Disruptive innovation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Disruptive_innovation 

 

An 1880 penny-farthing (left), and a 1886 Rover safety bicycle with gearing

Christensen's Types of Innovation

Sustaining An innovation that does not significantly affect existing markets. It may be either: Evolutionary An innovation that improves a product in an existing market in ways that customers are expecting (e.g., fuel injection for gasoline engines, which displaced carburetors.) Revolutionary (discontinuous but sustaining) An innovation that is unexpected, but nevertheless does not affect existing markets (e.g., the first automobiles in the late 19th century, which were expensive luxury items, and as such very few were sold) Disruptive An innovation that creates a new market or enters at the bottom of an existing market by providing a different set of values, which ultimately (and unexpectedly) overtakes incumbents (e.g., the lower-priced, affordable Ford Model T, which displaced horse-drawn carriages)

In business theory, disruptive innovation is innovation that creates a new market and value network or enters at the bottom of an existing market and eventually displaces established market-leading firms, products, and alliances. The concept was developed by the American academic Clayton Christensen and his collaborators beginning in 1995, and has been called the most influential business idea of the early 21st century. Lingfei Wu, Dashun Wang, and James A. Evans generalized this term to identify disruptive science and technological advances from more than 65 million papers, patents and software products that span the period 1954–2014. Their work was featured as the cover of the February 2019 issue of Nature and was selected as the Altmetric 100 most-discussed work in 2019.

Not all innovations are disruptive, even if they are revolutionary. For example, the first automobiles in the late 19th century were not a disruptive innovation, because early automobiles were expensive luxury items that did not disrupt the market for horse-drawn vehicles. The market for transportation essentially remained intact until the debut of the lower-priced Ford Model T in 1908. The mass-produced automobile was a disruptive innovation, because it changed the transportation market, whereas the first thirty years of automobiles did not.

Disruptive innovations tend to be produced by outsiders and entrepreneurs in startups, rather than existing market-leading companies. The business environment of market leaders does not allow them to pursue disruptive innovations when they first arise, because they are not profitable enough at first and because their development can take scarce resources away from sustaining innovations (which are needed to compete against current competition). Small teams are more likely to create disruptive innovations than large teams. A disruptive process can take longer to develop than by the conventional approach and the risk associated to it is higher than the other more incremental, architectural or evolutionary forms of innovations, but once it is deployed in the market, it achieves a much faster penetration and higher degree of impact on the established markets.

Beyond business and economics disruptive innovations can also be considered to disrupt complex systems, including economic and business-related aspects. Through identifying and analyzing systems for possible points of intervention, one can then design changes focused on disruptive interventions.

History and usage of the term

The term disruptive technologies was coined by Clayton M. Christensen and introduced in his 1995 article Disruptive Technologies: Catching the Wave, which he cowrote with Joseph Bower. The article is aimed at both management executives who make the funding or purchasing decisions in companies, as well as the research community, which is largely responsible for introducing the disruptive vector to the consumer market. He describes the term further in his book The Innovator's Dilemma. Innovator's Dilemma explored the case of the disk drive industry (the disk drive and memory industry, with its rapid technological evolution, is to the study of technology what fruit flies are to the study of genetics, as Christensen was told in the 1990s) and the excavating and Earth-moving industry (where hydraulic actuation slowly, yet eventually, displaced cable-actuated machinery). In his sequel with Michael E. Raynor, The Innovator's Solution, Christensen replaced the term disruptive technology with disruptive innovation because he recognized that most technologies are not intrinsically disruptive or sustaining in character; rather, it is the business model that identifies the crucial idea that potentiates profound market success and subsequently serves as the disruptive vector. However, comprehending Christensen's business model, which takes the disruptive vector from the idea borne from the mind of the innovator to a marketable product, is central to understanding how novel technology facilitates the rapid destruction of established technologies and markets by the disruptor. Christensen and Mark W. Johnson, who cofounded the management consulting firm Innosight, described the dynamics of "business model innovation" in the 2008 Harvard Business Review article "Reinventing Your Business Model". The concept of disruptive technology continues a long tradition of identifying radical technological change in the study of innovation by economists, and its implementation and execution by its management at a corporate or policy level.

