The influence of learning emotions on learning behaviors in college physical education is directly reflected in learning efficiency. Based on this research background, the paper applies a mathematical, statistical model to structural equation modeling of the emotional situation of students in a school's physical education curriculum. The study results found that college students have lower positive emotional attitudes towards college physical education courses. Thus, students’ mood is affected by the. For this reason, we should eliminate students’ resistance and fear of college physical education courses and promptly correct students’ unhealthy learning emotions about college physical education courses. In this way, it helps students correct their negative feelings about college physical education courses.

#### Keywords

- Learning mood
- college physical education curriculum
- structural equation
- mathematical-statistical model

#### MSC 2010

- 62J10

The current physical education curriculum, professional talent training, teaching reform research, students’ learning mood, and influencing factors of college physical education curriculum have attracted many scholars’ attention. Learning emotion refers to the psychological state of students’ understanding of learning and emotions. It has regulatory significance for learning behavior. Therefore, how to cultivate students to form correct and positive learning emotions and then improve their learning behaviors and results has always been a critical concern in pedagogy [1]. Unfortunately, there are still few domestic types of research on students’ emotions related to college physical education courses. The existing literature mainly focuses on studying related courses such as psychological statistics, medical statistics, and mathematics. Still, their analysis methods are mostly limited to descriptive analysis and regression analysis.

“Learning emotions” are not directly observable. Therefore, it needs to be measured by a scale. So far, there are more than ten kinds of leaves about learning emotions in college physical education courses. Foreign scholars widely adopt the SATS scale (Survey of Attitudes Toward Statistics). Schau proposed the scale in 1995, and initially, there were only 28 measurement items (SATS-28). Later the plate was added to 36 measurement items (SATS-36). This article will combine the SATS scale design questionnaire to carry out a field survey of college students in a sports college [2]. We measure their learning emotions towards college physical education courses. We build a learning emotion model based on the project packaging method and high-order factor structural equations to establish an overall understanding of college students’ learning emotions in college physical education courses. The thesis studies the influence path and influence intensity between college students’ college physical education curriculum learning emotions, physical ability, and expected mastery. The research conclusions of this article are expected to play a guiding role in the teaching activities of college physical education courses.

We use the SATS-36 scale to measure the emotions of college students in college physical education courses. The scale adopts a 7-level Likert plate (1 means “strongly disagree,” 7 means “strongly agree,” and so on). Thus, we use 6 dimensions, including emotion, cognitive ability, value, difficulty, interest, and effort, to measure the “emotion of college physical education courses.” In addition, the research also includes the factors of “sports ability” and “expected mastery” (as shown in Table 1).

Eight potential factors.

Latent factor | meaning | Number of Questions | Examples of measurement questions |
---|---|---|---|

emotion | Positive or negative feelings about the physical education curriculum | 6 | Q3: I will like sports. |

cognitive ability | Views on their ability to apply sports knowledge and skills | 6 | Q31: I can learn sports. |

value | Views on the value of sports knowledge in life and work | 9 | Q9: Physical education should be a necessary part of professional training. |

Difficulty | Awareness of the problem of physical education courses | 7 | Q6: The sports formula is easy to understand. |

interest | Interest in physical education | 4 | Q20: I am very interested in using sports. |

Work hard | The effort you intend to make for the physical education curriculum | 4 | Q14: I plan to study hard for every physical education test. |

Expected mastery | Anticipate their level of ability of sports knowledge | 2 | Q40: How confident are you in mastering the basics of sports? |

Physical ability | Past sports level | 2 | Q37: In the physical education courses you have taken before, how did you score? |

Structural equation modeling (SEM) is also often referred to as structural equation modeling. SEM is a statistical method to analyze the relationship between variables based on the covariance matrix of the variables. It can process latent variables and their measurement indicators at the same time. Simply put, the structural equation model can be divided into two parts: measurement model and structural model. The former describes the relationship between latent variables and measurement indicators, and the latter describes the relationship between latent variables [3]. The following measurement equation usually expresses the relationship between measurement indicators and latent variables:

Among them _{x} is the factor loading matrix of the exogenous measurement index on the exogenous latent variable, which represents the relationship between the exogenous measurement index and the exogenous latent variable. ∧_{y} is the factor loading matrix of the endogenous measurement index on the endogenous latent variable, representing the relationship between the endogenous measurement index and the endogenous latent variable.

