propagating disturbance in the ordering of a magnetic material. These low-lying collective excitations occur in magnetic lattices with continuous symmetry. From the equivalent quasiparticle point of view, spin waves are known as ...
equation
:
Note that this equation admits several integrable and non-integrable generalizations in 2 + 1 dimensions like the Ishimori equation The Ishimori equation is a partial differential equation proposed by the Japanese mathematician . Its interest is as the first example of a nonlinear spin-one field model in the plane that is integrable .
Equation
The Ishimori equation has the for ...
and so on.
Relation to vortices
showed that the work of on vortex filaments is closely related to the nonlinear Schrödinger equation. Subsequently, used this correspondence to show that breather solutions can also arise for a vortex filament.See also
* AKNS system *Eckhaus equation In mathematical physics, the Eckhaus equation – or the Kundu–Eckhaus equation – is a nonlinear partial differential equation within the nonlinear Schrödinger class:
:i \psi_t + \psi_ +2 \left( , \psi, ^2 \right)_x\, \psi + , \psi, ^4\, \psi ...
* Quartic interaction
In quantum field theory, a quartic interaction is a type of self-interaction in a scalar field. Other types of quartic interactions may be found under the topic of four-fermion interactions. A classical free scalar field \varphi satisfies the Kle ...
for a related model in quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
* Soliton (optics)
In optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and linear effects in the medium. There are two main kinds of solitons:
* spatial solitons: th ...
* Logarithmic Schrödinger equation In theoretical physics, the logarithmic Schrödinger equation (sometimes abbreviated as LNSE or LogSE) is one of the nonlinear modifications of Schrödinger's equation. It is a classical wave equation with applications to extensions of quantum mech ...
Notes
References
Notes
Other
* * * * *External links
*Tutorial lecture on Nonlinear Schrodinger Equation (video)
Nonlinear Schrodinger Equation with a Cubic Nonlinearity
at EqWorld: The World of Mathematical Equations.
Nonlinear Schrodinger Equation with a Power-Law Nonlinearity
at EqWorld: The World of Mathematical Equations.
Nonlinear Schrodinger Equation of General Form
at EqWorld: The World of Mathematical Equations.
Mathematical aspects of the nonlinear Schrödinger equation
at Dispersive Wiki {{DEFAULTSORT:Nonlinear Schrodinger equation Partial differential equations Exactly solvable models Schrödinger equation Integrable systems>\partial_x\psi, ^2+, \psi, ^4\right/math> with the
Poisson bracket
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ...
s
:
:
Unlike its linear counterpart, it never describes the time evolution of a quantum state.
The case with negative κ is called focusing and allows for bright soliton solutions (localized in space, and having spatial attenuation towards infinity) as well as breather solutions. It can be solved exactly by use of the inverse scattering transform In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to solv ...
, as shown by (see below). The other case, with κ positive, is the defocusing NLS which has dark soliton solutions (having constant amplitude at infinity, and a local spatial dip in amplitude).
Quantum mechanics
To get the quantized version, simply replace the Poisson brackets by commutators : andnormal order
In quantum field theory a product of quantum fields, or equivalently their creation and annihilation operators, is usually said to be normal ordered (also called Wick order) when all creation operators are to the left of all annihilation operator ...
the Hamiltonian
:
The quantum version was solved by Bethe ansatz by Lieb and Liniger. Thermodynamics was described by Chen-Ning Yang
Yang Chen-Ning or Chen-Ning Yang (; born 1 October 1922), also known as C. N. Yang or by the English name Frank Yang, is a Chinese Theoretical physics, theoretical physicist who made significant contributions to statistical mechanics, integrab ...
. Quantum correlation functions also were evaluated by Korepin in 1993. The model has higher conservation laws - Davies and Korepin in 1989 expressed them in terms of local fields.
Solving the equation
The nonlinear Schrödinger equation is integrable in 1d: solved it with theinverse scattering transform In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to solv ...
. The corresponding linear system of equations is known as the Zakharov–Shabat system:
:
where
:
The nonlinear Schrödinger equation arises as compatibility condition of the Zakharov–Shabat system:
:
By setting ''q'' = ''r''* or ''q'' = − ''r''* the nonlinear Schrödinger equation with attractive or repulsive interaction is obtained.
An alternative approach uses the Zakharov–Shabat system directly and employs the following Darboux transformation Darboux is a surname. Notable people with the surname include:
* Jean Gaston Darboux (1842–1917), French mathematician
* Lauriane Doumbouya (née Darboux), the current First Lady of Guinea since 5 September 2021
* Paul Darboux (1919–1982), ...