According to Christensen, "the term 'disruptive innovation' is misleading when it is used to refer to the derivative, or 'instantaneous value', of the market behavior of the product or service, rather than the integral, or 'sum over histories', of the product's market behavior."

In the late 1990s, the automotive sector began to embrace a perspective of "constructive disruptive technology" by working with the consultant David E. O'Ryan, whereby the use of current off-the-shelf technology was integrated with newer innovation to create what he called "an unfair advantage". The process or technology change as a whole had to be "constructive" in improving the current method of manufacturing, yet disruptively impact the whole of the business case model, resulting in a significant reduction of waste, energy, materials, labor, or legacy costs to the user.

In keeping with the insight that a persuasive advertising campaign can be just as effective as technological sophistication at bringing a successful product to market, Christensen's theory explains why many disruptive innovations are not advanced or useful technologies, which a default hypothesis would lead one to expect. Rather, they are often combinations of existing off-the-shelf components, applied shrewdly to a small, fledgling value network.

Online news site TechRepublic proposes an end using the term, and similar related terms, suggesting that, as of 2014, it is overused jargon.

What is (isn't) disruptive innovation

• Disruption is a process, not a product or service, that occurs from the nascent to the mainstream
• Originate in low-end (less demanding customers) or new market (where none existed) footholds
• New firms don't catch on with mainstream customers until quality catches up with their standards
• Success is not a requirement and some business can be disruptive but fail
• New firm's business model differs significantly from incumbent

Christensen continues to develop and refine the theory and has accepted that not all examples of disruptive innovation perfectly fit into his theory. For example, he conceded that originating in the low end of the market is not always a cause of disruptive innovation, but rather it fosters competitive business models, using Uber as an example. In an interview with Forbes magazine he stated:

"Uber helped me realize that it isn’t that being at the bottom of the market is the causal mechanism, but that it’s correlated with a business model that is unattractive to its competitor".

Theory

The current theoretical understanding of disruptive innovation is different from what might be expected by default, an idea that Clayton M. Christensen called the "technology mudslide hypothesis". This is the simplistic idea that an established firm fails because it doesn't "keep up technologically" with other firms. In this hypothesis, firms are like climbers scrambling upward on crumbling footing, where it takes constant upward-climbing effort just to stay still, and any break from the effort (such as complacency born of profitability) causes a rapid downhill slide. Christensen and colleagues have shown that this simplistic hypothesis is wrong; it doesn't model reality. What they have shown is that good firms are usually aware of the innovations, but their business environment does not allow them to pursue them when they first arise, because they are not profitable enough at first and because their development can take scarce resources away from that of sustaining innovations (which are needed to compete against current competition). In Christensen's terms, a firm's existing value networks place insufficient value on the disruptive innovation to allow its pursuit by that firm. Meanwhile, start-up firms inhabit different value networks, at least until the day that their disruptive innovation is able to invade the older value network. At that time, the established firm in that network can at best only fend off the market share attack with a me-too entry, for which survival (not thriving) is the only reward.

In the technology mudslide hypothesis, Christensen differentiated disruptive innovation from sustaining innovation. He explained that the latter's goal is to improve existing product performance. On the other hand, he defines a disruptive innovation as a product or service designed for a new set of customers.

Generally, disruptive innovations were technologically straightforward, consisting of off-the-shelf components put together in a product architecture that was often simpler than prior approaches. They offered less of what customers in established markets wanted and so could rarely be initially employed there. They offered a different package of attributes valued only in emerging markets remote from, and unimportant to, the mainstream.

