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Topic · Applied Mathematics

Nonlinear Partial Differential Equations

Nonlinear Partial Differential Equations is a research topic within Applied Mathematics. Science Explorer counts 25k research works in it since 1956. 9.9% of them reached the world's top 10% most cited for their field and year.

This cluster of papers focuses on the theory of fractional Laplacian operators, including topics such as Sobolev spaces, nonlinear equations, regularity theory, variational problems, critical exponents, ground state solutions, elliptic equations, nonlocal operators, and Hamiltonian estimates.

  • Fractional Laplacian
  • Sobolev Spaces
  • Nonlinear Equations
  • Regularity Theory
  • Variational Problems
  • Critical Exponents
  • Ground State Solutions
  • Elliptic Equations
  • Nonlocal Operators
  • Hamiltonian Estimates
Research works
25k
fractional, since 1956
In the world top 10%
2.5k
per year above
Top-10% rate
9.9%
share of its works in the world top 10%
Growth, 2018-21 → 2022-25
+12%
the tick is no change
Largest producer
China
47% of the 10 largest, 2022-now
Nonlinear Partial Differential Equations

Which countries lead Nonlinear Partial Differential Equations research?

By volume, China and Italy publish the most (1.9k and 439 works in 2022-now).

By volume, 2022-now

  1. 1 China 1.9k works
  2. 2 Italy 439 works
  3. 3 United States 400 works
  4. 4 Brazil 229 works
  5. 5 France 202 works
  6. 6 Germany 191 works
  7. 7 Morocco 177 works
  8. 8 Japan 174 works
  9. 9 Spain 135 works
  10. 10 India 129 works

How concentrated that is

The same countries as shares of everything the list above accounts for. A node where two countries do two thirds of the work and one spread evenly across twelve read alike as a ranking and not at all alike here.

China: 47.2%Italy: 11.2%United States: 10.2%Brazil: 5.8%6 others listed: 25.7%47%largest
China1,856 · 47.2%Italy439 · 11.2%United States400 · 10.2%Brazil229 · 5.8%6 others listed1,009 · 25.7%

Shares of the rows listed above, not of the whole node.

Which institutions lead Nonlinear Partial Differential Equations research?

By volume in 2022-now, Centre National de la Recherche Scientifique publishes the most Nonlinear Partial Differential Equations research, followed by Southwest University and Université Sultan Moulay Slimane.

By volume, 2022-now

  1. 1 Centre National de la Recherche ScientifiqueFrance 99 works
  2. 2 Southwest UniversityChina 56 works
  3. 3 Université Sultan Moulay SlimaneMorocco 50 works
  4. 4 Central South UniversityChina 49 works
  5. 5 Tsinghua UniversityChina 49 works
  6. 6 Beijing Normal UniversityChina 44 works
  7. 7 Russian Academy of SciencesRussia 40 works
  8. 8 Sapienza University of RomeItaly 38 works
  9. 9 Chinese Academy of SciencesChina 36 works
  10. 10 Zhejiang Normal UniversityChina 34 works

Who are the leading researchers in Nonlinear Partial Differential Equations?

The most-cited researchers publishing on Nonlinear Partial Differential Equations include Florian Fischer, R. W. L. Jones and Tasawar Hayat.

  1. 1 Florian Fischer Germany 257k citations
  2. 2 R. W. L. Jones United Kingdom 189k citations
  3. 3 Tasawar Hayat Pakistan 135k citations
  4. 4 Stanley Osher United States 119k citations
  5. 5 George Em Karniadakis United States 113k citations
  6. 6 Peng Shi Australia 89k citations
  7. 7 Dumitru Bǎleanu Türkiye 89k citations

Ranked by citations received across their whole record, among researchers with at least three works on this topic.

What are the most cited papers in Nonlinear Partial Differential Equations?

The most cited paper in Nonlinear Partial Differential Equations is "Dual variational methods in critical point theory and applications" (Antonio Ambrosetti, Paul H. Rabinowitz, 1973), with 4,419 citations. Next come "Hitchhikerʼs guide to the fractional Sobolev spaces" (4,369) and "Minimax Methods in Critical Point Theory with Applications to Differential Equations" (4,190).

Most cited papers
#PaperReleasedCitations
1 Dual variational methods in critical point theory and applications Antonio Ambrosetti, Paul H. Rabinowitz · Journal of Functional Analysis Open access Dec 1973 4,419
2 Hitchhikerʼs guide to the fractional Sobolev spaces Eleonora Di Nezza, Giampiero Palatucci, Enrico Valdinoci · Bulletin des Sciences Mathématiques Open access 29 Dec 2011 4,369
3 Minimax Methods in Critical Point Theory with Applications to Differential Equations Paul H. Rabinowitz · Regional conference series in mathematics Conference paper Jul 1986 4,190
4 Positive solutions of nonlinear elliptic equations involving critical sobolev exponents Haïm Brézis, Louis Nirenberg · Communications on Pure and Applied Mathematics Jul 1983 2,942
5 Symmetry and related properties via the maximum principle Basilis Gidas, Wei‐Ming Ni, Louis Nirenberg · Communications in Mathematical Physics Oct 1979 2,934
All 100 most cited papers, with filters and sorting

Where is Nonlinear Partial Differential Equations research done?