High-order confirmatory factor analysis (HCFA) is also known as the full Y model. The core idea is to coordinate the primary factors through high-level factors, thereby simplifying the correlation between the primary factors [5]. An HCFA model only needs to be determined by the following two equations:

When there is a consistent high correlation between the first-order factors, high-order confirmatory factor analysis can often achieve better results.

The item combination method is a process of reintegrating two or more question items in the same scale and using the composite score (total score or average) as the score of the new index to analyze. For example, a scale originally had 9 question items. We can use each 3 question item to calculate its composite score to obtain 3 new indicators and then use these 3 new indicators for subsequent analysis.

Some scholars have shown that the project portfolio method can correct the non-normality of the data to a certain extent. When we reduce the number of parameters to be estimated, we can effectively avoid problems such as multicollinearity between measurement indicators, excessive error correlation, and cross-loading of measurement indicators from adversely affecting the model. Furthermore, this method can improve the fitting effect of higher-order factor models. Some scholars have proposed a two-stage simplification method that combines project portfolio and high-order factor analysis [6]. This method can significantly simplify the model and obtain the perfect goodness of fit. It is especially suitable for the situation with many model parameters.

This article adopts the project portfolio method to form higher-order factors of learning emotion. This article has six potential first-order factors (emotion, cognitive ability, value, difficulty, interest, effort), as shown in Table 1. We take the arithmetic average of each factor's measured variables as the value of the element. The original six first-order factors become “explicit variables” (AFFPC, COGPC, VALPC, DIFPC, INTPC, and EFFPC) and jointly measure the “attitude “High-order factor (ATTITUDE). The “attitude” high-level factor will further affect the “expected mastery” factor (EXPMSTRY), and it will be affected by the “sports ability” factor (MATHCOMP). At the same time, “sports ability” will also directly affect the “expected mastery” (as shown in Figure 1).

In theory, the more active the learning mood of university physical education courses, the easier it is to expect that one has a good grasp of statistical knowledge. The stronger the physical ability, the more confident they are in their physical fitness [7]. This also often leads to the better the expected mastery of the physical education curriculum. In addition, the stronger the physical ability, the easier it is to get a sense of accomplishment from similar courses. The easier it is for students to take a positive attitude towards university physical education courses. Therefore, this article proposes the following hypotheses:

H1: “Attitude” positively affects “expected mastery.”

H2: “Sports ability” positively affects “expected mastery.”

H3: “Sports ability” positively affects “attitude.”

This study designed the questionnaire based on the English-Chinese version of the SATS-36 scale. The questionnaire consists of 47 items, including 36 Likert scale items and 11 background information items. The survey subjects are all the sophomore students who participated in physical education courses in a sports college [8]. The questionnaire was issued before the first class at the beginning of the semester. After preprocessing, such as deleting wrong values, filling in missing values, and deleting outliers, we finally get data containing 1074 sample points. Among them, girls accounted for 60.99%. Thus, the composition of the sample is consistent with the actual situation of sports colleges.

We use R software for statistical analysis. Table 2 gives the mean value of each factor. Among them, the value of each element itself is obtained by taking the mean value of its corresponding measurement item to get 1. It can be seen that the average values of the other factors except the difficulty factor are all greater than 4. That is to say, the overall emotions of the students towards the university physical education courses are positive [9]. Among them, the two factors of value and effort have the most considerable mean value. Students must recognize the importance of university physical education courses and are willing to make more extraordinary efforts. Students generally have a strong interest in university physical education courses and typically hold a more positive attitude towards their cognitive abilities. And students have positive feelings about college physical education courses. According to the calculation of the mean value of the difficulty factor, it is found that most students think college physical education courses are challenging. In addition, students also have a positive attitude towards their physical ability and expected mastery.

Descriptive analysis of potential factors.