Christensen also noted that products considered as disruptive innovations tend to skip stages in the traditional product design and development process to quickly gain market traction and competitive advantage. He argued that disruptive innovations can hurt successful, well-managed companies that are responsive to their customers and have excellent research and development. These companies tend to ignore the markets most susceptible to disruptive innovations, because the markets have very tight profit margins and are too small to provide a good growth rate to an established (sizable) firm. Thus, disruptive technology provides an example of an instance when the common business-world advice to "focus on the customer" (or "stay close to the customer", or "listen to the customer") can be strategically counterproductive.

While Christensen argued that disruptive innovations can hurt successful, well-managed companies, O'Ryan countered that "constructive" integration of existing, new, and forward-thinking innovation could improve the economic benefits of these same well-managed companies, once decision-making management understood the systemic benefits as a whole.

How low-end disruption occurs over time.

Christensen distinguishes between "low-end disruption", which targets customers who do not need the full performance valued by customers at the high end of the market, and "new-market disruption", which targets customers who have needs that were previously unserved by existing incumbents.

Low-end disruption

"Low-end disruption" occurs when the rate at which products improve exceeds the rate at which customers can adopt the new performance. Therefore, at some point the performance of the product overshoots the needs of certain customer segments. At this point, a disruptive technology may enter the market and provide a product that has lower performance than the incumbent but that exceeds the requirements of certain segments, thereby gaining a foothold in the market.

In low-end disruption, the disruptor is focused initially on serving the least profitable customer, who is happy with a good enough product. This type of customer is not willing to pay premium for enhancements in product functionality. Once the disruptor has gained a foothold in this customer segment, it seeks to improve its profit margin. To get higher profit margins, the disruptor needs to enter the segment where the customer is willing to pay a little more for higher quality. To ensure this quality in its product, the disruptor needs to innovate. The incumbent will not do much to retain its share in a not-so-profitable segment, and will move up-market and focus on its more attractive customers. After a number of such encounters, the incumbent is squeezed into smaller markets than it was previously serving. And then, finally, the disruptive technology meets the demands of the most profitable segment and drives the established company out of the market.

New market disruption

"New market disruption" occurs when a product fits a new or emerging market segment that is not being served by existing incumbents in the industry. Some scholars note that the creation of a new market is a defining feature of disruptive innovation, particularly in the way it tend to improve products or services differently in comparison to normal market drivers. It initially caters to a niche market and proceeds on defining the industry over time once it is able to penetrate the market or induce consumers to defect from the existing market into the new market it created.

The extrapolation of the theory to all aspects of life has been challenged, as has the methodology of relying on selected case studies as the principal form of evidence. Jill Lepore points out that some companies identified by the theory as victims of disruption a decade or more ago, rather than being defunct, remain dominant in their industries today (including Seagate Technology, U.S. Steel, and Bucyrus). Lepore questions whether the theory has been oversold and misapplied, as if it were able to explain everything in every sphere of life, including not just business but education and public institutions.

Disruptive technology

In 2009, Milan Zeleny described high technology as disruptive technology and raised the question of what is being disrupted. The answer, according to Zeleny, is the support network of high technology. For example, introducing electric cars disrupts the support network for gasoline cars (network of gas and service stations). Such disruption is fully expected and therefore effectively resisted by support net owners. In the long run, high (disruptive) technology bypasses, upgrades, or replaces the outdated support network.

Questioning the concept of a disruptive technology, Haxell (2012) questions how such technologies get named and framed, pointing out that this is a positioned and retrospective act.

Technology, being a form of social relationship, always evolves. No technology remains fixed. Technology starts, develops, persists, mutates, stagnates, and declines, just like living organisms. The evolutionary life cycle occurs in the use and development of any technology. A new high-technology core emerges and challenges existing technology support nets (TSNs), which are thus forced to coevolve with it. New versions of the core are designed and fitted into an increasingly appropriate TSN, with smaller and smaller high-technology effects. High technology becomes regular technology, with more efficient versions fitting the same support net. Finally, even the efficiency gains diminish, emphasis shifts to product tertiary attributes (appearance, style), and technology becomes TSN-preserving appropriate technology. This technological equilibrium state becomes established and fixated, resisting being interrupted by a technological mutation; then new high technology appears and the cycle is repeated.