:
:
which leaves the system invariant.
Here, ''φ'' is another invertible matrix solution (different from ''ϕ'') of the Zakharov–Shabat system with spectral parameter Ω:
:
Starting from the trivial solution ''U'' = 0 and iterating, one obtains the solutions with ''n'' soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium ...
s.
The NLS equation is a partial differential equation like the Gross–Pitaevskii equation The Gross–Pitaevskii equation (GPE, named after Eugene P. Gross and Lev Petrovich Pitaevskii) describes the ground state of a quantum system of identical bosons using the Hartree–Fock approximation and the pseudopotential interaction model.
A ...
. Usually it does not have analytic solution and the same numerical methods used to solve the Gross–Pitaevskii equation, such as the split-step Crank–Nicolson and Fourier spectral methods, are used for its solution. There are different Fortran and C programs for its solution.
Galilean invariance
The nonlinear Schrödinger equation is Galilean invariant in the following sense: Given a solution ''ψ''(''x, t'') a new solution can be obtained by replacing ''x'' with ''x'' + ''vt'' everywhere in ψ(''x, t'') and by appending a phase factor of : :The nonlinear Schrödinger equation in fiber optics
Inoptics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
, the nonlinear Schrödinger equation occurs in the Manakov system, a model of wave propagation in fiber optics. The function ψ represents a wave and the nonlinear Schrödinger equation describes the propagation of the wave through a nonlinear medium. The second-order derivative represents the dispersion, while the ''κ'' term represents the nonlinearity. The equation models many nonlinearity effects in a fiber, including but not limited to self-phase modulation Self-phase modulation (SPM) is a nonlinear optical effect of light–matter interaction.
An ultrashort pulse of light, when travelling in a medium, will induce a varying refractive index of the medium due to the optical Kerr effect. This variatio ...
, four-wave mixing Four-wave mixing (FWM) is an intermodulation phenomenon in nonlinear optics, whereby interactions between two or three wavelengths produce two or one new wavelengths. It is similar to the third-order intercept point in electrical systems. Four-wave ...
, second-harmonic generation
Second-harmonic generation (SHG, also called frequency doubling) is a nonlinear optical process in which two photons with the same frequency interact with a nonlinear material, are "combined", and generate a new photon with twice the energy of ...
, stimulated Raman scattering
Raman scattering or the Raman effect () is the inelastic scattering of photons by matter, meaning that there is both an exchange of energy and a change in the light's direction. Typically this effect involves vibrational energy being gained by a ...
, optical solitons
In optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and linear effects in the medium. There are two main kinds of solitons:
* spatial solitons: th ...
,
ultrashort pulses, etc.
The nonlinear Schrödinger equation in water waves
For water waves
In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of ...
, the nonlinear Schrödinger equation describes the evolution of the envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a sh ...
of modulated
In electronics and telecommunications, modulation is the process of varying one or more properties of a periodic waveform, called the ''carrier signal'', with a separate signal called the ''modulation signal'' that typically contains informatio ...
wave groups. In a paper in 1968, Vladimir E. Zakharov describes the Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
structure of water waves. In the same paper Zakharov shows, that for slowly modulated wave groups, the wave amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
satisfies the nonlinear Schrödinger equation, approximately. The value of the nonlinearity parameter ''к'' depends on the relative water depth. For deep water, with the water depth large compared to the wave length
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
of the water waves, ''к'' is negative and envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a sh ...
soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium ...
s may occur. Furthermore, these envelope solitons may be accelerated under external time dependent water flow.
For shallow water, with wavelengths longer than 4.6 times the water depth, the nonlinearity parameter ''к'' is positive and ''wave groups'' with ''envelope'' solitons do not exist. In shallow water ''surface-elevation'' solitons or waves of translation do exist, but they are not governed by the nonlinear Schrödinger equation.
The nonlinear Schrödinger equation is thought to be important for explaining the formation of rogue wave
Rogue waves (also known as freak waves, monster waves, episodic waves, killer waves, extreme waves, and abnormal waves) are unusually large, unpredictable, and suddenly appearing surface waves that can be extremely dangerous to ships, even to lar ...
s.
The complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
field ''ψ'', as appearing in the nonlinear Schrödinger equation, is related to the amplitude and phase of the water waves. Consider a slowly modulated carrier wave
In telecommunications, a carrier wave, carrier signal, or just carrier, is a waveform (usually sinusoidal) that is modulated (modified) with an information-bearing signal for the purpose of conveying information. This carrier wave usually has a ...
with water surface elevation
The elevation of a geographic location is its height above or below a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface (see Geodetic datum § Vert ...