Latent factor | First-order factor/measurement item | The first-order factor/mean of the measurement items | Latent factor |
---|---|---|---|

Attitude | emotion | 4.649 | 4.816 |

cognitive ability | 4.854 | ||

value | 5.687 | ||

Difficulty | 3.088 | ||

interest | 4.919 | ||

Work hard | 5.67 | ||

Physical ability | Previous physical education results (Q37) | 4.28 | 4.252 |

Sports level (Q38) | 4.224 | ||

Expected mastery | Confidence in mastering sports knowledge (Q40) | 4.858 | 4.482 |

What kind of results are expected (Q44) | 4.105 |

We perform confirmatory factor analysis on the “attitude” factor measurement model, and the initial fitting results are given in the “initial results” row of Table 3. The standardized chi-square and each appropriate index have not reached the ideal outcome. Therefore, we need to modify the initial model: add error-related terms according to the adjustment index. The “adjusted result” row of Table 3 shows the fit of the adjusted model. The standardized chi-square value has been dramatically improved, and each fitting index meets the requirements of empirical criteria. Sports ability and expected mastery have to sound appropriate effects.

The initial fit index of the “attitude” factor measurement model.

Fitting effect | Initial result | Adjusted result | Rule of thumb | |
---|---|---|---|---|

Index | Chisq. | 503.816 | 11.146 | – |

df. | 9 | 3 | – | |

Chisq./df. | 55.98 | 3.715 | <3 | |

SRMR | 0.108 | 0.02 | <0.1 | |

RMSEA | 0.226 | 0.05 | <0.05 | |

CFI | 0.771 | 0.996 | >0.9 | |

GFI | 0.846 | 0.997 | >0.9 | |

NFI | 0.768 | 0.995 | >0.9 | |

IFI | 0.771 | 0.996 | >0.9 |

Table 4, the “Standardized Load Factor” column, gives the estimated results of the adjusted measurement model. In terms of the positive and negative coefficients, except for the difficulty factor, the other five first-order factors have positive loading coefficients on the “attitude” high-order factors. This result indicates that the more positive learning emotions mean, the more positive emotions are, the more willing to work hard to study university physical education courses [10]. The negative loading of the difficulty factor means a more positive attitude towards university physical education courses. The more students who often accompany this result agree with the difficulty of college physical education courses.

Confirmatory factor analysis estimation results.

Latent factor | First-order factor/measurement item | Standardized load factor |
---|---|---|

Attitude | Emotion | 0.678^{***} |

Cognitive ability | 0.745^{***} | |

Value | 0.706^{***} | |

Difficulty | −0.152^{***} | |

Interest | 0.601^{***} | |

Work hard | 0.524^{***} | |

Sports performance | Previous physical education results (Q37) | 0.917^{***} |

Sports level (Q38) | 0.951^{***} | |

Expected mastery | Confidence in mastering statistical knowledge (Q40) | 0.748^{***} |

What kind of results are expected (Q44) | 0.660^{***} |

Note:

means p <0.001.

From the significance of the coefficients, all loads are significant. Except for the first-order factor of difficulty, the absolute values of the load coefficients of the other six first-order factors on the second-order factors of attitude are more significant than 0.50. Among them, philosophy has the most muscular interpretation of cognitive ability, while understanding value, emotion, interest, and effort weakened in turn.

Table 5 shows the reliability and validity test of the questionnaire data. Attitude and sports ability belong to high reliability. The Cronbach alpha coefficient value of the expected mastery level is slightly lower but still acceptable. We then test the convergent validity and discriminative validity of the three factors. First, according to the study in Table 4, it is found that the load coefficients of each measurement index and its corresponding element are all significant, indicating that the questionnaire data has good convergence validity [11]. Secondly, from the average variation extraction amount (AVE) in Table 5, the sports ability factor is more significant than 0.5, and the mastery degree factor is expected to be close to 0.5. Thus, although the attitude factor is slightly lower, the overall convergence validity of the model is better. Finally, the correlation coefficient between attitude and expected mastery is somewhat higher than the

Reliability and validity test indicators.