Regarding this evolving process of technology, Christensen said:

The technological changes that damage established companies are usually not radically new or difficult from a technological point of view. They do, however, have two important characteristics: First, they typically present a different package of performance attributes—ones that, at least at the outset, are not valued by existing customers. Second, the performance attributes that existing customers do value improve at such a rapid rate that the new technology can later invade those established markets.

The World Bank's 2019 World Development Report on The Changing Nature of Work examines how technology shapes the relative demand for certain skills in labor markets and expands the reach of firms - robotics and digital technologies, for example, enable firms to automate, replacing labor with machines to become more efficient, and innovate, expanding the number of tasks and products. Joseph Bower explained the process of how disruptive technology, through its requisite support net, dramatically transforms a certain industry.

When the technology that has the potential for revolutionizing an industry emerges, established companies typically see it as unattractive: it’s not something their mainstream customers want, and its projected profit margins aren’t sufficient to cover big-company cost structure. As a result, the new technology tends to get ignored in favor of what’s currently popular with the best customers. But then another company steps in to bring the innovation to a new market. Once the disruptive technology becomes established there, smaller-scale innovation rapidly raise the technology’s performance on attributes that mainstream customers’ value.

For example, the automobile was high technology with respect to the horse carriage; however, it evolved into technology and finally into appropriate technology with a stable, unchanging TSN. The main high-technology advance in the offing is some form of electric car—whether the energy source is the sun, hydrogen, water, air pressure, or traditional charging outlet. Electric cars preceded the gasoline automobile by many decades and are now returning to replace the traditional gasoline automobile. The printing press was a development that changed the way that information was stored, transmitted, and replicated. This allowed empowered authors but it also promoted censorship and information overload in writing technology.

Milan Zeleny described the above phenomenon. He also wrote that:

Implementing high technology is often resisted. This resistance is well understood on the part of active participants in the requisite TSN. The electric car will be resisted by gas-station operators in the same way automated teller machines (ATMs) were resisted by bank tellers and automobiles by horsewhip makers. Technology does not qualitatively restructure the TSN and therefore will not be resisted and never has been resisted. Middle management resists business process reengineering because BPR represents a direct assault on the support net (coordinative hierarchy) they thrive on. Teamwork and multi-functionality is resisted by those whose TSN provides the comfort of narrow specialization and command-driven work.

Social media could be considered a disruptive innovation within sports. More specifically, the way that news in sports circulates nowadays versus the pre-internet era where sports news was mainly on T.V., radio, and newspapers. Social media has created a new market for sports that was not around before in the sense that players and fans have instant access to information related to sports.

High-technology effects

High technology is a technology core that changes the very architecture (structure and organization) of the components of the technology support net. High technology therefore transforms the qualitative nature of the TSN's tasks and their relations, as well as their requisite physical, energy, and information flows. It also affects the skills required, the roles played, and the styles of management and coordination—the organizational culture itself.

This kind of technology core is different from regular technology core, which preserves the qualitative nature of flows and the structure of the support and only allows users to perform the same tasks in the same way, but faster, more reliably, in larger quantities, or more efficiently. It is also different from appropriate technology core, which preserves the TSN itself with the purpose of technology implementation and allows users to do the same thing in the same way at comparable levels of efficiency, instead of improving the efficiency of performance.

As for the difference between high technology and low technology, Milan Zeleny once said:

The effects of high technology always breaks the direct comparability by changing the system itself, therefore requiring new measures and new assessments of its productivity. High technology cannot be compared and evaluated with the existing technology purely on the basis of cost, net present value or return on investment. Only within an unchanging and relatively stable TSN would such direct financial comparability be meaningful. For example, you can directly compare a manual typewriter with an electric typewriter, but not a typewriter with a word processor. Therein lies the management challenge of high technology.