The largest centres of Nonlinear Partial Differential Equations research in 2022-now are Beijing (China), Chongqing (China), Shanghai (China) and Wuhan (China). Among places with at least 20 works in it, it is an unusually large share of all research in Beni Mellal, Qazvin and Oujda.

Largest cities, 2022-now

  1. 1 BeijingChina 305 works
  2. 2 ChongqingChina 94 works
  3. 3 ShanghaiChina 93 works
  4. 4 WuhanChina 82 works
  5. 5 ChangshaChina 74 works
  6. 6 GuangzhouChina 73 works
  7. 7 ParisFrance 68 works
  8. 8 NanjingChina 66 works
  9. 9 RomeItaly 56 works
  10. 10 MoscowRussia 56 works

Where it is the local speciality

  1. Beni MellalMA · 50.3 works68×
  2. QazvinIR · 21.3 works33×
  3. OujdaMA · 21.0 works24×
← less than its size predicts1×more →

Location quotient: how much more of its research is in Nonlinear Partial Differential Equations than the world average.

See Nonlinear Partial Differential Equations on the map

Where is the best place to study Nonlinear Partial Differential Equations?

Among universities, judged by research, Université Sultan Moulay Slimane, Beijing Normal University and Tsinghua University score highest, combining excellence, specialisation, size, growth and international reach. Research strength is one signal when choosing where to study; it does not measure teaching.

0%20%40%mean 22.61%fractional works in this field (log) →share of work in the world top 10% →Université Sultan Mo…Beijing Normal Unive…Tsinghua UniversityZhejiang Normal Univ…
above the meannear itbelow it

One dot per university in the first ten of the table below, at the default weights. The upper left is the interesting corner: small places doing unusually strong work.

Excellenceshare of its work in the world top 10%, 2022-now and all time averaged
40%
Specialisationhow much of its work is in this field, counted up to 10× the world share
10%
Sizeits output in this field, on a log scale
30%
Growthoutput growth, counted from -20% to +20%
10%
Internationalshare of its work with a co-author abroad
10%
Each slider sets a weight; the share it gets of the score is beside it.
#UniversityScoreTop 10%SpecialisationWorksGrowth
1Université Sultan Moulay SlimaneMorocco 82.026.3%79.5×50 +572.7%
2Beijing Normal UniversityChina 71.619.5%11.9×44 +34.3%
3Tsinghua UniversityChina 65.815.8%4.6×49 +68.6%
4Zhejiang Normal UniversityChina 63.615.2%25.7×34 +84.3%
5University of ParmaItaly 62.125.6%11.0×11 +72.9%
6Southwest UniversityChina 61.77.7%22.5×56 +44.4%
7Bielefeld UniversityGermany 61.327.8%13.1×14 +40.7%
8AGH University of KrakowPoland 61.330.1%6.1×11 -1.2%
9Guangzhou UniversityChina 60.818.2%14.7×25 +23.1%
10Technische Universität BerlinGermany 60.439.9%6.0×9 +75.8%

Universities only. The score blends excellence (40%), size (30%), specialisation (10%), growth (10%), international reach (10%), 2022-now; growth compares 2018-21 with 2022-25 and counts up to +20%, since a jump from a small base is mostly noise.

Every institution in Nonlinear Partial Differential Equations, country by country

Is Nonlinear Partial Differential Equations research growing?

Output in 2022-2025 was 12% higher than in 2018-2021, peaking in 2022. The fastest-growing topics are Nonlinear Partial Differential Equations.

still counting19801990200020102020
grewheldshrank

The same series as a ribbon: one cell per year, darker for more. The line above answers how much; this answers when.

Its share of all research

Growing output can still be a shrinking share: all of science grows. This is Nonlinear Partial Differential Equations as a percentage of every work published that year, 0.019% in 2024.

still counting0.010%0.020%0.030%Nonlinear Partial Differential Equations0.020% in 202519801990200020102020

Fractional works of Nonlinear Partial Differential Equations over all fractional works that year. How the balance of science has shifted.

How much of it reaches the world top 10%

Year by year.

still counting5%10%15%Nonlinear Partial Differential Equations9.8% in 2025200020102020

Of each year's works in it, the share that reached the world's 10% most cited for their field and year. A field is the world for its own topics, so there is no other line to hold it against.

Which topics inside it are moving

Growth and decline on one axis around a shared zero. Two lists side by side hide the thing that matters: whether the growth dwarfs the decline, or the other way round.

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