Latent factor | Cronbach | AVE | Correlation coefficient | |||
---|---|---|---|---|---|---|

Attitude | Expected mastery | Physical ability | ||||

attitude | 0.739 | 0.362 | 0.602 | 1 | 0.789 | 0.434 |

Expected mastery | 0.624 | 0.498 | 0.705 | 0.789 | 1 | 0.81 |

Physical ability | 0.931 | 0.872 | 0.934 | 0.434 | 0.81 | 0 |

The structural equation fitting index of the theoretical model is shown in Table 6. Except for standardized chi-square and RMSEA, all other indicators are ideal. The estimated results of the structural model are shown in Table 7. First of all, from the positive and negative path coefficients, students’ learning emotions towards college physical education courses have a significant and greater positive impact on their expected mastery. The more positive the students are about college physical education courses, the better they will be expected. Sports ability has a significant and greater positive impact on the scheduled mastery and the learning mood of university physical education courses. The stronger the physical capacity, the easier it is for students to attitude towards university physical education courses positively. Secondly, students expect that they will also have a good grasp of physical education courses [12]. Therefore, the estimation result of the model supports the hypothesis H1~H3. Secondly, from the perspective of influence path and size, the attitude has a large and significant favorable influence on the degree of expectation mastery. Sports ability, directly and indirectly, affects the degree of expected knowledge through two paths. Its direct effect is 0.577, the indirect impact is 0.234 (0.434 × 0.539), and the total product is 0.810. This result shows more remarkable than the influence of attitude on the expected degree of mastery. This indicates that the students’ existing physical level is an essential factor in their judgment of the predicted mastery degree. The degree of influence of this factor is greater than the impact of students’ subjective attitudes towards college physical education courses.

Fitting index of structural equation model.

Fitting effect | Index | ||||||||
---|---|---|---|---|---|---|---|---|---|

Chisq. | df. | Chisq./df. | SRMR | RMSEA | CFI | GFI | NFI | IFI | |

Value | 245.15 | 26 | 9.429 | 0.06 | 0.089 | 0.957 | 0.957 | 0.953 | 0.957 |

Rule of thumb | – | – | <3 or <5 | <0.1 | <0.05 | >0.9 | >0.9 | >0.9 | >0.9 |

Estimation results of the structural equation model.

path | Standardized path coefficient | test result |
---|---|---|

Attitude β Sports Ability | 0.434^{***} | H1 established |

Expected mastery β sports ability | 0.577^{***} | H2 established |

Expected mastery β attitude | 0.539^{***} | H3 established |

Note:

means p <0.001.

With the development of data science, university physical education courses have become compulsory courses for finance and economics majors. Through this survey, it can be seen that most of the students have favorable emotions towards the study of university physical education courses and recognize the value of statistical knowledge [13]. Students have a strong interest in learning physical education and are willing to make more extraordinary efforts. They hold a positive attitude towards their cognitive abilities and expect to achieve better results. At the same time, it is undeniable that most physical education students think that college physical education courses are demanding, which may be related to the thinking mode and reasoning ability involved in the learning process. Therefore, the teaching process should focus on the students’ cognition of the difficulty of college physical education courses. We need to adopt appropriate teaching methods to make it easier and to eliminate students’ fear of college physical education courses as much as possible.

When students recognize the value of university physical education courses and have more positive emotions for studying university physical education courses, students will have the stronger motivation and a more positive attitude to research university physical education courses. At the same time, when students are more affirmed of their cognitive abilities and are more willing to work hard, it also means that students have a more positive attitude towards studying university physical education courses. Therefore, in the teaching process, teachers should emphasize to students the application value of physical fitness in real life and realize the importance of sports knowledge [14]. At the same time, teachers should adopt vivid teaching methods to mobilize students’ interest in learning and enhance students’ positive attitudes towards their cognitive abilities. In this way, teachers can encourage students to learn sports knowledge with a more positive attitude.

Students’ learning emotions towards college physical education courses positively influence their expected mastery. When students have a more positive attitude towards the study of university physical education courses, the more interested they are in university physical education courses, the more willing they are to make efforts–more excellent results.

Students’ physical quality significantly positively affects their learning mood of university physical education courses and their expected mastery of university physical education courses. The higher the student's material quality, the easier it is to obtain a sense of accomplishment from the study of physical education courses and then have a more superior sense of superiority in the face of college physical education courses [15]. The student's physical fitness will also directly affect the student's expected mastery. The stronger the physical ability, the more students will hope to achieve good university physical education courses.

Past physical exercise learning experience affects students’ expectations of college physical education courses. Therefore, on the one hand, teachers need to correct students’ emotions about physical education as soon as possible, and help them gain a sense of accomplishment from physical education learning will help cultivate students’ positive emotions for college physical education courses. However, on the other hand, teachers only can students improve their opposing expectations of college physical education courses.