However, not all modern technologies are high technologies. They have to be used as such, function as such, and be embedded in their requisite TSNs. They have to empower the individual because only through the individual can they empower knowledge. Not all information technologies have integrative effects. Some information systems are still designed to improve the traditional hierarchy of command and thus preserve and entrench the existing TSN. The administrative model of management, for instance, further aggravates the division of task and labor, further specializes knowledge, separates management from workers, and concentrates information and knowledge in centers.

As knowledge surpasses capital, labor, and raw materials as the dominant economic resource, technologies are also starting to reflect this shift. Technologies are rapidly shifting from centralized hierarchies to distributed networks. Nowadays knowledge does not reside in a super-mind, super-book, or super-database, but in a complex relational pattern of networks brought forth to coordinate human action.

Internal auditor response

Internal audit plays a critical role maintaining effective control mitigating emerging risks. Businesses will increase risk or bypass opportunity if auditors do not address disruption-related risks. Alles has discussed that Big Data is a disruptive innovation that auditors must incorporate in practice. A 2019 study, Internal Auditors' Response to Disruptive Innovation, reports on the evolution of internal audit to react to changes. Disruptions examined include data analytics, agile processes, cloud computing, robotic process automation, continuous auditing, regulatory change, and artificial intelligence.

A proactive approach to disruptive innovation

A proactive approach to addressing the challenge posited by disruptive innovations has been debated by scholars. Petzold criticized the lack of acknowledgment of underlying process of the change to study the disruptive innovation over time from a process view and complexify the concept to support the understanding of its unfolding and advance its manageability. Keeping in view the multidimensional nature of disruptive innovation a measurement framework has been developed by Guo to enable a systemic assessment of disruptive potential of innovations, providing insights for the decisions in product/service launch and resource allocation. Middle managers play an important role in long term sustainability of any firm and thus have been studied to have a proactive role in exploitation of the disruptive innovation process.

Example of disruption

In the practical world, the popularization of personal computers illustrates how knowledge contributes to the ongoing technology innovation. The original centralized concept (one computer, many persons) is a knowledge-defying idea of the prehistory of computing, and its inadequacies and failures have become clearly apparent. The era of personal computing brought powerful computers "on every desk" (one person, one computer). This short transitional period was necessary for getting used to the new computing environment, but was inadequate from the vantage point of producing knowledge. Adequate knowledge creation and management come mainly from networking and distributed computing (one person, many computers). Each person's computer must form an access point to the entire computing landscape or ecology through the Internet of other computers, databases, and mainframes, as well as production, distribution, and retailing facilities, and the like. For the first time, technology empowers individuals rather than external hierarchies. It transfers influence and power where it optimally belongs: at the loci of the useful knowledge. Even though hierarchies and bureaucracies do not innovate, free and empowered individuals do; knowledge, innovation, spontaneity, and self-reliance are becoming increasingly valued and promoted.

Amazon Alexa and Airbnb are some other examples of disruption.

Uber is not an example of disruption because it did not originate in a low-end or new market footholds. One of the conditions for the business to be considered disruptive according to Clayton M. Christensen is that the business should originate on a) low-end or b) new-market footholds. Instead, Uber was launched in San Francisco, a large urban city with an established taxi service and did not target low-end customers or created a new market (from the consumer perspective). In contrast, UberSELECT, an option that provides luxurious cars such as limousine at a discounted price, is an example of disruption innovation because it originates from low-end customers segment - customers who would not have entered the traditional luxurious market.

Nonlinear Schrödinger equation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Nonlinear_Schr%C3%B6dinger_equation 

Absolute value of the complex envelope of exact analytical breather solutions of the nonlinear Schrödinger (NLS) equation in nondimensional form. (A) The Akhmediev breather; (B) the Peregrine breather; (C) the Kuznetsov–Ma breather.

In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. 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 condensates confined to highly anisotropic cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves 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. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model.

In quantum mechanics, the 1D NLSE is a special case of the classical nonlinear Schrödinger field, 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. 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 (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, 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 R. Y. Chiao, E. Garmire, and C. H. Townes (1964, equation (5)) 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 a nonlinear partial differential equation, applicable to classical and quantum mechanics.