''η'' of the form:
:
where ''a''(''x''0, ''t''0) and ''θ''(''x''0, ''t''0) are the slowly modulated amplitude and phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
. Further ''ω''0 and ''k''0 are the (constant) angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
and wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
of the carrier waves, which have to satisfy the dispersion
Dispersion may refer to:
Economics and finance
*Dispersion (finance), a measure for the statistical distribution of portfolio returns
*Price dispersion, a variation in prices across sellers of the same item
*Wage dispersion, the amount of variatio ...
relation ''ω''0 = Ω(''k''0). Then
:
So its modulus , ''ψ'', is the wave amplitude ''a'', and its argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
arg(''ψ'') is the phase ''θ''.
The relation between the physical coordinates (''x''0, ''t''0) and the (''x, t'') coordinates, as used in the nonlinear Schrödinger equation given above, is given by:
:
Thus (''x, t'') is a transformed coordinate system moving with the group velocity
The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space.
For example, if a stone is thrown into the middl ...
Ω'(''k''0) of the carrier waves,
The dispersion-relation curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
Ω"(''k''0) – representing group velocity dispersion
In optics, group velocity dispersion (GVD) is a characteristic of a dispersive medium, used most often to determine how the medium will affect the duration of an optical pulse traveling through it. Formally, GVD is defined as the derivative of the ...
– is always negative for water waves under the action of gravity, for any water depth.
For waves on the water surface of deep water, the coefficients of importance for the nonlinear Schrödinger equation are:
: so
where ''g'' is the acceleration due to gravity at the Earth's surface.
In the original (''x''0, ''t''0) coordinates the nonlinear Schrödinger equation for water waves reads:
:
with (i.e. the complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of ) and So for deep water waves.
Gauge equivalent counterpart
NLSE (1) is gauge equivalent to the following isotropic Landau-Lifshitz equation (LLE) orHeisenberg ferromagnet
A spin wave is a propagating disturbance in the ordering of a magnetic material. These low-lying collective excitations occur in magnetic lattices with continuous symmetry. From the equivalent quasiparticle point of view, spin waves are known as ...
equation
:
Note that this equation admits several integrable and non-integrable generalizations in 2 + 1 dimensions like the Ishimori equation The Ishimori equation is a partial differential equation proposed by the Japanese mathematician . Its interest is as the first example of a nonlinear spin-one field model in the plane that is integrable .
Equation
The Ishimori equation has the for ...
and so on.
Relation to vortices
showed that the work of on vortex filaments is closely related to the nonlinear Schrödinger equation. Subsequently, used this correspondence to show that breather solutions can also arise for a vortex filament.See also
* AKNS system *Eckhaus equation In mathematical physics, the Eckhaus equation – or the Kundu–Eckhaus equation – is a nonlinear partial differential equation within the nonlinear Schrödinger class:
:i \psi_t + \psi_ +2 \left( , \psi, ^2 \right)_x\, \psi + , \psi, ^4\, \psi ...
* Quartic interaction
In quantum field theory, a quartic interaction is a type of self-interaction in a scalar field. Other types of quartic interactions may be found under the topic of four-fermion interactions. A classical free scalar field \varphi satisfies the Kle ...
for a related model in quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
* Soliton (optics)
In optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and linear effects in the medium. There are two main kinds of solitons:
* spatial solitons: th ...
* Logarithmic Schrödinger equation In theoretical physics, the logarithmic Schrödinger equation (sometimes abbreviated as LNSE or LogSE) is one of the nonlinear modifications of Schrödinger's equation. It is a classical wave equation with applications to extensions of quantum mech ...
Notes
References
Notes
Other
* * * * *External links
*Tutorial lecture on Nonlinear Schrodinger Equation (video)
Nonlinear Schrodinger Equation with a Cubic Nonlinearity
at EqWorld: The World of Mathematical Equations.
Nonlinear Schrodinger Equation with a Power-Law Nonlinearity
at EqWorld: The World of Mathematical Equations.
Nonlinear Schrodinger Equation of General Form
at EqWorld: The World of Mathematical Equations.
Mathematical aspects of the nonlinear Schrödinger equation
at Dispersive Wiki {{DEFAULTSORT:Nonlinear Schrodinger equation Partial differential equations Exactly solvable models Schrödinger equation Integrable systems