Combined with the SATS-36 scale, we conducted a field investigation on the learning mood of college physical education courses of college students in a sports college in China. We use the project-based packaging method and the high-order factor method to construct a structural equation model about learning mood, sports level, and expected mastery. According to our research, the overall attitude of students towards college physical education courses is positive, but college physical education courses are considered problematic. Students’ attitudes towards college physical education courses are mainly reflected in five dimensions: emotion, cognitive ability, value, interest, and effort. Students’ learning emotions towards college physical education courses have a strong positive influence on their expected mastery. In addition, a student's physical fitness level can directly or indirectly affect their expected knowledge of college physical education courses.

#### Descriptive analysis of potential factors.

Latent factor | First-order factor/measurement item | The first-order factor/mean of the measurement items | Latent factor |
---|---|---|---|

Attitude | emotion | 4.649 | 4.816 |

cognitive ability | 4.854 | ||

value | 5.687 | ||

Difficulty | 3.088 | ||

interest | 4.919 | ||

Work hard | 5.67 | ||

Physical ability | Previous physical education results (Q37) | 4.28 | 4.252 |

Sports level (Q38) | 4.224 | ||

Expected mastery | Confidence in mastering sports knowledge (Q40) | 4.858 | 4.482 |

What kind of results are expected (Q44) | 4.105 |

#### Eight potential factors.

Latent factor | meaning | Number of Questions | Examples of measurement questions |
---|---|---|---|

emotion | Positive or negative feelings about the physical education curriculum | 6 | Q3: I will like sports. |

cognitive ability | Views on their ability to apply sports knowledge and skills | 6 | Q31: I can learn sports. |

value | Views on the value of sports knowledge in life and work | 9 | Q9: Physical education should be a necessary part of professional training. |

Difficulty | Awareness of the problem of physical education courses | 7 | Q6: The sports formula is easy to understand. |

interest | Interest in physical education | 4 | Q20: I am very interested in using sports. |

Work hard | The effort you intend to make for the physical education curriculum | 4 | Q14: I plan to study hard for every physical education test. |

Expected mastery | Anticipate their level of ability of sports knowledge | 2 | Q40: How confident are you in mastering the basics of sports? |

Physical ability | Past sports level | 2 | Q37: In the physical education courses you have taken before, how did you score? |

#### Reliability and validity test indicators.

Latent factor | Cronbach | AVE | Correlation coefficient | |||
---|---|---|---|---|---|---|

Attitude | Expected mastery | Physical ability | ||||

attitude | 0.739 | 0.362 | 0.602 | 1 | 0.789 | 0.434 |

Expected mastery | 0.624 | 0.498 | 0.705 | 0.789 | 1 | 0.81 |

Physical ability | 0.931 | 0.872 | 0.934 | 0.434 | 0.81 | 0 |

#### Estimation results of the structural equation model.

path | Standardized path coefficient | test result |
---|---|---|

Attitude β Sports Ability | 0.434^{***} | H1 established |

Expected mastery β sports ability | 0.577^{***} | H2 established |

Expected mastery β attitude | 0.539^{***} | H3 established |

#### The initial fit index of the “attitude” factor measurement model.

Fitting effect | Initial result | Adjusted result | Rule of thumb | |
---|---|---|---|---|

Index | Chisq. | 503.816 | 11.146 | – |

df. | 9 | 3 | – | |

Chisq./df. | 55.98 | 3.715 | <3 | |

SRMR | 0.108 | 0.02 | <0.1 | |

RMSEA | 0.226 | 0.05 | <0.05 | |

CFI | 0.771 | 0.996 | >0.9 | |

GFI | 0.846 | 0.997 | >0.9 | |

NFI | 0.768 | 0.995 | >0.9 | |

IFI | 0.771 | 0.996 | >0.9 |

#### Confirmatory factor analysis estimation results.

Latent factor | First-order factor/measurement item | Standardized load factor |
---|---|---|

Attitude | Emotion | 0.678^{***} |

Cognitive ability | 0.745^{***} | |

Value | 0.706^{***} | |

Difficulty | −0.152^{***} | |

Interest | 0.601^{***} | |

Work hard | 0.524^{***} | |

Sports performance | Previous physical education results (Q37) | 0.917^{***} |

Sports level (Q38) | 0.951^{***} | |

Expected mastery | Confidence in mastering statistical knowledge (Q40) | 0.748^{***} |