Classical equation

The classical field equation (in dimensionless form) is:

Nonlinear Schrödinger equation (Classical field theory)

${\displaystyle i\partial _{t}\psi =-{1 \over 2}\partial _{x}^{2}\psi +\kappa |\psi |^{2}\psi }$

for the complex field ψ(x,t).

This equation arises from the Hamiltonian

${\displaystyle H=\int \mathrm {d} x\left[{1 \over 2}|\partial _{x}\psi |^{2}+{\kappa \over 2}|\psi |^{4}\right]}$

with the Poisson brackets

${\displaystyle \{\psi (x),\psi (y)\}=\{\psi ^{*}(x),\psi ^{*}(y)\}=0\,}$

${\displaystyle \{\psi ^{*}(x),\psi (y)\}=i\delta (x-y).\,}$

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, as shown by Zakharov & Shabat (1972) (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

${\displaystyle {\begin{aligned}{}[\psi (x),\psi (y)]&=[\psi ^{*}(x),\psi ^{*}(y)]=0\\{}[\psi ^{*}(x),\psi (y)]&=-\delta (x-y)\end{aligned}}}$

and normal order the Hamiltonian

${\displaystyle H=\int dx\left[{1 \over 2}\partial _{x}\psi ^{\dagger }\partial _{x}\psi +{\kappa \over 2}\psi ^{\dagger }\psi ^{\dagger }\psi \psi \right].}$

The quantum version was solved by Bethe ansatz by Lieb and Liniger. Thermodynamics was described by Chen-Ning Yang. 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: Zakharov and Shabat (1972) solved it with the inverse scattering transform. The corresponding linear system of equations is known as the Zakharov–Shabat system:

${\displaystyle {\begin{aligned}\phi _{x}&=J\phi \Lambda +U\phi \\\phi _{t}&=2J\phi \Lambda ^{2}+2U\phi \Lambda +\left(JU^{2}-JU_{x}\right)\phi ,\end{aligned}}}$

where

${\displaystyle \Lambda ={\begin{pmatrix}\lambda _{1}&0\\0&\lambda _{2}\end{pmatrix}},\quad J=i\sigma _{z}={\begin{pmatrix}i&0\\0&-i\end{pmatrix}},\quad U=i{\begin{pmatrix}0&q\\r&0\end{pmatrix}}.}$

The nonlinear Schrödinger equation arises as compatibility condition of the Zakharov–Shabat system:

${\displaystyle \phi _{xt}=\phi _{tx}\quad \Rightarrow \quad U_{t}=-JU_{xx}+2JU^{2}U\quad \Leftrightarrow \quad {\begin{cases}iq_{t}=q_{xx}+2qrq\\ir_{t}=-r_{xx}-2qrr.\end{cases}}}$

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:

${\displaystyle {\begin{aligned}\phi \to \phi [1]&=\phi \Lambda -\sigma \phi \\U\to U[1]&=U+[J,\sigma ]\\\sigma &=\varphi \Omega \varphi ^{-1}\end{aligned}}}$

which leaves the system invariant.

Here, φ is another invertible matrix solution (different from ϕ) of the Zakharov–Shabat system with spectral parameter Ω:

${\displaystyle {\begin{aligned}\varphi _{x}&=J\varphi \Omega +U\varphi \\\varphi _{t}&=2J\varphi \Omega ^{2}+2U\varphi \Omega +\left(JU^{2}-JU_{x}\right)\varphi .\end{aligned}}}$

Starting from the trivial solution U = 0 and iterating, one obtains the solutions with n solitons.

The NLS equation is a partial differential equation like the Gross–Pitaevskii equation. 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 ${\displaystyle e^{-iv(x+vt/2)}\,}$:

${\displaystyle \psi (x,t)\mapsto \psi _{[v]}(x,t)=\psi (x+vt,t)\;e^{-iv(x+vt/2)}.}$

The nonlinear Schrödinger equation in fiber optics

In optics, 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, four-wave mixing, second-harmonic generation, stimulated Raman scattering, optical solitons, ultrashort pulses, etc.

The nonlinear Schrödinger equation in water waves

A hyperbolic secant (sech) envelope soliton for surface waves on deep water.
Blue line: water waves.
Red line: envelope soliton.

For water waves, the nonlinear Schrödinger equation describes the evolution of the envelope of modulated wave groups. In a paper in 1968, Vladimir E. Zakharov describes the Hamiltonian structure of water waves. In the same paper Zakharov shows, that for slowly modulated wave groups, the wave amplitude 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 of the water waves, к is negative and envelope solitons 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 waves.

The complex 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 with water surface elevation η of the form:

${\displaystyle \eta =a(x_{0},t_{0})\;\cos \left[k_{0}\,x_{0}-\omega _{0}\,t_{0}-\theta (x_{0},t_{0})\right],}$

where a(x₀, t₀) and θ(x₀, t₀) are the slowly modulated amplitude and phase. Further ω₀ and k₀ are the (constant) angular frequency and wavenumber of the carrier waves, which have to satisfy the dispersion relation ω₀ = Ω(k₀). Then

${\displaystyle \psi =a\;\exp \left(i\theta \right).}$

So its modulus |ψ| is the wave amplitude a, and its argument arg(ψ) is the phase θ.

The relation between the physical coordinates (x₀, t₀) and the (x, t) coordinates, as used in the nonlinear Schrödinger equation given above, is given by:

${\displaystyle x=k_{0}\left[x_{0}-\Omega '(k_{0})\;t_{0}\right],\quad t=k_{0}^{2}\left[-\Omega ''(k_{0})\right]\;t_{0}}$

Thus (x, t) is a transformed coordinate system moving with the group velocity Ω'(k₀) of the carrier waves, The dispersion-relation curvature Ω"(k₀) – representing group velocity dispersion – 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:

${\displaystyle \kappa =-2k_{0}^{2},\quad \Omega (k_{0})={\sqrt {gk_{0}}}=\omega _{0}\,\!}$   so   ${\displaystyle \Omega '(k_{0})={\frac {1}{2}}{\frac {\omega _{0}}{k_{0}}},\quad \Omega ''(k_{0})=-{\frac {1}{4}}{\frac {\omega _{0}}{k_{0}^{2}}},\,\!}$

where g is the acceleration due to gravity at the Earth's surface.

In the original (x₀, t₀) coordinates the nonlinear Schrödinger equation for water waves reads:

${\displaystyle i\,\partial _{t_{0}}A+i\,\Omega '(k_{0})\,\partial _{x_{0}}A+{\tfrac {1}{2}}\Omega ''(k_{0})\,\partial _{x_{0}x_{0}}A-\nu \,|A|^{2}\,A=0,}$

with ${\displaystyle A=\psi ^{*}}$ (i.e. the complex conjugate of ${\displaystyle \psi }$) and ${\displaystyle \nu =\kappa \,k_{0}^{2}\,\Omega ''(k_{0}).}$ So ${\displaystyle \nu ={\tfrac {1}{2}}\omega _{0}k_{0}^{2}}$ for deep water waves.

Gauge equivalent counterpart

NLSE (1) is gauge equivalent to the following isotropic Landau-Lifshitz equation (LLE) or Heisenberg ferromagnet equation

${\displaystyle {\vec {S}}_{t}={\vec {S}}\wedge {\vec {S}}_{xx}.\qquad }$

Note that this equation admits several integrable and non-integrable generalizations in 2 + 1 dimensions like the Ishimori equation and so on.

Relation to vortices

Hasimoto (1972) showed that the work of da Rios (1906) on vortex filaments is closely related to the nonlinear Schrödinger equation. Subsequently, Salman (2013) used this correspondence to show that breather solutions can also arise for a vortex filament.