What kind of results are expected (Q44) | 0.660^{***} |

#### Fitting index of structural equation model.

Fitting effect | Index | ||||||||
---|---|---|---|---|---|---|---|---|---|

Chisq. | df. | Chisq./df. | SRMR | RMSEA | CFI | GFI | NFI | IFI | |

Value | 245.15 | 26 | 9.429 | 0.06 | 0.089 | 0.957 | 0.957 | 0.953 | 0.957 |

Rule of thumb | – | – | <3 or <5 | <0.1 | <0.05 | >0.9 | >0.9 | >0.9 | >0.9 |

Regarding new wave distributions of the non-linear integro-partial Ito differential and fifth-order integrable equations Nonlinear Mathematical Modelling of Bone Damage and Remodelling Behaviour in Human Femur Value Creation of Real Estate Company Spin-off Property Service Company Listing Entrepreneur's Passion and Entrepreneurial Opportunity Identification: A Moderated Mediation Effect Model Applications of the extended rational sine-cosine and sinh-cosh techniques to some nonlinear complex models arising in mathematical physics Study on the Classification of Forestry Infrastructure from the Perspective of Supply Based on the Classical Quartering Method A Modified Iterative Method for Solving Nonlinear Functional Equation A comprehensive evaluation of county economies in the Beijing-Tianjin-Hebei Region based on entropy TOPSIS analysis New Principles of Non-Linear Integral Inequalities on Time Scales Has the belt and road initiative boosted the resident consumption in cities along the domestic route? – evidence from credit card consumption Analysis of the agglomeration of Chinese manufacturing industries and its effect on economic growth in different regions after entering the new normal Study on the social impact Assessment of Primary Land Development: Empirical Analysis of Public Opinion Survey on New Town Development in Pinggu District of Beijing Possible Relations between Brightest Central Galaxies and Their Host Galaxies Clusters and Groups Attitude control for the rigid spacecraft with the improved extended state observer An empirical investigation of physical literacy-based adolescent health promotion MHD 3-dimensional nanofluid flow induced by a power-law stretching sheet with thermal radiation, heat and mass fluxes The research of power allocation algorithm with lower computational complexity for non-orthogonal multiple access Research on the normalisation method of logging curves: taking XJ Oilfield as an example A Method of Directly Defining the inverse Mapping for a HIV infection of CD4+ T-cells model On the interaction of species capable of explosive growth Research on Evaluation of Intercultural Competence of Civil Aviation College Students Based on Language Operator Combustion stability control of gasoline compression ignition (GCI) under low-load conditions: A review Research on the Psychological Distribution Delay of Artificial Neural Network Based on the Analysis of Differential Equation by Inequality Expansion and Contraction Method The Comprehensive Diagnostic Method Combining Rough Sets and Evidence Theory Study on Establishment and Improvement Strategy of Aviation Equipment Design of software-defined network experimental teaching scheme based on virtualised Environment Research on Financial Risk Early Warning of Listed Companies Based on Stochastic Effect Mode System dynamics model of output of ball mill The Model of Sugar Metabolism and Exercise Energy Expenditure Based on Fractional Linear Regression Equation Constructing Artistic Surface Modeling Design Based on Nonlinear Over-limit Interpolation Equation Statistical analysis of typical elevator accidents in China from 2002 to 2019 Optimal allocation of microgrid using a differential multi-agent multi-objective evolution algorithm About one method of calculation in the arbitrary curvilinear basis of the Laplace operator and curl from the vector function Numerical Simulation Analysis Mathematics of Fluid Mechanics for Semiconductor Circuit Breaker Cartesian space robot manipulator clamping movement in ROS simulation and experiment Effects of internal/external EGR and combustion phase on gasoline compression ignition at low-load condition Research of urban waterfront space planning and design based on children-friendly idea Characteristics of Mathematical Statistics Model of Student Emotion in College Physical Education Human Body Movement Coupling Model in Physical Education Class in the Educational Mathematical Equation of Reasonable Exercise Course Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Impact of Web Page House Listing Cues on Internet Rental Research on management and control strategy of E-bikes based on attribute reduction method A study of aerial courtyard of super high-rise building based on optimisation of space structure Exact solutions of (2 + 1)-Ablowitz-Kaup-Newell-Segur equation